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2.8 Proving Angle Relationships

2.8 Proving Angle Relationships. Objectives. Write proofs involving supplementary and complementary angles Write proofs involving congruent and right angles. Postulates. Postulates 2.10 (Protractor Rule) All angles have measures between 0° and 180°.

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2.8 Proving Angle Relationships

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  1. 2.8 Proving Angle Relationships

  2. Objectives • Write proofs involving supplementary and complementary angles • Write proofs involving congruent and right angles

  3. Postulates Postulates 2.10 (Protractor Rule) All angles have measures between 0° and 180°. Postulate 2.11 (Angle Addition Postulate) R is in the interior of PQS iffmPQR + mRQS = mPQS. P R Q S

  4. Postulates and Theorems Copy all of the Theorems from Section 2.8. Theorems 2.3 thru 2.13

  5. TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? If the second hand stops where the angle is bisected, then the angle between the minute and second hands is one-half the measure of the angle formed by the hour and minute hands, or . Example 1:

  6. Example 1: By the Angle Addition Postulate, the sum of the two angles is 120, so the angle between the second and hour hands is also 60º. Answer: They are both 60º by the definition of angle bisector and the Angle Addition Postulate.

  7. QUILTING The diagram below shows one square for a particular quilt pattern. If and is a right angle, find Your Turn: Answer: 50

  8. form a linear pair and If and find Example 2: Supplement Theorem Subtraction Property Answer: 14

  9. If are complementary angles and . and find Your Turn: Answer: 28

  10. In the figure, form a linear pair, and Prove that are congruent. and and Given: form a linear pair. Prove: Example 3:

  11. Proof: Statements Reasons 1. 1. Given 2. 2. Linear pairs are supplementary. 3. 3. Definition of supplementary angles 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Definition of congruent angles Example 3:  1 &  4 linear pair;

  12. Your Turn: In the figure, NYR and RYA form a linear pair,AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXYare congruent.

  13. Proof: Statements Reasons 1. 1. Given linear pairs. 2. 2.If two s form a linear pair, then they are suppl. s. 3. 3.Given 4. 4. Your Turn:

  14. If 1 and 2are vertical angles and m1 andm2 find m1 and m2. 1 2 Vertical Angles Theorem m1 m2 Definition of congruent angles Example 4: Substitution Add 2d to each side. Add 32 to each side. Divide each side by 3.

  15. Example 4: Answer: m1 = 37 and m2 = 37

  16. If and are vertical angles and and find and Your Turn: Answer: mA= 52; mZ= 52

  17. Assignment • Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32 • Pre-AP Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32 , 34, 36, 38

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