Lesson One · Part 2
Relations between Angles
Two angles can share a position (adjacent) or share a total measure (complementary and supplementary). Combining both ideas gives the rules used to prove that rays lie on a straight line or are perpendicular.
1. Adjacent Angles
Adjacent angles are two angles in the same plane, with a common vertex and a common side, where the other two sides lie on opposite sides of the common side.
In the figure, ∠ABD and ∠CBD are adjacent because:
- • They share the vertex B and the side BD.
- • The other sides BA and BC lie on opposite sides of BD.
Not adjacent — case 1
∠ABD and ∠ABC are not adjacent, because the sides BD and BC lie on the same side of the common side BA (one angle is inside the other).
Not adjacent — case 2
∠ABC and ∠CDE are not adjacent, because they do not share a common vertex or a common side.
2. Complementary Angles
Complementary angles are two angles whose measures have a sum of 90°.
55° + 35° = 90° → they are complementary
They do not have to be adjacent: two separate angles measuring 55° and 35° are still complementary.
Notes on complementary angles
1. They are either two acute angles, or one is a zero angle and the other is a right angle.
2. Complements of the same angle (or of angles with equal measure) are equal. If ∠A complements ∠B and ∠C complements ∠B, then m(∠A) = m(∠C).
3. Supplementary Angles
Supplementary angles are two angles whose measures have a sum of 180°.
143° + 37° = 180° → they are supplementary
Notes on supplementary angles
1. They are either one obtuse and one acute angle, two right angles, or one zero angle and one straight angle.
2. Supplements of the same angle (or of angles with equal measure) are equal. If ∠A supplements ∠B and ∠C supplements ∠B, then m(∠A) = m(∠C).
| The measure of the angle | 35° | 90° | 0° | 48° | 60° |
|---|---|---|---|---|---|
| The measure of the complementary angle | 55° | 0° | 90° | 42° | 30° |
| The measure of the supplementary angle | 145° | 90° | 180° | 132° | 120° |
4. The Outer Sides of Two Adjacent Angles
Adjacent supplementary angles
If the outer sides of two adjacent angles lie on one straight line, the two angles are supplementary.
If BA and BC are opposite rays → m(∠ABD) + m(∠DBC) = 180°
The converse is also used: if the sum is 180°, the outer sides form a straight line.
Adjacent complementary angles
If the outer sides of two adjacent angles are perpendicular, the two angles are complementary.
If BA ⊥ BC → m(∠ABD) + m(∠DBC) = 90°
The converse is also used: if the sum is 90°, the outer sides are perpendicular.
Bisector reminder
The bisector of an angle is the ray that divides it into two angles of equal measure. If BD bisects ∠ABE and m(∠ABE) = 50°, then m(∠ABD) = m(∠DBE) = 25°.
Quick check: is OA ∪ OB a straight line?
144° + 36°
= 180° → yes, straight line
65° + 48° + 55°
= 168° ≠ 180° → no
49° + 42° + 89°
= 180° → yes, straight line
See these rules used in full solutions
Eight worked examples with every step written out.
Visual summary
Complementary pair totals 90°
Supplementary pair totals 180°
Angle + reflex angle totals 360°