34
Angles formed by two parallel lines and a transversal
Textbook: pp. 88–89
GoalUse the properties of angles formed by parallel lines and a transversal.
New words
parallel lines · parallel to‘g‘ri chiziqlaralternate interior angles · ichki almashinuvchi burchaklarcorresponding angles · mos burchaklarsame-side angles · bir tomonli burchaklar
Explanation
The theorems in this lesson are converses of the tests for parallel lines. Theorem 1: two parallel lines and a transversal form equal alternate interior angles. Proof by contradiction: if they were unequal we could build an equal angle at the point and get a second parallel, contradicting the parallel axiom. Theorem 2: corresponding angles formed by parallel lines and a transversal are equal. Theorem 3: same-side interior angles add up to 180°. Corollary: a line perpendicular to one of two parallel lines is perpendicular to the other. When a ∥ b is given, use these properties to find unknown angles.
Worked examples
If a ∥ b and an angle is 65°, the alternate interior angle is 65° and the same-side interior angle is 180° − 65° = 115°.
a ∥ b, a pair of same-side interior angles x and 3x: 4x = 180°, x = 45°; the angles are 45° and 135°.
Class activity
“All from one”: for parallel lines and a transversal one angle is given; groups find the other 7 angles.
Practice
1
a ∥ b. One interior angle with the transversal is 72°. Its alternate interior angle (degrees)?
72
2
a ∥ b. One same-side interior angle is 128°. The other (degrees)?
52
3
a ∥ b, corresponding angles 5x and 3x + 40°. Find x.
20
4
Why is a line perpendicular to one of two parallel lines perpendicular to the other too?
If one corresponding angle is 90°, the other corresponding angle is equal, so also 90°.