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Proving and Using Theorems on Parallel Lines and Transversals

Learn how to prove and use theorems about the angles formed by parallel lines and a transversal. Practice finding angle measures using postulates and theorems.

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Proving and Using Theorems on Parallel Lines and Transversals

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  1. Warm Up Identify each angle pair. 1.1 and 3 2. 3 and 6 3. 4 and 5 4. 6 and 7 corr. s alt. int. s alt. ext. s same-side int s

  2. Learning Target Prove and use theorems about the angles formed by parallel lines and a transversal.

  3. POSTULATE

  4. 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°

  5. Check It Out! Example 1 Find mQRS. x = 118 Corr. s Post. mQRS + x = 180° Def. of Linear Pair Subtract x from both sides. mQRS = 180° – x = 180° – 118° Substitute 118° for x. = 62°

  6. Helpful Hint If a transversal is perpendicular to two parallel lines, all eight angles are congruent.

  7. 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.

  8. Theorem 3-2-2: Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Theorem 3-2-3: Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem 3-2-4: Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary.

  9. Example 2: Finding Angle Measures Find each angle measure. A. mEDG mEDG = 75° Alt. Ext. s Thm. B. mBDG x – 30° = 75° Alt. Ext. s Thm. x = 105 Add 30 to both sides. mBDG = 105°

  10. 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.

  11. Example 3: Music Application 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

  12. 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°.

  13. Homework: Page 158 – 159, #6 – 23.

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