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Transcript
2.5 Proving Angles Congruent SWBAT… • Identify angles • To prove and apply theorems about angles Definitions Vertical angles: two angles whose sides form two pairs of opposite rays Adjacent angles: two coplanar angels with a common side, a common vertex, and no common interior points. Complementary angles: two angles whose measures have sum 90 Supplementary angels: two angles whose measures have sum of 180 Id angles pairs • In the diagram identity pairs of numbered angles that are related as follows. a. Complementary b. Supplementary c. Vertical 2 1 5 3 4 FYI • Things we can conclude from a diagram • Adjacent angles • Adjacent supplementary angles • Vertical angles • Things we can NOT conclude • Angels or segments are congruent • An angle is a right angle • Lines are parallel or perpendicular Supplemental Theorem • If two angles form a linear pair, then they are supplementary Complement Theorem • If the noncommon sides of two adjacent angels form a right angle, then the angles are complimentary angles. Theorem 2-2 • If two angles are supplements of the same angle (or of congruent angles), then the two angels are congruent. • If m<1 + m<2 = 180 and m<2 + m<3 = 180 then m<1 = m<3 Example 3: Proof of Theorem 2.2 • Given <1 and <2 are supplementary <2 and <3 are supplementary • Prove: <1 = <3 • 1. 2. Statements <1 and <2 are supplementary; <2 and <3 are supplementary m<1 + m<2 = 180 m<2 + m<3 = 180 3. 4. m<1 = m<3 5. <1 congruent <3 1. Given 2. 3. Substitution 4. 5. Theorem 2-3 • Angles complementary to the same angle or to congruent angles are congruent