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Chapter 2 Section 6

Chapter 2 Section 6. Verifying Angle Relations. Warm-Up. 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

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Chapter 2 Section 6

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  1. Chapter 2Section 6 Verifying Angle Relations

  2. Warm-Up • 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

  3. Vocabulary • 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.

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

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

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

  7. 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. x degrees 30 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 180. 30 + x = 180 x = 150

  8. 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 + 26 2x + 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 = 10 + 26 m<8 = 36 Vertical angles are congruent Substitution Property of Equality Subtraction Property of Equality Subtraction Property of Equality Division Property of Equality

  9. 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 6in 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

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