Statements Goals Identify and evaluate conditional statements. Identify converses and biconditionals. Drafting, Sports, Geography

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Statements Goals Identify and evaluate conditional statements. Identify converses and biconditionals. Drafting, Sports, Geography"

Transcription

1 3-6 Conditional Statements Goals Identify and evaluate conditional statements. Identify converses and biconditionals. Applications Drafting, Sports, Geography Do you think each statement is true or false? Explain your reasoning. a. Denver is the capital of Colorado. b. For all x, x 2 x. c. Lines m and n are parallel. BUILD UNDERSTANDING Many of the statements in this chapter are written in if then form. Statements like these are called conditional statements, or simply conditionals. The clause following if is called the hypothesis of the conditional. The clause following then is called the conclusion. For example, the parallel lines postulate was presented as a conditional. Denver, C If two parallel lines are cut by a transversal, hypothesis then corresponding angles are equal in measure. conclusion A conditional is either true or false. When a conditional is true, you can justify it in a variety of ways. For instance, you may be able to show that the conditional is true because it follows directly from a definition. When a conditional is a postulate, such as the parallel lines postulate, it is assumed to be true. Still other conditionals are theorems, and these must be proved true. To demonstrate that a conditional is false, you need to find only one example for which the hypothesis is true but the conclusion is false. An example like this is called a counterexample. E x a m p l e 1 Tell whether each conditional is true or false. a. If two lines are parallel, then they are coplanar. b. If two lines do not intersect, then they are parallel. a. arallel lines are defined as coplanar lines that do not intersect. So, the conditional is true. b. Consider skew lines k and shown. By the definition of skew lines, k and do not intersect, and so the hypothesis is true. However, also by the definition of skew lines, k and are noncoplanar. Lines k and cannot be parallel, and so the conclusion is false. Therefore, lines k and are a counterexample, and the conditional is false. k 128 Chapter 3 Geometry and Reasoning

2 The converse of a conditional is formed by interchanging the hypothesis and the conclusion. The fact that a conditional is true is no guarantee that its converse is true. Reading Math E x a m p l e 2 DRAFTING eople who draw plans must apply this true statement: If two lines are parallel, then they do not intersect. Write the converse of the statement. Is it also true? Interchange the hypothesis and the conclusion of the given statement. Statement: If two lines are parallel, then they do not intersect. hypothesis conclusion Converse: If two lines do not intersect, then they are parallel. Many statements about everyday situations can be expressed as conditionals. For instance, the following is a true conditional. If it is raining, then it is cloudy. Its converse is false. If it is cloudy, then it is raining. hypothesis conclusion By definition, parallel lines do not intersect, and so the given statement is true. art b of Example 1 demonstrated that lines that do not intersect are not necessarily parallel, and so the converse is false. The converse of the parallel lines postulate also is assumed to be true. It is stated as the corresponding angles postulate in the following manner. ostulate 10 The Corresponding Angles ostulate If two lines are cut by a transversal so that a pair of corresponding angles are equal in measure, then the lines are parallel. When a statement and its converse are both true, they can be combined into an if and only if statement. This type of statement is called a biconditional statement, or simply a biconditional. Every definition can be written as a biconditional. E x a m p l e 3 Write this definition as two conditionals and as a single biconditional. A right angle is an angle whose measure is 90. The definition leads to two true conditionals. If an angle is a right angle, then its measure is 90. If the measure of an angle is 90, then it is a right angle. These can be combined into a single biconditional as follows. An angle is a right angle if and only if its measure is 90. roblem Solving Tip When writing the converse of a conditional, you may need to change the wording of the hypotheses and the conclusion to make the converse read clearly. mathmatters3.com/extra_examples Lesson 3-6 Conditional Statements 129

3 TRY THESE EXERCISES 1. TALK ABUT IT Decide whether this conditional is true or false. If two lines are each perpendicular to a third line, then they are parallel to each other. Discuss your reasoning with a classmate. 2. Write the converse of this statement. If two angles are vertical angles, then they are equal in measure. Are the given statement and its converse true or false? 3. Write this definition as two conditionals and as a single biconditional. The bisector of an angle is the ray that divides the angle into two adjacent angles that are equal in measure. 4. NUMBER SENSE Tell whether this conditional is true or false. If a number is less than 1, the number is a proper fraction. Write the converse of the statement. Is the converse true or false? 5. SRTS If a shortstop makes a bad throw to first base, the error is charged to the shortstop. This statement is true. Write the converse of the statement. Is it true or false? 6. GEGRAHY If a point is located north of the equator, it has a northern latitude. This statement is true. Write the converse of the statement. Is it true or false? RACTICE EXERCISES For Extra ractice, see page 672. Sketch a counterexample that shows why each conditional is false. 7. If line t intersects lines g and h, then line t is a transversal. 8. If Q QR, then point Q is the midpoint of R. 9. If points A, B, and C are collinear, then BA and BC are opposite rays. 10. If two angles share a common side and a common vertex, then they are adjacent angles. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 11. If points J, K, and L are coplanar, then they are collinear. 12. If point Y is the midpoint of X Z, then XY YZ XZ. 13. If the sum of the measures of two angles is 90, then the angles are complementary. 14. If two angles are supplementary, then the sum of their measures is greater than If two lines are perpendicular, then they do not intersect. 16. If m QRS m SRT, then RS bisects QRT. 130 Chapter 3 Geometry and Reasoning

4 Write each definition as two conditionals and as a single biconditional. 17. The midpoint of a segment is the point that divides it into two segments of equal length. 18. erpendicular lines are two lines that intersect to form right angles. 19. A transversal is a line that intersects two or more coplanar lines in different points. 20. Vertical angles are two angles whose sides form two pairs of opposite rays. GEMETRIC CNSTRUCTIN The corresponding angles postulate provides a method for constructing parallel lines. In the figure at the right, you see the finished construction of a line parallel to line through point. Trace line and point onto a sheet of paper and repeat the construction. Then complete the statements below that outline the steps of the construction. 21. Step 1: Using a straightedge, draw any line? through point intersecting line. Label the intersection point?. 22. Step 2: With the compass point at?, draw an arc intersecting lines t and. Label the intersection points? and?. 23. Step 3: Using the same radius as in Step 2, place the compass point at point? and draw an arc intersecting line t. Label the intersection point?. 24. Step 4: lace the compass point at point? and the pencil at point?. Using this radius, draw an arc that intersects line. 25. Step 5: Using the same radius as in Step 4, place the compass point at point? and draw an arc that intersects the arc you drew in Step 3. Label the intersection point?. 26. Step 6: Draw line? through points and Y. m? m?, and so??. m t X Y A Q B EXTENDED RACTICE EXERCISES WRITING MATH Explain why each of the following is not a good definition. 27. Vertical angles are two angles whose sides form opposite rays. 28. A line segment is part of a line. 29. Complementary angles are adjacent angles whose exterior sides form a right angle. 30. Skew lines are noncoplanar lines that do not intersect. MIXED REVIEW EXERCISES Find each length. (Lesson 3-1) 31. In the figure below, AC 130. Find BC. 32. In the figure below, LM 94. Find LN. 3x 4 4x 7p 14 3p L M A B C N 33. In the figure below, JL 88. Find KL. 34. In the figure below, QS 41. Find QR. 3y 8 9y 2z 14 3z 18 J K L Q R S mathmatters3.com/self_check_quiz Lesson 3-6 Conditional Statements 131

5 Name Date RETEACHING 3-6 CNDITINAL STATEMENTS Conditional statements are written in an if-then (hypothesis/conclusion) form. The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion. A counterexample proves a conditional or converse is false. E x a m p l e Write the converse of this statement: If RS TU, then RS, TU, and RT are coplanar. Then decide whether the statement and its converse are true or false. If false, give a counterexample. Converse: If R S, T U and R T are coplanar, then R S T U. riginal statement is true, since parallel lines are coplanar by definition and for any two points in a plane, the line joining them lies in the plane (ostulate 3). Converse is false. Counterexample: Each pair of coplanar lines could intersect to form a triangle. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If false, give a counterexample. 1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D are collinear. 2. If m IQJ m HQJ 180, then IQJ and HQJ are obtuse angles. 3. If three lines have one point in common, then they are coplanar. 4. If two lines are skew, then they are not coplanar. Glencoe/McGraw-Hill 88 MathMatters 3

6 Name Date EXTRA RACTICE 3-6 CNDITINAL STATEMENTS Sketch a counterexample that shows why each conditional is false. Use your own paper. 1. If point A is the midpoint of C D, then C A D A. 2. If XY and XZ are opposite rays, then point X is the midpoint of Y Z. 3. If RST and TSV are congruent, then ST bisects RSV. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 4. If points A, B, and C are collinear, then B is the midpoint of A C. 5. If point X is the vertex of 1 and 2, then 1 and 2 are adjacent angles. 6. If two angles are complementary, then both of the angles are acute. 7. If two lines intersect, then they are parallel. 8. If an angle is obtuse, then its supplement is acute. Write each definition as two conditionals and as a single biconditional. 9. Coplanar points are points that lie in the same plane. 10. A segment is a part of a line that begins at one endpoint and ends at another. Glencoe/McGraw-Hill 89 MathMatters 3

7 Review and ractice Your Skills RACTICE LESSN 3-5 Draw the next figure in each pattern. Then describe the tenth figure in the pattern RACTICE LESSN 3-6 Sketch a counterexample to show why each conditional is false. 9. If ABC and DEF are supplements, then m ABC m DEF. 10. If three points are coplanar, then they are collinear. 11. If two lines are skew, then they intersect. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 12. If two lines intersect, then they are perpendicular. 13. If C is the midpoint of A B, then AB 2(AC ). 14. If two angles are vertical angles, then their supplements are equal. Write each definition as two conditionals and as a single biconditional. 15. erpendicular lines are lines that intersect to form right angles. 16. Skew lines are noncoplanar lines. 132 Chapter 3 Geometry and Reasoning

8 Name Date ENRICHMENT 3-6 CATEGRICAL RSITINS A categorical proposition is a statement about an entire category or class of things. There are four different standard forms of categorical propositions. All S is. No S is. Some S is. Some S is not. All dogs are friendly. No dogs are friendly. Some dogs are friendly. Some dogs are not friendly. Venn diagrams can be used to illustrate categorical propositions. E x a m p l e 1 Diagram All S is. E x a m p l e 2 Diagram Some S is not. The shading shows that this part of the diagram has no members. The X shows that this part of the diagram has at least one member. Draw a Venn diagram for each categorical proposition. 1. No S is. 2. Some S is. Write the converse of each of the four standard forms. Then draw a Venn diagram for each one Which of the standard forms are logically equivalent to their converses? Glencoe/McGraw-Hill 90 MathMatters 3

9 Lesson lanning NCTM Standards/Strands Number & perations Algebra Geometry roblem Solving Reasoning & roof Communication Connections Representation Vocabulary conditional conclusion biconditional Materials Needed paper/pencil Lesson Resources Warm-Up Transparency 8 Reteaching 3-6 Extra ractice 3-6 Enrichment 3-6 Getting Started hypothesis counterexample 3-6 Conditional Statements Goals Identify and evaluate conditional statements. Identify converses and biconditionals. Do you think each statement is true or false? Explain your reasoning. True; this is a a. Denver is the capital of Colorado. fact. b. For all x, x 2 x. False; 1 < c. Lines m and n are parallel. Cannot tell; there is no information given to identify line m and n. BUILD UNDERSTANDING Many of the statements in this chapter are written in if then form. Statements like these are called conditional statements, or simply conditionals. The clause following if is called the hypothesis of the conditional. The clause following then is called the conclusion. For example, the parallel lines postulate was presented as a conditional. If two parallel lines are cut by a transversal, hypothesis then corresponding angles are equal in measure. conclusion A conditional is either true or false. When a conditional is true, you can justify it in a variety of ways. For instance, you may be able to show that the conditional is true because it follows directly from a definition. When a conditional is a postulate, such as the parallel lines postulate, it is assumed to be true. Still other conditionals are theorems, and these must be proved true. To demonstrate that a conditional is false, you need to find only one example for which the hypothesis is true but the conclusion is false. An example like this is called a counterexample. E x a m p l e 1 Applications Drafting, Sports, Geography Denver, C 5-MINUTE WARM-U Is each statement true or false? 1. If a figure is a square, then it has four sides. true 2. If a figure has four sides, then it is a square. false Tell whether each conditional is true or false. a. If two lines are parallel, then they are coplanar. b. If two lines do not intersect, then they are parallel. a. arallel lines are defined as coplanar lines that do not intersect. So, the conditional is true. Introduction to Lesson 3-6 Have students work in small groups to answer the questions. oint out that while the first two statements are either true or false, the third cannot be determined without knowing more about lines m and n. 128 b. Consider skew lines k and shown. By the definition of skew lines, k and do not intersect, and so the hypothesis is true. However, also by the definition of skew lines, k and are noncoplanar. Lines k and cannot be parallel, and so the conclusion is false. Therefore, lines k and are a counterexample, and the conditional is false. Chapter 3 Geometry and Reasoning k ADDITINAL ANSWERS 3. If an angle is bisected by a ray, then the two adjacent angles formed are equal in measure. If an angle is divided by a ray into two adjacent angles that are equal in measure, then the ray bisects the angle. An angle is bisected by a ray if and only if the two adjacent angles formed are equal in measure. 128 Chapter 3 Geometry and Reasoning h g t B A C 10. There are two possible counterexamples. In the following figure, AXB and AXC share a common side and a common vertex, but they also have interior points in common. Therefore, they are not adjacent angles. 5 Q 5 R X A B C In the figure below, MN and N share a common side and a common vertex, but they do not lie in the M same plane. N Therefore, they are not adjacent.

10 The converse of a conditional is formed by interchanging the hypothesis and the conclusion. The fact that a conditional is true is no guarantee that its converse is true. E x a m p l e 2 DRAFTING eople who draw plans must apply this true statement: If two lines are parallel, then they do not intersect. Write the converse of the statement. Is it also true? Interchange the hypothesis and the conclusion of the given statement. Statement: If two lines are parallel, then they do not intersect. hypothesis conclusion Converse: If two lines do not intersect, then they are parallel. hypothesis conclusion By definition, parallel lines do not intersect, and so the given statement is true. art b of Example 1 demonstrated that lines that do not intersect are not necessarily parallel, and so the converse is false. The converse of the parallel lines postulate also is assumed to be true. It is stated as the corresponding angles postulate in the following manner. ostulate 10 The Corresponding Angles ostulate If two lines are cut by a transversal so that a pair of corresponding angles are equal in measure, then the lines are parallel. Reading Math Many statements about everyday situations can be expressed as conditionals. For instance, the following is a true conditional. If it is raining, then it is cloudy. Its converse is false. If it is cloudy, then it is raining. Chalkboard Examples Supplementary Example 1 Tell whether this conditional is true or false: If two lines are coplanar, then they are not skew lines. true Supplementary Example 2 Write the converse of this statement: If a number is divisible by six, then it is divisible by three. Then decide whether the statement and its converse are true or false. If a number is divisible by three, then it is divisible by six. The statement is true; the converse is false. Supplementary Example 3 Write this definition as two conditionals and a single biconditional: A triangle is a polygon with three sides. If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is a triangle. A polygon is a triangle if and only if it has three sides. When a statement and its converse are both true, they can be combined into an if and only if statement. This type of statement is called a biconditional statement, or simply a biconditional. Every definition can be written as a biconditional. E x a m p l e 3 Write this definition as two conditionals and as a single biconditional. A right angle is an angle whose measure is 90. The definition leads to two true conditionals. If an angle is a right angle, then its measure is 90. If the measure of an angle is 90, then it is a right angle. These can be combined into a single biconditional as follows. An angle is a right angle if and only if its measure is 90. mathmatters3.com/extra_examples roblem Solving Tip When writing the converse of a conditional, you may need to change the wording of the hypotheses and the conclusion to make the converse read clearly. Lesson 3-6 Conditional Statements 129 Reteaching Worksheet 3-6 Name RETEACHING 3-6 Date CNDITINAL STATEMENTS Conditional statements are written in an if-then (hypothesis/conclusion) form. The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion. A counterexample proves a conditional or converse is false. E x a m p l e Write the converse of this statement: If RS TU, then RS, TU, and RT are coplanar. Then decide whether the statement and its converse are true or false. If false, give a counterexample. Converse: If R S, T U and R T are coplanar, then R S T U. riginal statement is true, since parallel lines are coplanar by definition and for any two points in a plane, the line joining them lies in the plane (ostulate 3). Converse is false. Counterexample: Each pair of coplanar lines could intersect to form a triangle. 11. Converse; If points J, K, and L are collinear, then they are coplanar. The given statement is false. Its converse is true. 12. Converse: If XY YZ XZ, then point Y is the midpoint of XZ. The given statement is true. Its converse is false. 13. Converse: If two angles are complementary, then the sum of their measures is 90º. Both the given statement and its converse are true. 14. Converse: If the sum of the measures of two angles is greater than 90º, then the angles are supplementary. The given statement is true. Its converse is false. 15. Converse: If two lines do not intersect, then they are perpendicular. Both the given statement and its converse are false. 16. Converse: If RS bisects QRT, then m QRS m SRT. The given statement is false. Its converse is true. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If false, give a counterexample. 1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D are collinear. If points A, B, C, and D are collinear, then points A, B, C, and D lie in both plane L and plane M.; true; false; Counterexample: AD lies on plane L and plane L is parallel to plane M. 2. If m IQJ m HQJ 180, then IQJ and HQJ are obtuse angles. If IQJ and HQJ are obtuse angles, then m IQJ m HQJ 180.; false; Counterexample: If m IQJ 30 and m HQJ 170, then m IQJ m HQJ ; true 3. If three lines have one point in common, then they are coplanar. If three lines are coplanar, then they have one point in common. false; Counterexample: 2 lines can be coplanar and a third line can intersect that plane at only one point.; false; 3 parallel lines 4. If two lines are skew, then they are not coplanar. If two lines are not coplanar, then they are skew.; true; true Lesson 3-6 Conditional Statements 129

11 Lesson Wrap-up QUICK ASSESSMENT Ask the following questions to determine if students understand the content presented in this lesson. 1. When is a conditional statement true? when it is always true; when it can be shown that if the hypothesis is true, then the conclusion must also be true 2. Give an example of a mathematical statement that is true although its converse is false. Answers may vary. 3. Explain how a definition and a biconditional statement are related. Any definition can be written as two true conditional statements (the statement and its converse), which can be written as a biconditional. ASSIGNMENT GUIDE Basic: 1 26, 31, 32 Enriched: 1 32 ADDITINAL ANSWERS 17. Conditionals: If a point is the midpoint of a segment, then it divides the segment into two segments of equal length; if a point divides a segment into two segments of equal length, then it is the midpoint of the segment. Biconditional: A point is the midpoint of a segment if and only if it divides the segment into two segments of equal length. 18. Conditionals: If two lines are perpendicular, then they intersect to form right angles; if two lines intersect to form right angles, then they are perpendicular. Biconditional: Two lines are perpendicular if and only if they intersect to form right angles. 19. Conditionals: If a line is a transversal, then it intersects two or more coplanar lines in different points; if a line intersects two or more coplanar lines in different points, then it is a transversal. Biconditional: A line is a transversal if and only if it intersects 130 Chapter 3 Geometry and Reasoning 130 TRY THESE EXERCISES 1. TALK ABUT IT Decide whether this conditional is true or false. False; it is possible that the two lines are noncoplanar. If two lines are each perpendicular to a third line, then they are parallel to each other. Discuss your reasoning with a classmate. 2. Write the converse of this statement. If two angles are equal in measure, then they are vertical angles. If two angles are vertical angles, then they are equal in measure. Are the given statement and its converse true or false? Given statement is true. Converse is false. 3. Write this definition as two conditionals and as a single biconditional. See additional answers. The bisector of an angle is the ray that divides the angle into two adjacent angles that are equal in measure. 4. NUMBER SENSE Tell whether this conditional is true or false. False. Negative integers are less than 1. If a number is less than 1, the number is a proper fraction. Write the converse of the statement. Is the converse true or false? Converse: If the number is a proper fraction, then it is less than 1. True. 5. SRTS If a shortstop makes a bad throw to first base, the error is charged to the shortstop. This statement is true. Write the converse of the statement. Is it true or false? If an error is charged to the shortstop, then the shortstop made a bad throw to first base. False. 6. GEGRAHY If a point is located north of the equator, it has a northern latitude. This statement is true. Write the converse of the statement. Is it true or false? If a point has a northern latitude, then the point is located north of the equator. True. RACTICE EXERCISES For Extra ractice, see page 672. Sketch a counterexample that shows why each conditional is false. See additional answers. 7. If line t intersects lines g and h, then line t is a transversal. 8. If Q QR, then point Q is the midpoint of R. 9. If points A, B, and C are collinear, then BA and BC are opposite rays. 10. If two angles share a common side and a common vertex, then they are adjacent angles. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. See additional answers. 11. If points J, K, and L are coplanar, then they are collinear. 12. If point Y is the midpoint of X Z, then XY YZ XZ. 13. If the sum of the measures of two angles is 90, then the angles are complementary. 14. If two angles are supplementary, then the sum of their measures is greater than If two lines are perpendicular, then they do not intersect. 16. If m QRS m SRT, then RS bisects QRT. Chapter 3 Geometry and Reasoning two or more coplanar lines in different points. 20. Conditionals: If two angles are vertical angles, then their sides form two pairs of opposite rays; if the sides of two angles form two pairs of opposite rays, then they are vertical angles. Biconditional: Two angles are vertical angles if and only if their sides form two pairs of opposite rays. 27. Write the given definition as two conditional statements: If two angles are vertical angles, then their sides form opposite rays is true. However, If the sides of two angles form opposite rays, then the angles are vertical angles is false. Here is a counterexample in which the sides of 1 and 2 form a pair of opposite rays, but the angles are not vertical angles. For this reason, it is necessary to define vertical angles as two angles 1 2 whose sides form two pairs of opposite rays.

12 Write each definition as two conditionals and as a single biconditional. See additional answers. 17. The midpoint of a segment is the point that divides it into two segments of equal length. 18. erpendicular lines are two lines that intersect to form right angles. 19. A transversal is a line that intersects two or more coplanar lines in different points. 20. Vertical angles are two angles whose sides form two pairs of opposite rays. GEMETRIC CNSTRUCTIN The corresponding angles postulate provides a method for constructing parallel lines. In the figure at the right, you see the finished construction of a line parallel to line through point. Trace line and point onto a sheet of paper and repeat the construction. Then complete the statements below that outline the steps of the construction. 21. Step 1: Using a straightedge, draw any line? through point intersecting line. Label the intersection point?. t, Q 22. Step 2: With the compass point at?, draw an arc intersecting lines t and. Label the intersection points? and?. Q, A, B 23. Step 3: Using the same radius as in Step 2, place the compass point at point? and draw an arc intersecting line t. Label the intersection point?., X 24. Step 4: lace the compass point at point? and the pencil at point?. Using this radius, draw an arc that intersects line. 25. Step 5: Using the same radius as in Step 4, place the compass point at point? and draw an arc that intersects the arc you drew in Step 3. Label the intersection point?. X, Y 26. Step 6: Draw line? through points and Y. m? m?, and so??. m, XY, AQB, m, EXTENDED RACTICE EXERCISES A, B m t X Y A Q B Extra ractice Worksheet 3-6 Name EXTRA RACTICE 3-6 CNDITINAL STATEMENTS Sketch a counterexample that shows why each conditional is false. Use your own paper. Check students drawings. 1. If point A is the midpoint of C D, then C A D A. 2. If XY and XZ are opposite rays, then point X is the midpoint of Y Z. 3. If RST and TSV are congruent, then ST bisects RSV. Date Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 4. If points A, B, and C are collinear, then B is the midpoint of A C. If B is the midpoint of A C, then points A, B, and C are collinear; false; true 5. If point X is the vertex of 1 and 2, then 1 and 2 are adjacent angles. If 1 and 2 are adjacent angles, then they have a common vertex; false; true 6. If two angles are complementary, then both of the angles are acute. If two angles are acute, then the angles are complementary.; true; false 7. If two lines intersect, then they are parallel. If two lines are parallel, then the line intersect.; false; false 8. If an angle is obtuse, then its supplement is acute. If the supplement of an angle is acute, then the angle is obtuse.; true; true Write each definition as two conditionals and as a single biconditional. 9. Coplanar points are points that lie in the same plane. If points are coplanar, then they lie in the same plane. If points lie in the same plane, then the points are coplanar. oints are coplanar if and only if they lie in the same plane 10. A segment is a part of a line that begins at one endpoint and ends at another. If a figure is a segment, then it is a part of a line that begins at one endpoint and ends at another. If a figure is a part of a line that begins at one endpoint and ends at another, then the figure is a segment. A figure is a segment if and only if it is part of a line that begins at one endpoint and ends at another. WRITING MATH Explain why each of the following is not a good definition. See additional answers. 27. Vertical angles are two angles whose sides form opposite rays. 28. A line segment is part of a line. 29. Complementary angles are adjacent angles whose exterior sides form a right angle. 30. Skew lines are noncoplanar lines that do not intersect. MIXED REVIEW EXERCISES Find each length. (Lesson 3-1) 31. In the figure below, AC 130. Find BC In the figure below, LM 94. Find LN. 70 3x 4 4x 7p 14 3p L M A B C N 33. In the figure below, JL 88. Find KL In the figure below, QS 41. Find QR. 32 3y 8 9y 2z 14 3z 18 J K L Q R S mathmatters3.com/self_check_quiz Lesson 3-6 Conditional Statements 131 Enrichment Worksheet 3-6 Name ENRICHMENT 3-6 CATEGRICAL RSITINS A categorical proposition is a statement about an entire category or class of things. There are four different standard forms of categorical propositions. All S is. No S is. Some S is. Some S is not. All dogs are friendly. No dogs are friendly. Some dogs are friendly. Some dogs are not friendly. Venn diagrams can be used to illustrate categorical propositions. E x a m p l e 1 Diagram All S is. The shading shows that this part of the diagram has no members. E x a m p l e 2 Date Diagram Some S is not. The X shows that this part of the diagram has at least one member. 28. Write the given definition as two conditionals: If a figure is a line segment, then it is part of a line is true. However, If a figure is part of a line, then it is a line segment is false. A ray also is part of a line. It is necessary to specify that a line segment is part of a line that begins at one endpoint and ends at another. 29. Write the given definition as two conditionals: If two angles are adjacent angles whose exterior sides form a right angle, then they are complementary is true. However, If two angles are complementary, then they are adjacent angles whose exterior sides form a right angle is false. Complementary angles are not necessarily adjacent. 30. The given definition is a true statement, but it contains too much information to be a good definition. It is not necessary to include the phrase that do not intersect because the term noncoplanar lines already indicates that the two lines do not intersect. Draw a Venn diagram for each categorical proposition. 1. No S is. 2. Some S is. S Write the converse of each of the four standard forms. Then draw a Venn diagram for each one. 3. All is S. 4. No is S. S 5. Some is S. 6. Some is not S. 7. Which of the standard forms are logically equivalent to their converses? No S is. No is S. and Some S is. Some is S. Lesson 3-6 Conditional Statements 131 S S S X X

13 Skills ractice Vocabulary Review Lesson 3-5 inductive reasoning conjecture sequence Lesson 3-6 conditional conclusion biconditional ASSIGNMENT GUIDE All students: 1 24 hypothesis counterexample ADDITINAL ANSWERS Review and ractice Your Skills RACTICE LESSN 3-5 Draw the next figure in each pattern. Then describe the tenth figure in the pattern. See additional answers for drawings units horizontal; 10 units vertical units long same orientation as second figure, but 512 individual angles, each with ten lines in the interior same as second figure 512 units each as long as original line. same as second figure 5. RACTICE LESSN sided polygon with all diagonals C D A B E F Y X Z Sketch a counterexample to show why each conditional is false. See additional answers for sample answers. 9. If ABC and DEF are supplements, then m ABC m DEF. 10. If three points are coplanar, then they are collinear. 11. If two lines are skew, then they intersect. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If two lines are perpendicular, then they 12. If two lines intersect, then they are perpendicular. intersect. false, true 13. If C is the midpoint of A B, then AB 2(AC ). If AB 2(AC), then C is the midpoint of A B. true, false 14. If two angles are vertical angles, then their supplements are equal. If the supplements of two angles are equal, then they are vertical angles. true, false Write each definition as two conditionals and as a single biconditional. 15. erpendicular lines are lines that intersect to form right angles. See additional answers. 16. Skew lines are noncoplanar lines. If lines are skew, they are noncoplanar. If lines are noncoplanar, then they are skew. Two lines are skew if and only if they are noncoplanar. Chapter 3 Geometry and Reasoning 15. If two lines are perpendicular, then they intersect to form right angles. If two lines intersect to form right angles, then they are perpendicular. Two lines are perpendicular if and only if they intersect to form right angles. 132 Chapter 3 Geometry and Reasoning

14 Answers (Lesson 3-5 and 3-6) Name Date ENRICHMENT 3-5 ATTERNS WITH DISSECTINS There are five ways to dissect a convex five-sided polygon into three triangles. 1. Show all the different ways to dissect a convex four-sided polygon into 2 triangles. How many are there? 2 ways 2. Show all the different ways to dissect a convex six-sided polygon into 4 triangles. How many are there? 14 ways Study the patterns in this chart. Then fill in the missing values Number Number of of Sides Dissections Formula for D n 3 D D 4 2 D 3 D 3 5 D 5 5 D 4 D 3 D 3 D 4 6 D 6 14 D 5 D 3 D 4 D 4 D 3 D 5 7 D 7 42 D 6 D 3 D 5 D 4 D 4 D 5 D 3 D 6 8 D D 7 D 3 D 6 D 4 D 5 D 5 D 4 D 6 D 3 D 7 9 D D 8 D 3 D 7 D 4 D 6 D 5 D 5 D 6 D 4 D 7 D 3 D 8 10 D D 9 D 3 D 8 D 4 D 7 D 5 D 6 D 6 D 5 D 7 D 4 D 8 D 3 D 9 Glencoe/McGraw-Hill 87 MathMatters 3 Name Date RETEACHING 3-6 CNDITINAL STATEMENTS Conditional statements are written in an if-then (hypothesis/conclusion) form. The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion. A counterexample proves a conditional or converse is false. E x a m p l e Write the converse of this statement: If RS TU, then RS, TU, and RT are coplanar. Then decide whether the statement and its converse are true or false. If false, give a counterexample. Converse: If R S, T U and R T are coplanar, then R S T U. riginal statement is true, since parallel lines are coplanar by definition and for any two points in a plane, the line joining them lies in the plane (ostulate 3). Converse is false. Counterexample: Each pair of coplanar lines could intersect to form a triangle. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If false, give a counterexample. 1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D are collinear. If points A, B, C, and D are collinear, then points A, B, C, and D lie in both plane L and plane M.; true; false; Counterexample: AD lies on plane L and plane L is parallel to plane M. 2. If m IQJ m HQJ 180, then IQJ and HQJ are obtuse angles. If IQJ and HQJ are obtuse angles, then m IQJ m HQJ 180.; false; Counterexample: If m IQJ 30 and m HQJ 170, then m IQJ m HQJ ; true 3. If three lines have one point in common, then they are coplanar. If three lines are coplanar, then they have one point in common. false; Counterexample: 2 lines can be coplanar and a third line can intersect that plane at only one point.; false; 3 parallel lines 4. If two lines are skew, then they are not coplanar. If two lines are not coplanar, then they are skew.; true; true Glencoe/McGraw-Hill 88 MathMatters 3 Glencoe/McGraw-Hill A9 MathMatters 3

15 Answers (Lesson 3-6) Name Date EXTRA RACTICE 3-6 CNDITINAL STATEMENTS Sketch a counterexample that shows why each conditional is false. Use your own paper. Check students drawings. 1. If point A is the midpoint of C D, then C A D A. 2. If XY and XZ are opposite rays, then point X is the midpoint of Y Z. 3. If RST and TSV are congruent, then ST bisects RSV. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 4. If points A, B, and C are collinear, then B is the midpoint of A C. midpoint of A C, then points A, B, and C are collinear; false; true 5. If point X is the vertex of 1 and 2, then 1 and 2 are adjacent angles. and 2 are adjacent angles, then they have a common vertex; false; true 6. If two angles are complementary, then both of the angles are acute. angles are acute, then the angles are complementary.; true; false 7. If two lines intersect, then they are parallel. the line intersect.; false; false 8. If an angle is obtuse, then its supplement is acute. angle is acute, then the angle is obtuse.; true; true If B is the If two If 1 If two lines are parallel, then If the supplement of an Write each definition as two conditionals and as a single biconditional. 9. Coplanar points are points that lie in the same plane. If points are coplanar, then they lie in the same plane. If points lie in the same plane, then the points are coplanar. oints are coplanar if and only if they lie in the same plane 10. A segment is a part of a line that begins at one endpoint and ends at another. If a figure is a segment, then it is a part of a line that begins at one endpoint and ends at another. If a figure is a part of a line that begins at one endpoint and ends at another, then the figure is a segment. A figure is a segment if and only if it is part of a line that begins at one endpoint and ends at another. Glencoe/McGraw-Hill 89 MathMatters 3 Name Date ENRICHMENT 3-6 CATEGRICAL RSITINS A categorical proposition is a statement about an entire category or class of things. There are four different standard forms of categorical propositions. All S is. All dogs are friendly. No S is. No dogs are friendly. Some S is. Some dogs are friendly. Some S is not. Some dogs are not friendly. Venn diagrams can be used to illustrate categorical propositions. E x a m p l e 1 E x a m p l e 2 Diagram All S is. Diagram Some S is not. The shading shows that this part of the diagram has no members. The X shows that this part of the diagram has at least one member. Draw a Venn diagram for each categorical proposition. 1. No S is. 2. Some S is. S S X Write the converse of each of the four standard forms. Then draw a Venn diagram for each one. All is S S No is S. S Some is S Some is not S. S X 7. Which of the standard forms are logically equivalent to their converses? No S is. No is S. and Some S is. Some is S. Glencoe/McGraw-Hill 90 MathMatters 3 Glencoe/McGraw-Hill A10 MathMatters 3

1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above? 1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

More information

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle. Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

More information

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures. Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

More information

2.1. Inductive Reasoning EXAMPLE A

2.1. Inductive Reasoning EXAMPLE A CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers

More information

Duplicating Segments and Angles

Duplicating Segments and Angles CONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson, you Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using patty

More information

Chapter 1: Essentials of Geometry

Chapter 1: Essentials of Geometry Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

More information

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1 Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the

More information

Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math. Semester 1 Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

More information

This is a tentative schedule, date may change. Please be sure to write down homework assignments daily.

This is a tentative schedule, date may change. Please be sure to write down homework assignments daily. Mon Tue Wed Thu Fri Aug 26 Aug 27 Aug 28 Aug 29 Aug 30 Introductions, Expectations, Course Outline and Carnegie Review summer packet Topic: (1-1) Points, Lines, & Planes Topic: (1-2) Segment Measure Quiz

More information

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points. Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

More information

GEOMETRY - QUARTER 1 BENCHMARK

GEOMETRY - QUARTER 1 BENCHMARK Name: Class: _ Date: _ GEOMETRY - QUARTER 1 BENCHMARK Multiple Choice Identify the choice that best completes the statement or answers the question. Refer to Figure 1. Figure 1 1. What is another name

More information

1.1 Identify Points, Lines, and Planes

1.1 Identify Points, Lines, and Planes 1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms - These words do not have formal definitions, but there is agreement aboutwhat they mean.

More information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line. Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

More information

Find the measure of each numbered angle, and name the theorems that justify your work.

Find the measure of each numbered angle, and name the theorems that justify your work. Find the measure of each numbered angle, and name the theorems that justify your work. 1. The angles 2 and 3 are complementary, or adjacent angles that form a right angle. So, m 2 + m 3 = 90. Substitute.

More information

Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1. Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

More information

A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.

A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon. hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.

More information

GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd: GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Chapter Two. Deductive Reasoning

Chapter Two. Deductive Reasoning Chapter Two Deductive Reasoning Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply

More information

GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY. Constructions OBJECTIVE #: G.CO.12 GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

More information

Geometry: 2.1-2.3 Notes

Geometry: 2.1-2.3 Notes Geometry: 2.1-2.3 Notes NAME 2.1 Be able to write all types of conditional statements. Date: Define Vocabulary: conditional statement if-then form hypothesis conclusion negation converse inverse contrapositive

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

More information

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram

More information

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

More information

Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: Activity 24

Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: Activity 24 Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 24 esources: Springoard- Geometry Unit Overview In this unit, students will study formal definitions of basic figures,

More information

Chapter 6 Notes: Circles

Chapter 6 Notes: Circles Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment

More information

Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning:

Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Conjecture: Advantages: can draw conclusions from limited information helps us to organize

More information

How Do You Measure a Triangle? Examples

How Do You Measure a Triangle? Examples How Do You Measure a Triangle? Examples 1. A triangle is a three-sided polygon. A polygon is a closed figure in a plane that is made up of segments called sides that intersect only at their endpoints,

More information

Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 6 and 7 Notes: Circles, Locus and Concurrence Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture. CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion

More information

Final Review Geometry A Fall Semester

Final Review Geometry A Fall Semester Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over

More information

4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem. 14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

More information

A geometric construction is a drawing of geometric shapes using a compass and a straightedge.

A geometric construction is a drawing of geometric shapes using a compass and a straightedge. Geometric Construction Notes A geometric construction is a drawing of geometric shapes using a compass and a straightedge. When performing a geometric construction, only a compass (with a pencil) and a

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

3.1. Angle Pairs. What s Your Angle? Angle Pairs. ACTIVITY 3.1 Investigative. Activity Focus Measuring angles Angle pairs

3.1. Angle Pairs. What s Your Angle? Angle Pairs. ACTIVITY 3.1 Investigative. Activity Focus Measuring angles Angle pairs SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Use Manipulatives Two rays with a common endpoint form an angle. The common endpoint is called the vertex. You can use a protractor to draw and measure

More information

Lesson 3.1 Duplicating Segments and Angles

Lesson 3.1 Duplicating Segments and Angles Lesson 3.1 Duplicating Segments and ngles In Exercises 1 3, use the segments and angles below. Q R S 1. Using only a compass and straightedge, duplicate each segment and angle. There is an arc in each

More information

Definitions, Postulates and Theorems

Definitions, Postulates and Theorems Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

More information

Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

More information

5.1 Midsegment Theorem and Coordinate Proof

5.1 Midsegment Theorem and Coordinate Proof 5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects

More information

Incenter Circumcenter

Incenter Circumcenter TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

More information

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,

More information

Intermediate Math Circles October 10, 2012 Geometry I: Angles

Intermediate Math Circles October 10, 2012 Geometry I: Angles Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,

More information

10-4 Inscribed Angles. Find each measure. 1.

10-4 Inscribed Angles. Find each measure. 1. Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

More information

Mathematics Geometry Unit 1 (SAMPLE)

Mathematics Geometry Unit 1 (SAMPLE) Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This

More information

Foundations of Geometry 1: Points, Lines, Segments, Angles

Foundations of Geometry 1: Points, Lines, Segments, Angles Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

More information

Lesson 10.1 Skills Practice

Lesson 10.1 Skills Practice Lesson 0. Skills Practice Name_Date Location, Location, Location! Line Relationships Vocabulary Write the term or terms from the box that best complete each statement. intersecting lines perpendicular

More information

Unit 6 Grade 7 Geometry

Unit 6 Grade 7 Geometry Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson

More information

Show all work for credit. Attach paper as needed to keep work neat & organized.

Show all work for credit. Attach paper as needed to keep work neat & organized. Geometry Semester 1 Review Part 2 Name Show all work for credit. Attach paper as needed to keep work neat & organized. Determine the reflectional (# of lines and draw them in) and rotational symmetry (order

More information

Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18 Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 19, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications

More information

Circle Name: Radius: Diameter: Chord: Secant:

Circle Name: Radius: Diameter: Chord: Secant: 12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

More information

Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

More information

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle. DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles that are between parallel lines, but on opposite sides of a transversal. GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

More information

Geometry Enduring Understandings Students will understand 1. that all circles are similar.

Geometry Enduring Understandings Students will understand 1. that all circles are similar. High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

More information

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures 8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

More information

GEOMETRY CONCEPT MAP. Suggested Sequence:

GEOMETRY CONCEPT MAP. Suggested Sequence: CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

More information

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

Unit 2 - Triangles. Equilateral Triangles

Unit 2 - Triangles. Equilateral Triangles Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics

More information

Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period

Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period Unit 8 Quadrilaterals Academic Geometry Spring 2014 Name Teacher Period 1 2 3 Unit 8 at a glance Quadrilaterals This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,

More information

Chapter 1. Reasoning in Geometry. Section 1-1 Inductive Reasoning

Chapter 1. Reasoning in Geometry. Section 1-1 Inductive Reasoning Chapter 1 Reasoning in Geometry Section 1-1 Inductive Reasoning Inductive Reasoning = Conjecture = Make a conjecture from the following information. 1. Eric was driving his friends to school when his car

More information

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades. Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

ABC is the triangle with vertices at points A, B and C

ABC is the triangle with vertices at points A, B and C Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

More information

Semester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Semester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question. Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No,

More information

Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key

Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,

More information

Lesson 18: Looking More Carefully at Parallel Lines

Lesson 18: Looking More Carefully at Parallel Lines Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

More information

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts: Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

More information

PROVING STATEMENTS IN GEOMETRY

PROVING STATEMENTS IN GEOMETRY CHAPTER PROVING STATEMENTS IN GEOMETRY After proposing 23 definitions, Euclid listed five postulates and five common notions. These definitions, postulates, and common notions provided the foundation for

More information

2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?

2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE? MATH 206 - Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of

More information

New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

More information

Session 4 Angle Measurement

Session 4 Angle Measurement Key Terms in This Session Session 4 Angle Measurement New in This Session acute angle adjacent angles central angle complementary angles congruent angles exterior angle interior (vertex) angle irregular

More information

A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

More information

Geometry Chapter 5 Relationships Within Triangles

Geometry Chapter 5 Relationships Within Triangles Objectives: Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 To use properties of midsegments to solve problems. To use properties of perpendicular bisectors and angle bisectors. To identify

More information

Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3

Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3 Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3 Main ideas: Identify and use perpendicular bisectors and angle bisectors in triangles. Standard: 12.0 A perpendicular bisector of a

More information

Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will

Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles

More information

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than

More information

Conjectures. Chapter 2. Chapter 3

Conjectures. Chapter 2. Chapter 3 Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

More information

Measure and classify angles. Identify and use congruent angles and the bisector of an angle. big is a degree? One of the first references to the

Measure and classify angles. Identify and use congruent angles and the bisector of an angle. big is a degree? One of the first references to the ngle Measure Vocabulary degree ray opposite rays angle sides vertex interior exterior right angle acute angle obtuse angle angle bisector tudy ip eading Math Opposite rays are also known as a straight

More information

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT!

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! FINDING THE DISTANCE BETWEEN TWO POINTS DISTANCE FORMULA- (x₂-x₁)²+(y₂-y₁)² Find the distance between the points ( -3,2) and

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

Chapter 5: Relationships within Triangles

Chapter 5: Relationships within Triangles Name: Chapter 5: Relationships within Triangles Guided Notes Geometry Fall Semester CH. 5 Guided Notes, page 2 5.1 Midsegment Theorem and Coordinate Proof Term Definition Example midsegment of a triangle

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

Geometry Module 4 Unit 2 Practice Exam

Geometry Module 4 Unit 2 Practice Exam Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

More information

A summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:

A summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: efinitions: efinition of mid-point and segment bisector M If a line intersects another line segment

More information

POTENTIAL REASONS: Definition of Congruence:

POTENTIAL REASONS: Definition of Congruence: Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides

More information

Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1.

Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Name: lass: ate: I: hapter 1 Exam Multiple hoice Identify the choice that best completes the statement or answers the question. 1. bisects, m = (7x 1), and m = (4x + 8). Find m. a. m = c. m = 40 b. m =

More information

0810ge. Geometry Regents Exam 0810

0810ge. Geometry Regents Exam 0810 0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify

More information

Hon Geometry Midterm Review

Hon Geometry Midterm Review Class: Date: Hon Geometry Midterm Review Multiple Choice Identify the choice that best completes the statement or answers the question. Refer to Figure 1. Figure 1 1. Name the plane containing lines m

More information

alternate interior angles

alternate interior angles alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

More information

Most popular response to

Most popular response to Class #33 Most popular response to What did the students want to prove? The angle bisectors of a square meet at a point. A square is a convex quadrilateral in which all sides are congruent and all angles

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

Unit 6 Grade 7 Geometry

Unit 6 Grade 7 Geometry Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson

More information

Mathematics Task Arcs

Mathematics Task Arcs Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number

More information

Quadrilaterals GETTING READY FOR INSTRUCTION

Quadrilaterals GETTING READY FOR INSTRUCTION Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper

More information