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Name __________________________________ Date ___________________ LESSON 2.7 Study Guide For use with pages 122–131 GOAL Use properties of special pairs of angles. Vocabulary Theorem 2.3 Right Angles Congruence Theorem: All right angles are congruent. Theorem 2.4 Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. Theorem 2.5 Congruent Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then they are congruent. Postulate 12 Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. Theorem 2.6 Vertical Angles Congruence Theorem: Vertical angles are congruent. EXAMPLE 1 Find angle measures Complete the statement given that mAGF = 90. a. mCGD __?__ b. If mBGF =113, then mDGE = __?__. Solution a. Because CGD and AGF are vertical angles, CGD ≅ AGF. By the definition of congruent angles, mCGD = mAGF. So, mCGD = 90. b. By the Angle Addition Postulate, mBGF = mAGF + mAGB. Substitute to get 113° = 90° + mAGB. By the Subtraction Property of Equality, mAGB = 23°. Because DGE and AGB are vertical angles, DGE ≅ AGB. By the definition of congruent angles, mDGE = mAGB. So, mDGE = 23°. Exercises for Example 1 Complete the statement given that mBHD = mCHE = 90. 1. 2. 3. 4. 5. mAHG =__?__ mCHA = __?__ If mCHD = 31, then mEHF = __?__. If mBHG = 125, then mCHF = __?__. If mEHF = 38, then mBHC = __?__. Name __________________________________ Date ___________________ LESSON 2.7 Study Guide continued For use with pages 122–131 EXAMPLE 2 Find angle measures If mBGD = 90 and mCGD = 26, find ml, m2, and m3. Solution BGC and CGD are complementary. So, m1 = 90 26 = 64°. AGB and BGD are supplementary. So, m2 = 180° 90° = 90°. AGF and CGD are vertical angles. So, m3 = 26°. Exercises for Example 2 In Exercises 6 and 7, refer to Example 2. 6. Find mFGE. 7. Find mDGE. EXAMPLE 3 Use algebra Solve for x in the diagram. Solution Because AEB and BEC form a linear pair, the sum of their measures is 180. So, you can solve for x as follows: (2x + 3) + 25 = 180 2x + 28 = 180 2x = 152 x = 76 Definition of supplementary angles. Combine like terms. Subtract 28 from both sides, Divide each side by 2. Exercises for Example 3 Solve for x in the diagram. 8. 9.