2.8 Proving Angle Relationships

Objectives Write proofs involving supplementary and complementary angles Write proofs involving congruent and right angles

Postulates and Theorems Copy all of the postulates and Theorems from Section 2.8. Postulates 2.10 and 2.11 Theorems 2.3 thru 2.13

Example 1: 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: 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.

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

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

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

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

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

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 AXY are congruent.

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

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

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

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

Assignment Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32 Pre-AP Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32