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Warm Up

Warm Up. Lesson Presentation. Lesson Quiz. Warm Up Identify each of the following. 1. points that lie in the same plane 2. two angles whose sum is 180° 3. the intersection of two distinct intersecting lines 4. a pair of adjacent angles whose non-common sides are opposite rays.

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Warm Up

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  1. Warm Up Lesson Presentation Lesson Quiz

  2. Warm Up • Identify each of the following. • 1.points that lie in the same plane • 2. two angles whose sum is 180° • 3. the intersection of two distinct intersecting lines • 4. a pair of adjacent angles whose non-common sides are opposite rays coplanar points supplementary angles point linear pair

  3. Objective Prove and use theorems about the angles formed by parallel lines and a transversal. Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal.

  4. Vocabulary parallel lines perpendicular lines skew lines parallel planes transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles

  5. Classifying Pairs of Angles Give an example of each angle pair. A. corresponding angles 1 and 5 B. alternate interior angles 3 and 5 C. alternate exterior angles 1 and 7 D. same-side interior angles 3 and 6

  6. TEACH! Classifying Pairs of Angles Give an example of each angle pair. A. corresponding angles 1 and 3 B. alternate interior angles 2 and 7 C. alternate exterior angles 1 and 8 D. same-side interior angles 2 and 3

  7. Using the Corresponding Angles Postulate Find each angle measure. A. mECF mECF= 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°

  8. TEACH! Using the Corresponding Angles Postulate 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°

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

  10. 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.

  11. 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°

  12. TEACH! Finding Angle Measures Find mABD. 2x + 10° = 3x – 15° Alt. Int. s Thm. Subtract 2x and add 15 to both sides. x = 25 Substitute 25 for x. mABD = (2x + 10) = 2(25) + 10 = 60°

  13. Identifying Angle Pairs and Transversals Identify the transversal and classify each angle pair. A. 1 and 3 transversal l corr. s B. 2 and 6 transversal n alt. int s C. 4 and 6 transversal m alt. ext s

  14. TEACH! Identifying Angle Pairs and Transversals Identify the transversal and classify the angle pair 2 and 5 in the diagram. transversal n same-side int. s.

  15. Piano Strings Angles Modeling 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

  16. The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Pressing a key on the piano's keyboard causes a hammer to strike steel strings. The hammers rebound, allowing the strings to continue vibrating at their resonant frequency.

  17. TEACH! Example 4 > Piano Strings Find the measures of the acute angles in the diagram of Bass & Treble strings.. 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°.

  18. Lesson Quiz State the theorem or postulate that is related to the measures of the angles in each pair. Then find the unknown angle measures. 1. m1 = 120°, m2 = (60x)° 2. m2 = (75x – 30)°, m3 = (30x + 60)° Alt. Ext. s Thm.; m2 = 120° Corr. s Post.; m2 = 120°, m3 = 120° 3. m3 = (50x + 20)°, m4= (100x – 80)° 4. m3 = (45x + 30)°, m5 = (25x + 10)° Alt. Int. s Thm.; m3 = 120°, m4 =120° Same-Side Int. s Thm.; m3 = 120°, m5 =60°

  19. Helpful Hint To determine which line is the transversal for a given angle pair, locate the line that connects the vertices.

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