☰ Contents · Geometry

Tests for parallel lines

Lessons 31–32 · 2 lessons · Geometry Grade 7, corrected and expanded 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2017 (approved by the Ministry of Public Education)
32

Tests for parallel lines (continued)

Textbook: pp. 84–85
GoalWork with the corollaries of the test (corresponding and same-side interior angles).
New words
corollary · natijacorresponding angles · mos burchaklarsame-side interior angles · ichki bir tomonli burchaklarsum of 180° · yig‘indisi 180°
Explanation

A statement that we get straight from a proved theorem, with almost no extra reasoning, is called a corollary. Corollary 1: if a pair of corresponding angles formed by two lines and a transversal is equal, the lines are parallel, because a corresponding angle equals a vertical angle that is an alternate interior angle. Corollary 2: if a pair of same-side interior angles adds up to 180°, the lines are parallel, because adjacent angles add to 180°, which makes the alternate interior angles equal. Checking one pair of angles is enough: the other pairs follow from it. In problems, find the unknown angle from the figure and decide which corollary applies.

Worked examples
Corresponding angles 80° and 80° ⇒ a ∥ b (Corollary 1).
Same-side interior angles 108° and 72°: 108° + 72° = 180° ⇒ a ∥ b (Corollary 2).
Class activity

“Pick the corollary”: the teacher names angles formed by two lines and a transversal; pupils answer “Corollary 1”, “Corollary 2” or “not parallel”.

Practice
1
Same-side interior angles are 115° and 65°. Are the lines parallel?
2
Corresponding angles are 50° and 55°. Parallel?
3
Same-side interior angles are 4x and 5x. For the lines to be parallel, find x (degrees).
4
Why do same-side interior angles summing to 180° make the alternate interior angles equal?