GEOMETRY R Unit 2: Angles and Parallel Lines Day Classwork Homework Friday 9/16 Unit 1 Test Monday 9/19 Tuesday 9/20 Angle Relationships HW 2.1 Angle Relationships with Transversals HW 2.2 Wednesday 9/21 Proving Lines are Parallel and Constructing Parallel Lines HW 2.3 Thursday 9/22 Interior and Exterior Angle Theorems HW 2.4 Friday 9/23 More Unknown Angle Problems with Justifications Unit 2 Quiz 1 HW 2.5 Monday 9/26 Tuesday 9/27 Unknown Angle Proofs HW 2.6 Proofs with Auxiliary Lines HW 2.7 Wednesday 9/28 Thursday 10/29 Friday 9/30 Review Unit 2 Quiz 2 Review Unit 2 Test Review Packet Study!
Angles [1] Vocabulary Term Example Adjacent Angles - Two angles in the same plane that share a common vertex and a common ray, but share no common interior points. Linear Pair - Two adjacent angles with non-common sides that are opposite rays. Complementary Angles Two angles with measures that have a sum of 90 degrees. Supplementary Angles Two angles with measures that have a sum of 180 degrees. Angle Addition Postulate: If two adjacent angles combine to form a whole angle, the measure of the whole angle equals the sum of the measures of the two adjacent angles. Supplement Theorem Two angles that form a linear pair are supplementary angles. Examples: 1. Find the measure of a. 2. Find the measure of f. A B
3. Find the measure of c. 4. Find the value of x. 5. Two angles are complementary. One angle is six less than twice the other angle. Find both angles. Vertical Angles - Two non-adjacent angles formed by two intersecting lines. Vertical Angles Theorem - Vertical angles are congruent. Examples: 2 1. If m 1 x 4x and m 2 x 36, find m 3. 3 1 2
2. Find the value of x and y. Find the values of the missing variables. 3. 4. 5. 2x + 1 3x + 10 y 3 y 8 9 x + 10
6. 3x + 6 x - 10 y x - 11 7. 8.
ANGLES AND PARALLEL LINES [2] Vocabulary Transversal - A line that intersects two or more coplanar lines. Examples Corresponding Angles - Non-adjacent angles that lie on the same side of the transversal, one exterior and one interior. Alternate Interior Angles - Non-adjacent interior angles that lie on opposite sides of a transversal. Alternate Exterior Angles - Non-adjacent exterior angles that lie on opposite sides of a transversal. Theorems Words Diagram Corresponding Angles Postulate If 2 parallel lines are cut by a transversal, then corresponding angles are congruent. Alternate Interior Angles Theorem If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. Consecutive Interior Angles Theorem If 2 parallel lines are cut by a transversal, then consecutive interior angles are supplementary Alternate Exterior Angles Theorem If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Finding Values of Variables The special relationships between the angles formed by two parallel lines and a transversal can be used to find unknown values. 1. If m 4 = 2x 17 and m 1 = 85, find x. 2. Find y if m 3 = 4y + 30 and m 7 = 7y + 6. 3. If m 1 = 4x + 7 and m 5 = 5x 13, find x. 4. Find y if m 7 = 68 and m 6 = 3y 2. An is sometimes useful when solving for unknown angles. In this figure, we can use the auxiliary line to find the measure of W. What is the measure of W? Exercises Find the unknown angles that are labeled. Justify each step. 1.
2. 3. 4. 5.
6. 7. 8. 9.
Find the unknown (labeled) angles. Give reasons for your solutions. 1. 2. 3. 4.
PROVING LINES ARE PARALLEL [3] Words Diagram Converse of the Corresponding Angles Postulate If 2 lines are cut by a transversal such that corresponding angles are congruent, then the lines are parallel. Converse of the Alternate Interior Angles Theorem If 2 lines are cut by a transversal such that alternate interior angles are congruent, then the lines are parallel. Converse of the Consecutive Interior Angles Theorem If 2 lines are cut by a transversal such that consecutive interior angles are supplementary, then the lines are parallel. Converse of the Alternate Exterior Angles Theorem If 2 lines are cut by a transversal such that alternate exterior angles are congruent, then the lines are parallel. Converse of the Perpendicular Transversal Theorem If 2 lines are perpendicular to the same line, then they are parallel to each other. Examples 1. Given the following information, is it possible to prove that any of the lines shown are parallel? Justify your reasoning. a. 2 8 c. 1 15 b. 3 11 d. 8 6 e. 12 14
2. Find m MRQ so that a b. Show your work. 3. Use the converse of the corresponding angles postulate to construct a line parallel to AB, passing through point C. C A B 4. Use the converse of the alternate interior angles theorem to construct a line parallel to AB, passing through point C. C A B
ANGLES OF TRIANGLES [4 & 5] The sum of the interior angles of a triangle is 180 degrees. The acute angles of a right triangle are complementary. Each angle of an equilateral triangle measures 60 degrees. The sum of the interior angles of a quadrilateral is 360 degrees. Examples 1. The vertex angle of an isosceles triangle exceeds the measure of a base angle by 30. Find the measure of the vertex angle. 2. Classify triangle ABC based on its sides and angles if m A = (9x), m B=(3x-6), and m C = (11x + 2). Theorem Words Example The Exterior Angle Theorem An exterior angle of a triangle equals the sum of its two remote interior angles Proof of Exterior Angle Theorem:
Examples 1. Find x: 2 x x 12 5x 2. Find m a and m b in the figure to the right. Justify your results. In each figure, determine the measures of the unknown (labeled) angles. Give reasons for each step of your calculations. 3. 4. 5.
6. 7. 8.
9. 10. 11.
12. 13. 14.
UNKNOWN ANGLE PROOFS [6] Reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction is called inductive reasoning. These conclusions lack logical certainty. The system of reasoning that uses facts, rules, definitions, or properties to reach logical conclusions is called deductive reasoning. 1. Angles that form a linear pair are supplementary. Use this fact to prove that vertical angles are equal in measure. 2. Given the diagram at the right, prove that m w + m x + m z = 180. 3. Given the diagram at the right, prove that m w = m y + m z.
4. In the diagram at the right, prove that m y + m z = m w + m x. 5. In the figure at the right, AB CD and BC. DE Prove that ABC CDE. 6. In the figure at the right, prove that the sum of the angles marked by arrows is 900.
7. In the figure at the right, prove that DC EF. Z 8. In the labeled figure on the right, prove that m n.
9. In the diagram on the right, prove that the sum of the angles marked by arrows is 360. 10. In the diagram at the right, prove that m a + m d m b = 180. a d b c
MORE UNKNOWN ANGLE PROBLEMS [7] A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true. A theorem is a statement or conjecture that can be proven true by undefined terms, definitions, and postulates. 1. In the figure on the right, AB DE and BC EF. Prove that B E. (Hint: Extend BC and ) ED 2. Given: AB CD Prove: m z = m x + m y 3. In the figure, AB CD and BC DE. Prove that ABC CDE.
4. In the figure, AB CD and BC DE. Prove that m B + m D = 180. 5. In the figure, prove that m ADC = m A + m B + m C. 6. Use the theorems involving parallel lines discussed to prove that the three angles of a triangle sum to 180. For this proof, you will need to draw an auxiliary line, parallel to one of the triangle s sides and passing through the vertex opposite that side. Add any necessary labels and write out your proof.
7. A theorem states that in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other of the two parallel lines. Prove this theorem. (a) Construct and label an appropriate figure, (b) state the given information and the theorem to be proved, then (c) list the necessary steps to demonstrate the proof. 8. In the figure, AB DE and BC EF. Prove that ABC DEF. 9. In the figure, AB CD. Prove that m AEC = m A + m C.
MORE EUCLIDEAN PROOFS 1. Given: 1 2 Prove: line l line m 2. Given:WX YZ, 2 3 Prove: WY XZ 3. Given: 1 6, 3 6 Prove: DC AB
D 4. Given: AE bisects DAC 2 B Prove: AE BC A 1 2 E B 3 C 5. Given: CB bisects ACD ; 1 3 Prove: AB CD A 3 4 B C 1 2 D 6. Given: 1 2 Prove: 3 4 2 3 k 1 4 m