Parallel Lines and Transversals. Students will learn how to find the measures of alternate interior angles and same-side interior angles.
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1 Parallel Lines and Transversals Students will learn how to find the measures of alternate interior angles and same-side interior angles.
2 Parallel Lines and Transversals When a pair of parallel lines are cut by a transversal, the angle pairs formed are either congruent or supplementary. These are congruent: Corresponding angles Alternate interior angles Alternate exterior angles These are supplementary: Same-side interior angles Which ones are which? FHS Unit C 2
3 Vertical Angles Theorem We need to remember the Vertical Angles Theorem from Chapter 1: If two angles are vertical angles then they are congruent. If A and B are vertical angles, then A B. A B FHS Unit C 3
4 Corresponding Angles Postulate If two parallel lines are intersected by a transversal, then the pairs of corresponding angles are congruent. If k l, then 1 5, 2 6, 3 7, and 4 8 k 1 3 l FHS Unit C 4
5 Alternate Interior Angles Theorem If two parallel lines are intersected by a transversal, then the pairs of alternate interior angles are congruent. k 1 3 l If k l, then 3 6 and 4 5. FHS Unit C 5
6 Alternate Exterior Angles Theorem If two parallel lines are intersected by a transversal, then the pairs of alternate exterior angles are congruent. If k l, then 1 8 and 2 7. k 1 3 l FHS Unit C 6
7 Same-side Interior Angles Theorem If two parallel lines are intersected by a transversal, then the pairs of same-side interior angles are supplementary. If k l, then 2 m 3 + m 5 = and k 3 m 4 + m 6 = 180 l FHS Unit C 7
8 Example 1: Using the Corresponding Angles Postulate A. Find m ECF m GBC = 70 m ECF = 70 B. Find m DCE 5x = 4x + 22 x = 22 FHS Unit C 8 A B G (4x + 22)º 70º m DCE = 5x = 5(22) = 110º C F D (5x)º E
9 Example 2: Using the Alternate Interior Angle Theorem A. Find m ABD 2x + 10 = 3x = x m ABD = (2x + 10) = 2(25) + 10 = 60 m ABD = 60º A B C (2x + 10)º (3x 15)º D E FHS Unit C 9
10 Example 3: Using the Alternate Exterior Angle Theorem A. Find m EDG m EDG = 85 B. Find x and m BDG x 30 = 85 x = 115 m BDG = F A B 85º C (2x 135)º G (x 30)º D E = 95 FHS Unit C 10
11 Example 4: Using the Same-Side Interior Angle Theorem In trapezoid ABCD, find the measure of B and D. m A + m D = 180º 80º + m D = 180º A 80º B m D = 100º m C + m B = 180º 40º + m B = 180º m B = 140º 40º FHS Unit C 11 D C
12 Lesson Quiz - Part I Find the measure of each angle. 1. m 1 = 2. m 2 = 130º 50º º 3. m 3 = 50º 4. m 4 = 5. m 5 = 50º 130º 4 5 FHS Unit C 12
13 Lesson Quiz - Part II 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) AEA Thm.; m 2 = m 2 = (75x 30), m 3 = (30x + 60) CA Post.; m 2 = 120, m 3 = FHS Unit C 13
14 Conditions for Parallel Lines Students will learn how to apply the converses of theorems about parallel lines and transversals.
15 Parallel Postulate Through a point P not on line l, there is exactly one (meaning one and only one) line parallel to l. k P l line l k. FHS Unit C 15
16 Converse of the Corresponding Angles Postulate If two lines are intersected by a transversal and the corre-sponding angles are congruent, then the lines are parallel. l k If 1 5, 2 6, 3 7, or 4 8, then line l k. FHS Unit C 16
17 Converse of the Alternate Interior (or Exterior) Angles Theorem If two lines are intersected by a transversal and the alternate interior (or exterior) angles are congruent, then the lines are parallel. l k If 1 8 or 3 6, then line l k. FHS Unit C 17
18 Converse of the Same-Side Interior Angles Theorem If two lines are intersected by a transversal and same-side interior angles are supplementary, then the lines are parallel. l k If m 2 + m 5 = 180º or m 4 + m 7 = 180º, then line l k. FHS Unit C 18
19 Are the lines parallel? Why? Parallel because same-side interior angles are supplementary. Parallel because alternate interior angles are congruent. 65º 70º 100º Not parallel 115º because same-side interior angles are not supplementary. FHS Unit C 19
20 Example 8 Suppose the corresponding angles ( 1 and 2) of the boat pictured measure (4y 2) and (3y + 6). What value of y would prove that the oars are parallel. 4y 2 = 3y + 6 y = 8 If y = 8, the oars would be by the Converse of the CA Postulate. FHS Unit C 20
21 To Review: If two lines are intersected by a transversal and corresponding angles are congruent, then the lines are parallel. If two line are intersected by a transversal and alternate interior (or exterior) angles are congruent, then the lines are parallel. If two lines are intersected by a transversal and same-side interior angles are supplementary, then the lines are parallel. FHS Unit C 21
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