21.1 Objective Prove and use theorems about the angles formed by parallel lines and a transversal.

21.1 Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles postulate is given as a postulate, it can be used to prove the next three theorems.

21.1

21.1 Find mQRS. x = 118 Corr. s Post. mQRS + x = 180° *Def. of Linear Pair* mQRS = 180° – x Subtract x from both sides. = 180° – 118° Substitute 118° for x. = 62°

Example 1: Using the Corresponding Angles Postulate 21.1 Example 1: Using the Corresponding Angles Postulate Find each angle measure. A. mECF x = 70 Corr. s Post. mECF = 70° B. mDCE 5x = 4x + 22 Corr. s Post. x = 22 Subtract 4x from both sides. mDCE = 5x = 5(22) Substitute 22 for x. = 110°

21.1 Example 3 Find each angle measure. A. mEDG mEDG = 75° Alt. Ext. s are Congruent. B. mBDG x – 30° = 75° Alt. Ext. s are congruent. x = 105 Add 30 to both sides. mBDG = 105°

D B) m( DEF) E F x – 45 = 30 x = 75 m( DEF) = (x+30)0 = (75+30)0 corresponding angles x – 45 = 30 subtract x from both sides x = 75 add 45 to both sides m( DEF) = (x+30)0 = (75+30)0 = 1050 CONFIDENTIAL

R T B) m( TUS) S U 36x = 180 x = 5 m( TUS) = 23(5)0 = 1150 13x0 23x0 Same-side interior angles theorem 36x = 180 Combine like terms x = 5 divide both sides by 36 m( TUS) = 23(5)0 Substitute 5 for x = 1150 CONFIDENTIAL

Check It Out! Example 2 Find mABD. 2x + 10° = 3x – 15° Alt. Int. s Thm. Subtract 2x and add 15 to both sides. x = 25 mABD = 2(25) + 10 = 60° Substitute 25 for x.

21.1 Example 4 Find x and y in the diagram. By the Alternate Interior Angles Theorem, (5x + 4y)° = 55°. By the Corresponding Angles Postulate, (5x + 5y)° = 60°. 5x + 5y = 60 –(5x + 4y = 55) y = 5 Subtract the first equation from the second equation. Substitute 5 for y in 5x + 5y = 60. Simplify and solve for x. 5x + 5(5) = 60 x = 7, y = 5

Check It Out! Example 3 Find the measures of the acute angles in the diagram. By the Alternate Exterior Angles Theorem, (25x + 5y)° = 125°. By the Corresponding Angles Postulate, (25x + 4y)° = 120°. An acute angle will be 180° – 125°, or 55°. The other acute angle will be 180° – 120°, or 60°.