Complementary and Supplementary Angles
Angles on a straight line total 180 degrees; angles forming a right angle total 90. Plus why opposite angles at a crossing are always equal, and what perpendicular lines guarantee.
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Two angles that sit together often add to a fixed total. If they form a straight line the total is 180°; if they form a right angle it is 90°. Those two facts, together with the equality of vertical angles, settle most simple angle problems.
Concept
Supplementary angles
Two adjacent angles that form a straight line add to 180°.
These are called supplementary angles. Knowing one immediately gives the other by subtraction.
Concept
Complementary angles
Two adjacent angles that add to 90° are complementary.
The distinction is easy to hold onto: complementary goes with the corner of a right angle at 90°, while supplementary goes with the straight line at 180°.
Concept
Vertical angles
When two lines cross, four angles appear, and each pair of opposite angles is equal.
Widen one angle and the one facing it widens by the same amount, while the other two narrow equally. That is why opposite angles stay equal.
Concept
Perpendicular lines
If two lines cross at right angles, then all four angles measure 90°. Knowing that one is a right angle is enough to conclude it for the rest — a direct consequence of the vertical-angle rule.
All right angles are equal, wherever they appear and however the figure is drawn.
Adjacent angles at a perpendicular crossing are equal as well, since each is 90°.
There is also a useful converse: if two adjacent angles on a straight line are equal, each must be 90°, because they share the 180° evenly.
Example
Using both totals
One angle on a straight line measures 115°. Find its supplement. Then find the complement of a 35° angle.
Supplementary angles total 180°, so 
.
Complementary angles total 90°, so 
.
⟹ 65° and 55°
A 115° angle has no complement at all — it already exceeds 90°. Only angles smaller than 90° have one.
Note
Mistakes to avoid
Swapping the two words — complementary is 90°, supplementary is 180°.
Looking for the complement of an angle larger than 90°; it does not exist.
Assuming two angles are supplementary when they do not actually form a straight line.
Expecting vertical angles to add to 180°; they are equal, not supplementary.
Reading angle sizes from the drawing rather than calculating them.
Summary
- Supplementary angles add to 180° and form a straight line.
- Complementary angles add to 90° and form a right angle.
- When two lines cross, opposite angles are always equal.
- If lines are perpendicular, all four angles are 90°, and all right angles are equal.
- Two equal adjacent angles on a straight line must each be 90°.