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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Find each measure intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what is? intercepted arc. 126 intercepted arc. 72 esolutions Manual - Powered by Cognero Page 1

2 ALGEBRA Find each measure PROOF Write a two-column proof. Given: bisects. If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. So, Given: bisects If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. So, Proof: Statements (Reasons) 1. bisects. (Given) 2. (Def. of segment bisector) 3. intercepts. intercepts. (Def. of intercepted arc) 4. (Inscribed of same arc are.) 5. (Vertical are.) 6. (AAS) Given: bisects. 36 Proof: Statements (Reasons) 1. bisects. (Given) 2. (Def. of segment bisector) 3. intercepts. intercepts. (Def. of intercepted arc) 4. (Inscribed of same arc are.) 5. (Vertical are.) 6. (AAS) esolutions Manual - Powered by Cognero Page 2

3 CCSS STRUCTURE Find each value and An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Solve the two equations to find the values of x and y. 9. x is 180. So, Use the values of the variables to find. 122; 80 ALGEBRA Find each measure. and An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, Since The sum of the measures of the angles of a triangle is 180. So, intercepted arc esolutions Manual - Powered by Cognero Page 3

4 intercepted arc intercepted arc. 48 The sum of the measures of the central angles of a circle with no interior points in common is 360. So, intercepted arc. Here, The arc is a semicircle. So, intercepted arc. So, esolutions Manual - Powered by Cognero Page 4

5 Here, The arc is a semicircle. So, Then, intercepted arc If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. So, If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. So, 32 If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Since A and B both intercept arc CD, m A = m B. m A = 5(4) or esolutions Manual - Powered by Cognero Page 5

6 20. PROOF Write the specified type of proof. 21. paragraph proof Given: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Since C and D both intercept arc AB, m C = m D. m C = 5(10) 3 or Proof: Given means that. Since and, the equation becomes. Multiplying each side of the equation by 2 results in. Proof: Given means that. Since and, the equation becomes. Multiplying each side of the equation by 2 results in. esolutions Manual - Powered by Cognero Page 6

7 22. two-column proof Given: 24. An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle is 180. So, 23. x Statements (Reasons): 1. (Given) 2. (Inscribed intercepting same arc are.) 3. (Vertical are.) 4. (AA Similarity) Statements (Reasons): 1. (Given) 2. (Inscribed intercepting same arc are.) 3. (Vertical are.) 4. (AA Similarity) ALGEBRA Find each value. An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle is 180. So, 25. x An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle is 180. So, An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. So, The sum of the measures of the angles of a triangle is 180. So, esolutions Manual - Powered by Cognero Page 7

8 CCSS STRUCTURE Find each measure. CCSS STRUCTURE Find each measure If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. 135 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. 106 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary PROOF Write a paragraph proof for Theorem Given: Quadrilateral ABCD is inscribed in. and are supplementary. and are supplementary. Proof: By arc addition and the definitions of arc measure and the sum of central angles, esolutions Manual - Powered by Cognero Page 8

9 and. Since by Theorem 10.6 SIGNS: A stop sign in the shape of a regular octagon is inscribed in a circle. Find each, but, so or 180. This makes C and A supplementary. Because the sum of the measures of the interior angles of a quadrilateral is 360,. But, so, making them supplementary also. Given: Quadrilateral ABCD is inscribed in. and are supplementary. and are supplementary. 32. measure. Since all the sides of the stop sign are congruent chords of the circle, the corresponding arcs are congruent. There are eight adjacent arcs that make up the circle, so their sum is 360. Thus, the measure of each arc joining consecutive vertices is of 360 or 45. measure of arc NPQ is Proof: By arc addition and the definitions of arc measure and the sum of central angles, and. Since by Theorem 10.6, so, but or 180. This makes C and A supplementary. Because the sum of the measures of the interior angles of a quadrilateral is 360,. But, so, making them supplementary also. 33. Since all the sides of the stop sign are congruent chords of the circle, all the corresponding arcs are congruent. There are eight adjacent arcs that make up the circle, so their sum is 360. Thus, the measure of each arc joining consecutive vertices is of 360 or 45. Since RLQ is inscribed in the circle, its measure equals one half the measure of its intercepted arc QR. the measure of RLQ is esolutions Manual - Powered by Cognero Page 9

10 34. intercepted arc. So, The sum of the measures of the central angles of a circle with no interior points in common is 360. Here, 360 is divided into 8 equal arcs and each arc measures Then, c. d intercepted arc. So, The sum of the measures of the central angles of a circle with no interior points in common is 360. Here, 360 is divided into 8 equal arcs and each arc measures Then, ART Four different string art star patterns are shown. If all of the inscribed angles of each star shown are congruent, find the measure of each inscribed angle. a. b. a. The sum of the measures of the central angles of a circle with no interior points in common is 360. Here, 360 is divided into 5 equal arcs and each arc measures intercepted arc. So, the measure of each inscribed angle is 36. b. Here, 360 is divided into 6 equal arcs and each arc measures Each inscribed angle is formed by intercepting alternate vertices of the star. So, each intercepted arc measures 120. If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. So, the measure of each inscribed angle is 60. c. Here, 360 is divided into 7 equal arcs and each arc measures If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. So, the measure of each inscribed angle is about d. The sum of the measures of the central angles of a circle with no interior points in common is 360. Here, 360 is divided into 8 equal arcs and each arc measures Each inscribed angle is formed by intercepting alternate vertices of the star. So, each intercepted arc measures 90. If an angle is inscribed esolutions Manual - Powered by Cognero Page 10

12 38. Case 3 Given: P lies outside. is a diameter. Proof: Statements (Reasons) 1. m ABC = m DBC m DBA ( Addition Postulate, Subtraction Property of Equality) 2. m DBC = m(arc DC) m DBA = m(arc DA) (The measure of an inscribed whose side is a diameter is half the measure of the intercepted arc (Case 1)). 3. m ABC = m(arc DC) m(arc DA) (Substitution) 4. m ABC = [m(arc DC) m(arc DA)] (Factor) 5. m(arc DA) + m(arc AC) = m(arc DC) (Arc Addition Postulate) 6. m(arc AC) = m(arc DC) m(arc DA) (Subtraction Property of Equality) 7. m ABC = m(arc AC) (Substitution) Proof: Statements (Reasons) 1. m ABC = m DBC m DBA ( Addition Postulate, Subtraction Property of Equality) 2. m DBC = m(arc DC) m DBA = m(arc DA) (The measure of an inscribed whose side is a diameter is half the measure of the intercepted arc (Case 1)). 3. m ABC = m(arc DC) m(arc DA) (Substitution) 4. m ABC = [m(arc DC) m(arc DA)] (Factor) 5. m(arc DA) + m(arc AC) = m(arc DC) (Arc Addition Postulate) 6. m(arc AC) = m(arc DC) m(arc DA) (Subtraction Property of Equality) 7. m ABC = m(arc AC) (Substitution) PROOF Write the specified proof for each theorem. 39. Theorem 10.7, two-column proof Given: and are inscribed; Proof: Statements (Reasons) 1. and are inscribed; (Given) 2. ; (Measure of an inscribed 3. (Def. of arcs) = half measure of intercepted arc.) 4. (Mult. Prop. of Equality) 5. (Substitution) 6. (Def. of ) Given: and are inscribed; Proof: esolutions Manual - Powered by Cognero Page 12

13 Statements (Reasons) 1. and are inscribed; (Given) 2. ; (Measure of an inscribed 3. (Def. of arcs) = half measure of intercepted arc.) 4. (Mult. Prop. of Equality) 5. (Substitution) 6. (Def. of ) 40. Theorem 10.8, paragraph proof Part I: Given: Proof: Since then is a semicircle. is a right angle. is a semicircle, then. Since is an inscribed angle, or 90. So, by definition, is a right angle. Part II: Given: is a right angle. is a semicircle. Proof: Since is an inscribed angle, then and by the Multiplication Part I: Given: Proof: Since then Part II: Given: Proof: Since is a semicircle. is a right angle. is a semicircle, then. Since is an inscribed angle, is a right angle. is a semicircle. or 90. So, by definition, is a right angle. is an inscribed angle, then and by the Multiplication Property of Equality, m = 2m. Because is a right angle, m = 90. Then m = 2(90) or 180. So by definition, is a semicircle. Property of Equality, m = 2m. Because is a right angle, m = 90. Then m = 2(90) or 180. So by definition, is a semicircle. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate the relationship between the arcs of a circle that are cut by two parallel chords. a. GEOMETRIC Use a compass to draw a circle with parallel chords and. Connect points A and D by drawing segment. b. NUMERICAL Use a protractor to find and. Then determine and. What is true about these arcs? Explain. c. VERBAL Draw another circle and repeat parts a and b. Make a conjecture about arcs of a circle that are cut by two parallel chords. d. ANALYTICAL Use your conjecture to find and in the figure at the right. Verify by using inscribed angles to find the measures of the arcs. esolutions Manual - Powered by Cognero Page 13

14 , ; The arcs are congruent because they have equal measures. c. Sample answer: In a circle, two parallel chords cut congruent arcs. See students work. d. 70; 70 a. b. Sample answer:, ;, ; The arcs are congruent because they have equal measures. c. Sample answer: m A = 15, m D = 15, = 30, and = 30. The arcs are congruent. In a circle, two parallel chords cut congruent arcs. d. In a circle, two parallel chords cut congruent arcs. So, =. = 4(16) + 6 or 70 = 6(16) 26 or 70 the measure of arcs QS and PR are each 70. a. CCSS ARGUMENTS Determine whether the quadrilateral can always, sometimes, or never be inscribed in a circle. Explain your reasoning. 42. square Squares have right angles at each vertex, therefore each pair of opposite angles will be supplementary and inscribed in a circle. the statement is always true. Always; squares have right angles at each vertex, therefore each pair of opposite angles will be supplementary and inscribed in a circle. 43. rectangle Rectangles have right angles at each vertex, therefore each pair of opposite angles will be supplementary and inscribed in a circle. the statement is always true. Always; rectangles have right angles at each vertex, therefore each pair of opposite angles will be supplementary and inscribed in a circle. 44. parallelogram The opposite angles of a parallelogram are always congruent. They will only be supplementary when they are right angles. So, a parallelogram can be inscribed in a circle as long as it is a rectangle. the statement is sometimes true. Sometimes; a parallelogram can be inscribed in a circle as long as it is a rectangle. b. Sample answer:, ; esolutions Manual - Powered by Cognero Page 14

15 45. rhombus The opposite angles of a rhombus are always congruent. They will only be supplementary when they are right angles. So, a rhombus can be inscribed in a circle as long as it is a square. Since the opposite angles of rhombi that are not squares are not supplementary, they can not be inscribed in a circle. the statement is sometimes true. Sometimes; a rhombus can be inscribed in a circle as long as it is a square. Since the opposite angles of rhombi that are not squares are not supplementary, they can not be inscribed in a circle. 47. CHALLENGE A square is inscribed in a circle. What is the ratio of the area of the circle to the area of the square? A square with side s is inscribed in a circle with radius r. Using the Pythagorean Theorem, 46. kite Exactly one pair of opposite angles of a kite are congruent. To be supplementary, they must each be a right angle. If one pair of opposite angles for a quadrilateral is supplementary, the other pair must also be supplementary. So, as long as the angles that compose the pair of congruent opposite angles are right angles, a kite can be inscribed in a circle. the statement is sometimes true. Sometimes; as long as the angles that compose the pair of congruent opposite angles are right angles. The area of a square of side s is A = s 2 and the area of a circle of radius r is A =. the ratio of the area of the circle to the area of the inscribed square is esolutions Manual - Powered by Cognero Page 15

16 48. WRITING IN MATH A right triangle is inscribed in a circle. If the radius of the circle is given, explain how to find the lengths of the right triangle s legs. Sample answer: A triangle will have two inscribed angles of 45 and one of 90. The hypotenuse is across from the 90, or right angle. According to theorem 10.8, an inscribed angle of a triangle intercepts a diameter if the angle is a right angle. the hypotenuse is a diameter and has a length of 2r. Use trigonometry to find the length of the equal legs. the length of each leg of the triangle can be found by multiplying the radius of the circle in which it is inscribed by. Sample answer: According to theorem 10.8, an inscribed angle of a triangle intercepts a diameter if the angle is a right angle. the hypotenuse is a diameter and has a length of 2r. Using trigonometry, each leg = sin or. 49. OPEN ENDED Find and sketch a real-world logo with an inscribed polygon. See students work. See students work. 50. WRITING IN MATH Compare and contrast inscribed angles and central angles of a circle. If they intercept the same arc, how are they related? An inscribed angle has its vertex on the circle. A central angle has its vertex at the center of the circle. If a central angle intercepts arc AB, then the measure of the central angle is equal to m(arc AB). If an inscribed angle also intercepts arc AB, then the measure of the inscribed angle is equal to m(arc AB). So, if an inscribed angle and a central angle intercept the same arc, then the measure of the inscribed angle is one-half the measure of the central angle. An inscribed angle has its vertex on the circle. A central angle has its vertex at the center of the circle. If an inscribed angle and a central angle intercept the same arc, then the measure of the inscribed angle is one-half the measure of the central angle. 51. In the circle below, and. What is? A 42 B 61 C 80 D 84 intercepted arc. But, The correct choice is A. A esolutions Manual - Powered by Cognero Page 16

17 52. ALGEBRA Simplify 4(3x 2)(2x + 4) + 3x 2 + 5x 6. F 9x 2 + 3x 14 G 9x x 14 H 27x x 38 J 27x x 26 Use the Distributive Property to simplify the first term. 54. SAT/ACT The sum of three consecutive integers is 48. What is the least of the three integers? A 15 B 16 C 17 D 18 E 19 Let the three consecutive integers be x, x +1,and x + 2. Then, the correct choice is H. H 53. In the circle below, is a diameter, AC = 8 inches, and BC = 15 inches. Find the diameter, the radius, and the circumference of the circle. So, the three integers are 17, 16, and 15 and the least of these is 17. the correct choice is C. C In, FL = 24, HJ = 48, and. Find each measure. Use the Pythagorean Theorem to find the hypotenuse of the triangle which is also the diameter of the circle. The radius is half the diameter. So, the radius of the circle is about 8.5 in. The circumference C of a circle of a circle of diameter d is given by the circumference of the circle is 55. FG If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, bisects 48 d = 17 in., r = 8.5 in., or about 53.4 in. esolutions Manual - Powered by Cognero Page 17

18 NJ If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, bisects 65 If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, bisects 59. Find x. The sum of the measures of the central angles of a circle with no interior points in common is 360. So, If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, bisects 60. The sum of the measures of the central angles of a circle with no interior points in common is 360. So, esolutions Manual - Powered by Cognero Page 18

19 61. The sum of the measures of the central angles of a circle with no interior points in common is 360. So, 62. PHOTOGRAPHY In one of the first cameras invented, light entered an opening in the front. An image was reflected in the back of the camera, upside down, forming similar triangles. Suppose the image of the person on the back of the camera is 12 inches, the distance from the opening to the person is 7 feet, and the camera itself is 15 inches long. How tall is the person being photographed? 144 The height of the person being photographed is the length of the base of the larger triangle. The two triangles are similar. So, their corresponding sides will be proportional. One foot is equivalent to 12 in. Then, the height of the larger triangle is 7(12) = 84 in. Let x be the length of the base of the larger triangle. the person being photographed is 5.6 ft tall. 5.6 ft esolutions Manual - Powered by Cognero Page 19

20 ALGEBRA Suppose B is the midpoint of. Use the given information to find the missing measure. 63. AB = 4x 5, BC = x, AC =? Since B is the midpoint of 66. AB = 10s + 2, AC = 40, s =? Since B is the midpoint of AB = 6y 14, BC = 10 2y, AC =? Since B is the midpoint of BC = 6 4m, AC = 8, m =? Since B is the midpoint of esolutions Manual - Powered by Cognero Page 20

### 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

### 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

### 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. 3. PROOF Write a two-column proof to prove that if ABCD is a rhombus with diagonal. 1. If, find. A rhombus is a parallelogram with all

### 11-4 Areas of Regular Polygons and Composite Figures

1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,

### 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

### 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

### Test to see if ΔFEG is a right triangle.

1. Copy the figure shown, and draw the common tangents. If no common tangent exists, state no common tangent. Every tangent drawn to the small circle will intersect the larger circle in two points. Every

### 8-1 Geometric Mean. or Find the geometric mean between each pair of numbers and 20. similar triangles in the figure.

8-1 Geometric Mean or 24.5 Find the geometric mean between each pair of numbers. 1. 5 and 20 4. Write a similarity statement identifying the three similar triangles in the figure. numbers a and b is given

1. Determine whether each quadrilateral is a Justify your answer. 3. KITES Charmaine is building the kite shown below. She wants to be sure that the string around her frame forms a parallelogram before

### Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles

Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles IMPORTANT TERMS AND DEFINITIONS parallelogram rectangle square rhombus A quadrilateral is a polygon that has four sides. A parallelogram is

### Objectives. Cabri Jr. Tools

Activity 24 Angle Bisectors and Medians of Quadrilaterals Objectives To investigate the properties of quadrilaterals formed by angle bisectors of a given quadrilateral To investigate the properties of

### 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

### 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

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### 11-1 Areas of Parallelograms and Triangles. Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary.

Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary. 2. 1. Use the Pythagorean Theorem to find the height h, of the parallelogram. Each pair of opposite

### 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

### 10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: ANSWER: 2 SOLUTION: ANSWER:

Find x. Assume that segments that appear to be tangent are tangent. 1. 3. 2 5 2. 4. 6 13 esolutions Manual - Powered by Cognero Page 1 5. SCIENCE A piece of broken pottery found at an archaeological site

### 1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms.

Quadrilaterals - Answers 1. A 2. C 3. A 4. C 5. C 6. B 7. B 8. B 9. B 10. C 11. D 12. B 13. A 14. C 15. D Quadrilaterals - Explanations 1. An isosceles trapezoid does not have perpendicular diagonals,

### 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

### 11-2 Areas of Trapezoids, Rhombi, and Kites. Find the area of each trapezoid, rhombus, or kite. 1. SOLUTION: 2. SOLUTION: 3.

Find the area of each trapezoid, rhombus, or kite. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. OPEN ENDED Suki is doing fashion design at 4-H Club. Her first project is to make a simple A-line

### 10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: 2. SOLUTION: 3.

Find x. Assume that segments that appear to be tangent are tangent. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. 5. SCIENCE A piece of broken pottery found at an archaeological site is shown.

### Each pair of opposite sides of a parallelogram is congruent to each other.

Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary. 1. Use the Pythagorean Theorem to find the height h, of the parallelogram. 2. Each pair of opposite

### 10.1: Areas of Parallelograms and Triangles

10.1: Areas of Parallelograms and Triangles Important Vocabulary: By the end of this lesson, you should be able to define these terms: Base of a Parallelogram, Altitude of a Parallelogram, Height of a

### 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

# 30-60 right triangle, 441-442, 684 A Absolute value, 59 Acute angle, 77, 669 Acute triangle, 178 Addition Property of Equality, 86 Addition Property of Inequality, 258 Adjacent angle, 109, 669 Adjacent

### 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

### 1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area?

1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area? (a) 20 ft x 19 ft (b) 21 ft x 18 ft (c) 22 ft x 17 ft 2. Which conditional

### Sum of the interior angles of a n-sided Polygon = (n-2) 180

5.1 Interior angles of a polygon Sides 3 4 5 6 n Number of Triangles 1 Sum of interiorangles 180 Sum of the interior angles of a n-sided Polygon = (n-2) 180 What you need to know: How to use the formula

### **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle.

Geometry Week 7 Sec 4.2 to 4.5 section 4.2 **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle. Protractor Postulate:

### CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

### 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

### Geometry. Unit 6. Quadrilaterals. Unit 6

Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections

### 8-2 The Pythagorean Theorem and Its Converse. Find x.

Find x. 1. of the hypotenuse. The length of the hypotenuse is 13 and the lengths of the legs are 5 and x. 2. of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 12.

### 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,

### Unit 1: Similarity, Congruence, and Proofs

Unit 1: Similarity, Congruence, and Proofs This unit introduces the concepts of similarity and congruence. The definition of similarity is explored through dilation transformations. The concept of scale

### Geometry: Euclidean. Through a given external point there is at most one line parallel to a

Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,

### Geometry Credit Recovery

Geometry Credit Recovery COURSE DESCRIPTION: This is a comprehensive course featuring geometric terms and processes, logic, and problem solving. Topics include parallel line and planes, congruent triangles,

### (a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

### of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 459-2058 Mobile: (949) 510-8153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:

### Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

### 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.

### 4.1 Euclidean Parallelism, Existence of Rectangles

Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles Definition 4.1 Two distinct

### 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:

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100

Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score 1 25 2 25 3 25 4 25 Subtotal 100 Extra

### Geometry Essential Curriculum

Geometry Essential Curriculum Unit I: Fundamental Concepts and Patterns in Geometry Goal: The student will demonstrate the ability to use the fundamental concepts of geometry including the definitions

### 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

### 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,

### 4-1 Classifying Triangles. ARCHITECTURE Classify each triangle as acute, equiangular, obtuse, or right. 1. Refer to the figure on page 240.

ARCHITECTURE Classify each triangle as acute, equiangular, obtuse, or right. 1. Refer to the figure on page 240. Classify each triangle as acute, equiangular, obtuse, or right. Explain your reasoning.

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### 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

### BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

### 11-4 Areas of Regular Polygons and Composite Figures

1.In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. SOLUTION: Center: point F, radius:,

### are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

For Exercises 1 4, refer to. 1. Name the circle. The center of the circle is N. So, the circle is 2. Identify each. a. a chord b. a diameter c. a radius a. A chord is a segment with endpoints on the circle.

### 6-1 Angles of Polygons

Find the sum of the measures of the interior angles of each convex polygon. 1. decagon A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures.

### Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

### A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3

Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x - 1 D. y = 3x + 1

### 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,

### In the statement, each side is the same "XY = XY". The Reflexive Property justifies this statement.

State the property that justifies each statement. 1. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. There are two parts to the hypotheses. "If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. "The end of the first

### as a fraction and as a decimal to the nearest hundredth.

Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. sin A The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. So, 2. tan C The tangent of an

### 7-1 Ratios and Proportions

1. PETS Out of a survey of 1000 households, 460 had at least one dog or cat as a pet. What is the ratio of pet owners to households? The ratio of per owners to households is 23:50. 2. SPORTS Thirty girls

### Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.

Chapter 11: Areas of Plane Figures (page 422) 11-1: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses

CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning

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,

### Parallelogram Bisectors Geometry Final Part B Problem 7

Parallelogram Bisectors Geometry Final Part B Problem 7 By: Douglas A. Ruby Date: 11/10/2002 Class: Geometry Grades: 11/12 Problem 7: When the bisectors of two consecutive angles of a parallelogram intersect

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### 39 Symmetry of Plane Figures

39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

### 12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. SOLUTION: ANSWER: 1017.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 3. sphere: area of great circle = 36π yd 2 We know that the area of a great circle is r.. Find 1. Now find the surface area.

### 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

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 17 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### 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,

### Solutions Section J: Perimeter and Area

Solutions Section J: Perimeter and Area 1. The 6 by 10 rectangle below has semi-circles attached on each end. 6 10 a) Find the perimeter of (the distance around) the figure above. b) Find the area enclosed

### 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

### An inscribed polygon is a polygon whose vertices are points of a circle. in means in and scribed means written

Geometry Week 6 Sec 3.5 to 4.1 Definitions: section 3.5 A A B D B D C Connect the points in consecutive order with segments. C The square is inscribed in the circle. An inscribed polygon is a polygon whose

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, :15 a.m. SAMPLE RESPONSE SET

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Question 29................... 2 Question 30...................

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### 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,

### 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)

### Name Date Class. Lines and Segments That Intersect Circles. AB and CD are chords. Tangent Circles. Theorem Hypothesis Conclusion

Section. Lines That Intersect Circles Lines and Segments That Intersect Circles A chord is a segment whose endpoints lie on a circle. A secant is a line that intersects a circle at two points. A tangent

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### Lesson 28: Properties of Parallelograms

Student Outcomes Students complete proofs that incorporate properties of parallelograms. Lesson Notes Throughout this module, we have seen the theme of building new facts with the use of established ones.

### 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

### 13-3 Geometric Probability. 1. P(X is on ) SOLUTION: 2. P(X is on ) SOLUTION:

Point X is chosen at random on. Find the probability of each event. 1. P(X is on ) 2. P(X is on ) 3. CARDS In a game of cards, 43 cards are used, including one joker. Four players are each dealt 10 cards

### Math 311 Test III, Spring 2013 (with solutions)

Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam

### 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

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

COURSE OVERVIEW The geometry course is centered on the beliefs that The ability to construct a valid argument is the basis of logical communication, in both mathematics and the real-world. There is a need

### 1-4 Angle Measure. Use the figure at the right. 1. Name the vertex of SOLUTION: 2. Name the sides of SOLUTION: 3. What is another name for SOLUTION:

Use the figure at the right. 6. 1. Name the vertex of U 2. Name the sides of 7. AFD is an obtuse angle. The measure of AFD is 150. 3. What is another name for XYU, UYX 4. What is another name for 1, YXU

### Geometry Sample Problems

Geometry Sample Problems Sample Proofs Below are examples of some typical proofs covered in Jesuit Geometry classes. Shown first are blank proofs that can be used as sample problems, with the solutions

### Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

### 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

### 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

### 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

### 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,

### Identifying Triangles 5.5

Identifying Triangles 5.5 Name Date Directions: Identify the name of each triangle below. If the triangle has more than one name, use all names. 1. 5. 2. 6. 3. 7. 4. 8. 47 Answer Key Pages 19 and 20 Name