Pairs of angles
Many geometry problems are solved by spotting a pair of angles with a known relationship.
Pair | Definition | Example |
|---|---|---|
Complementary | two angles that add up to 90° | 35° and 55° |
Supplementary | two angles that add up to 180° | 110° and 70° |
Adjacent | share a vertex and a common arm, and lie on opposite sides of that arm | ∠AOB and ∠BOC |
Linear pair | adjacent angles whose outer arms form a straight line; they are always supplementary | 120° and 60° on a straight line |
Vertically opposite | the opposite angles formed when two lines cross; they are always equal | the two 40° angles at an X |
A memory trick: C comes before S in the alphabet, as 90 comes before 180. Complementary is 90°, Supplementary is 180°.
Example. The complement of 28° is 90° − 28° = 62°. The supplement of 28° is 180° − 28° = 152°.
Why vertically opposite angles are equal
Two straight lines cross and make angles a, b, c, d in order around the point.
a + b = 180° (a linear pair)
b + c = 180° (a linear pair)
So a + b = b + c, which gives a = c. In the same way, b = d.
Example. Two lines intersect and one angle is 72°. The angle opposite it is also 72°, and the other two angles are 180° − 72° = 108° each.
Transversals
A transversal is a line that crosses two or more lines at different points. A transversal crossing two lines makes eight angles.
t
/
──────/────── l
1 / 2
4 / 3
/
────/────────── m
5 / 6
8/ 7
/The eight angles fall into named pairs:
Name | Angles in the figure | Position |
|---|---|---|
Corresponding angles | 1 & 5, 2 & 6, 4 & 8, 3 & 7 | same position at each crossing |
Alternate interior angles | 3 & 5, 4 & 6 | between the lines, on opposite sides of the transversal |
Alternate exterior angles | 1 & 7, 2 & 8 | outside the lines, on opposite sides of the transversal |
Interior angles on the same side | 3 & 6, 4 & 5 | between the lines, on the same side of the transversal |
When the lines are parallel
If l and m are parallel:
Corresponding angles are equal.
Alternate interior angles are equal.
Interior angles on the same side add up to 180°.
The reverse is also true, and gives a test for parallel lines: if a transversal makes a pair of equal corresponding angles, or equal alternate interior angles, or same-side interior angles adding to 180°, the lines are parallel.
Example. l ∥ m, and the transversal makes an angle of 65° at l. The corresponding angle at m is 65°, and the interior angle on the same side as the 65° angle, at m, is 180° − 65° = 115°.
Solving for an unknown
Example. Two angles form a linear pair and one is 30° more than the other. Find them.
Let the smaller angle be x. Then x + (x + 30°) = 180°, so 2x = 150°, and x = 75°. The angles are 75° and 105°.
Summary
Complementary angles sum to 90°; supplementary angles sum to 180°.
A linear pair is supplementary; vertically opposite angles are equal.
With parallel lines, corresponding and alternate interior angles are equal, and same-side interior angles sum to 180°.