1. 3.2 Transversals and Angles
CHAPTER
3.2 Transversals and Angles
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Definitions
A transversal is a line that intersects two or more coplanar lines at different points. The figure below shows the eight angles formed by a transversal t and two lines / and m.
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Definitions
Angles 3, 4, 5, and 6 lie between l and m. They are interior angles. Angles 1, 2, 7, and 8 lie outside of l and m. They are exterior angles.
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Definitions
Alternate interior angles are nonadjacent interior angles that lie on opposite sides of the transversal.
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Definitions
Same-side interior angles are interior angles that lie on the same side of the transversal (sometimes called consecutive interior angles).
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Definitions
Corresponding angles lie on the same side of the transversal and in corresponding positions.
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Definitions
Alternate exterior angles are nonadjacent exterior angles that lie on opposite sides of the transversal.
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8. Identifying Angle Pairs
List all angle pairs in the figure. a. alternate interior b. corresponding Solution a.
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9. Classifying an Angle Pair
The photo shows the Hearst Building in New York City. The new tower (showing many triangles) was completed in Fill in the blank. a. 1 and 5 are _____________ angles. b. 2 and 7 are _____________ angles.
alternate exterior
same-side interior
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10. Theorem 3.2 Alternate Interior Angles Theorem
Theorem If two lines are cut by a transversal and a pair of alternate interior angles are congruent, then the two lines are parallel. If… Then…
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11. Theorem 3.3 Corresponding Angles Theorem
Theorem If two lines are cut by a transversal and a pair of corresponding angles are congruent, then the lines are parallel. If… Then…
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12. Theorem 3.4 Alternate Exterior Angles Theorem
Theorem If two lines are cut by a transversal and a pair of alternate exterior angles that are congruent, then the two lines are parallel. If… Then…
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13. Theorem 3.5 Same-Side Interior Angles Theorem
Theorem If two lines are cut by a transversal are two interior angles on the same side of the transversal are supplementary, then the two lines are parallel. If… Then…
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Theorem 3.6 Alternate Interior Angles Converse (Converse of Theorem 3.2)
Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If… Then…
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15. Identifying Parallel Lines
Which lines are parallel if Justify your answer. Solution are not formed by line l, so we concentrate on line a and line b with transversal m. are alternate interior angles. If , then a || b by the Alternate Interior Angles Theorem.
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16. Determining Whether Lines Are Parallel
The fence gate at the right is made up of pieces of wood arranged in various directions. Suppose Are lines r and s parallel? Explain. Solution Yes, r || s. are alternate exterior angles. If two lines and a transversal form congruent alternate exterior angles, then the lines are parallel (Alternate Exterior Angles Theorem).
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17. Using Algebra to Prove Lines Are Parallel
What is the value of x that makes a || b? Solution The two angles are same-side interior angles. By the Same-Side Interior Angles Theorem, a || b if the angles are supplementary.
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18. Using Algebra to Prove Lines Are Parallel
What is the value of x that makes a || b? (2x + 9) = 180 2x = 180 2x = 60 x = 30 Thus, if x = 30, then a || b.
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