☰ Contents · Geometry

Parallel lines and a transversal

Lessons 29–30 · 2 lessons · Geometry Grade 7, corrected and expanded 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2017 (approved by the Ministry of Public Education)
30

Angles formed by two lines and a transversal

Textbook: pp. 80–81
GoalName the pairs of angles formed by two lines and a transversal.
New words
transversal · kesuvchialternate interior angles · ichki almashinuvchi burchaklarcorresponding angles · mos burchaklarsame-side interior angles · ichki bir tomonli burchaklar
Explanation

When a third line c crosses two lines a and b, eight angles are formed. For definiteness, let c cross a at P and b at Q (a above b). At P the angles are ∠1 upper-left, ∠2 upper-right, ∠3 lower-right, ∠4 lower-left; at Q, in the same order, ∠5, ∠6, ∠7, ∠8. Alternate interior angles: ∠3 and ∠5, ∠4 and ∠6; same-side interior: ∠3 and ∠6, ∠4 and ∠5; corresponding: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8. Property 1: if one pair of alternate interior angles is equal, the other pair is equal too. Property 2: if corresponding angles are equal, the same-side interior angles add up to 180°. Property 3: if alternate interior angles are equal, the corresponding angles are equal too.

Worked examples
With the numbering above, if ∠4 = 70° and ∠6 = 70° (alternate interior), then ∠3 = ∠5 = 180° − 70° = 110°.
If ∠2 = ∠6 = 62° (corresponding angles), then ∠3 = 180° − ∠2 = 118° (adjacent to ∠2). The same-side interior pair gives ∠3 + ∠6 = 118° + 62° = 180°.
Class activity

“Eight angles”: draw two lines and a transversal, number the 8 angles and measure with a protractor to find equal pairs and pairs summing to 180°.

Practice
1
What kind of pair do ∠4 and ∠6 form (numbering above)?
2
Same numbering. What kind of pair are ∠1 and ∠5?
3
∠3 and ∠5 are alternate interior and equal; ∠3 = 105°. Find ∠5 (degrees).
4
Why does equality of one pair of alternate interior angles give equality of the other pair?