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Warm-Up

Warm-Up. Refer to the figure to identify the special name for the given pair of angles. 1) <1 and <8 2) <4 and <5 3) <1 and <5 4) <4 and <6 Draw a figure to illustrate each of the following. 5) Two perpendicular planes 6) Two parallel planes and a skew line. 2. 1. 4. 3. 6. 5. 8. 7.

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Warm-Up

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  1. Warm-Up Refer to the figure to identify the special name for the given pair of angles. 1) <1 and <8 2) <4 and <5 3) <1 and <5 4) <4 and <6 Draw a figure to illustrate each of the following. 5) Two perpendicular planes 6) Two parallel planes and a skew line 2 1 4 3 6 5 8 7

  2. Warm-Up Refer to the figure to identify the special name for the given pair of angles. 1) <1 and <8 Alternate Exterior Angels 2) <4 and <5 Alternate Interior Angles 3) <1 and <5 Corresponding Angles 4) <4 and <6 Consecutive Interior Angles 2 1 4 3 6 5 8 7

  3. Warm-Up Draw a figure to illustrate each of the following. 5) Two perpendicular planes 6) Two parallel planes and a skew line

  4. Chapter 3Section 2 Angles and Parallel Lines

  5. Vocabulary Corresponding Angles Postulate Theorem - If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent. <5 = <1, <7 = <3, <6 = <2, and <8 = <4 Alternate Interior Angles Theorem- If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. <7 = <2 and <8 = <1 Consecutive Interior Angles Theorem – If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. <7 + <1 = 180 and <8 + <2 = 180 t 5 6 l 7 8 1 2 m 3 4

  6. Vocabulary Cont. t Alternate Exterior Angles Theorem- If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. <5 = <4 and <6 = <3 Perpendicular Transversal Theorem- In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. 5 6 l 7 8 1 2 m 3 4

  7. Example 1: Use the figure at the right. • a) Find the m<3 if the m<6 = 116. • <3 and <6 are alternate interior angles so we know they are congruent. • m<6 = 116 = m<3 • b) Find the m<1 if the m<6 = 110. • <6 and <2 are corresponding angles so we know they are congruent. • m<6 = 110 = m<2 • Now <1 and <2 are supplementary so we know m<1 + m<2 = 180. • m<1 + m<2 = 180. • m<1 + 110 = 180 • m<1 = 70 1 2 3 4 5 6 7 8

  8. Example 2: Use the figure at the right. In the figure, MA HT and NG EL. Find the values of x, y, and z. N E Find x: <MAT and <ATC are alternate interior angles so they are congruent. m<MAT = m<ATC 2x = 72 x = 36 Find y: <ATC and <OHG are alternate exterior angles so they are congruent. m<ATC = m<OHC 72 = 5y + 2 70 = 5y 14 = y M A 4z 2x O H T C 72 (5y + 2) G L Find z: <HMA and <MAT are consecutive interior angles so they are supplementary. m<HMA + m<MAT = 180 4z + 2x = 180 4z + 2(36) = 180 4z + 72 = 180 4z = 108 z = 27

  9. Example 3: In the figure, l is parallel to m and c is parallel to d. Find the values of x, y, and z. Find y: 98 and (3y + 8) are alternate exterior angles so they are congruent. 98 = 3y + 8 90 = 3y 30 = y Find z: 14z and 98 are alternate interior angles so they are congruent. 98 = 14z 7 = z 14z (2x + 5) 98 (3y + 8) Find x: 14z and (2x + 5) are consecutive interior angles so they are supplementary. 14z + 2x + 5 = 180 14(7) + 2x + 5 = 180 98 + 2x + 5 = 180 103 + 2x = 180 2x = 77 x = 38.5

  10. Example 4: Find the values of x and y. Find x: (4x – 5) and (3x + 11) are corresponding angles so they are congruent. 4x – 5 = 3x = 11 x – 5 = 11 x = 16 Find y: (3x + 11) and (3y + 1) are consecutive interior angles so they are supplementary. 3x + 11 + 3y + 1 = 180 3(16) + 11 + 3y + 1 = 180 48 + 11 + 3y + 1 = 180 60 + 3y = 180 3y = 120 y = 40 (4x – 5) (3y + 1) (3x + 11)

  11. Example 5: Find the values of x and y. Find y: (13y – 10) and (6y) are consecutive interior angles so they are supplementary. 13y – 10 + 6y = 180 19y – 10 = 180 19y = 190 y = 10 Find x: (6y) and (9x + 12) are supplementary. 9x + 12 + 6y = 180 9x + 12 + 6(10) = 180 9x + 12 + 60 = 180 9x + 72 = 180 9x = 108 x = 12 (6y) (9x + 12) (13y - 10)

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