Verifying Angle Relations

Write the reason for each statement. 1) If AB is congruent to CD, then AB = CD Definition of congruent segments 2) If GH = JK, then GH + LM = JK + LM Addition Property of Equality 3) R, D, and S are collinear. RS = RD + DS Segment Addition Postulate 4) If WX = YZ, then WX – UV = YZ – UV Subtraction Property of Equality

Theorem 2-2 (Supplement Theorem)- If two angles form a linear pair, then they are supplementary angles. Theorem 2-3- Congruence of angles is reflexive, symmetric, and transitive. Theorem 2-4- Angles supplementary to the same angle or to congruent angles are congruent. Theorem 2-5- Angles complementary to the same angle or to congruent angles are congruent. Theorem 2-6- All right angles are congruent. Theorem 2-7- Vertical angles are congruent. Theorem 2-8- Perpendicular lines intersect to form four right angles.

Example 1) Given: <1 is congruent to <2 and <2 is congruent to <3 Prove: <1 is congruent <3 123 StatementsReasons <1 is congruent to <2; <2 is congruent to <3 m<1 =_____ m<2 =_____ m<1 = m<3 <1 is congruent <3 Given Definition of congruent angles Transitive Property of Equality Definition of congruent angles m<2 m<3

Example 2) Given: <1 and <3 are supplementary; <2 and <3 are supplementary Prove: <1 is congruent to <2 132 StatementsReasons <1 and <3 are supplementary; <2 and <3 are supplementary m<1 + m<3 = 180 ________ + ________ = 180 m<1 + _______ = _______ + m<3 m<1 = m<3 Given Definition of supplementary Subtraction Property of Equality m<2m<3 m<2 <1 is congruent to <2 Substitution Property of Equality Definition of congruent angles

Example 3) Given: <1 and <3 are vertical angles Prove: <1 is congruent to < StatementsReasons <1 and <3 are vertical angles <1 and ________ form a linear pair. <2 and ________ form a linear pair _____ and _____ are supplementary. <1 is congruent <3 Given Definition of linear pairs If 2 angles form a linear pair, then they are supplementary. Angles supplementary to the same angle are congruent. m<2 m<3 4 m<2 m<1

Example 4) The angle formed by a ladder and the ground measure 30 degrees. Find the measure x of the larger angle formed by the ladder and the ground. 30 degreesx degrees Since the two angles form a linear pair, we know they are also supplementary. Since they are supplementary, we know they add up to x = 180 x = 150

Example 5) If m<D = 30 and <D is congruent to <E and <E is congruent to <F, find m<F. By the transitive property of equality, since m<D = 30, we know m<E = 30, m<F = 30. Example 6) If <7 and <8 are vertical angles and m<7 = 3x + 6 and m<8 = x + 26, find m<7 and m<8. m<7 = m<8 3x + 6 = x x + 6 = 26 2x = 20 x = 10 Now plug 10 in to either equation to find the measures of the angles. m<8 = x + 26 m<8 = m<8 = 36 Vertical angles are congruent Substitution Property of Equality Subtraction Property of Equality Division Property of Equality

Example 7) If <A and <C are vertical angles and m<A = 3x – 2 and m<C = 2x + 4, find m<A and m<C. m<A = m<C 3x - 2= 2x + 4 3x = 2x + 6 x = 6 Now plug 6 in to either equation to find the measures of the angles. m<A = 3x - 2 m<A = 3(6) - 2 m<A = 18 – 2 m<A = 16 Vertical angles are congruent Substitution Property of Equality Addition Property of Equality Subtraction Property of Equality