Angle Pairs: Complementary, Supplementary, Vertical & Linear Pairs
Angles rarely show up alone. Two lines cross, or two angles share a side, and suddenly their measures are locked together by a rule: vertical angles are congruent, complementary angles add to , and supplementary angles add to . Those three facts are the entire topic.
They matter because they turn pictures into equations. A figure that gives you one angle measure secretly gives you several more — and once you can write the relationship as an equation, finding the missing angle is just arithmetic.
Vertical angles and linear pairs
When two lines intersect, they make four angles around the crossing point. The angles directly across from each other are vertical angles, and vertical angles are always congruent — same measure, no exceptions.
The angles next to each other are a linear pair: together they fill one side of a straight line, so their measures add to . In the figure below, the two angles are a vertical pair, the two angles are the other vertical pair, and any angle next to a angle is a linear pair since .
Notice that one given measure unlocks all four: the angle across from it matches it, and the two beside it are its supplements.
Complementary vs. supplementary
Complementary angles are two angles whose measures add to — together they make a corner. Supplementary angles are two angles whose measures add to — together they make a straight line. A linear pair is just a supplementary pair that happens to sit side by side.
The two angles do not have to touch. An angle of in one figure and an angle of somewhere else are still complementary, because the definition is only about the sum.
To keep the numbers straight: C comes before S in the alphabet, and comes before . Complementary is the corner, supplementary is the straight line.
Turning angle facts into equations
Every problem in this topic is one of three equations. Vertical angles: set the two expressions equal. Supplementary angles or a linear pair: set the sum equal to . Complementary angles: set the sum equal to .
Read the figure first and name the relationship out loud before writing anything. The most common mistake is using where the relationship is actually a corner, or setting a linear pair equal instead of supplementary.
Worked examples
Example 1: finding a complement
An angle measures . What is the measure of its complement?
Answer:
Example 2: finding a supplement
An angle measures . What is the measure of its supplement?
Answer:
Example 3: vertical angles with algebra
Two lines intersect. An angle measuring is vertical to an angle measuring . Find .
Answer:
Example 4: a supplementary pair with algebra
Angles measuring and are supplementary. Find both angle measures.
Answer: and
Try one yourself
Common questions
Do complementary or supplementary angles have to be next to each other?
No. The definitions only care about the sum of the measures — for complementary, for supplementary. When a supplementary pair does sit side by side along a straight line, it gets the extra name linear pair.
How do I remember which sum is 90 and which is 180?
C comes before S in the alphabet, and comes before . Complementary makes a corner (); supplementary makes a straight line ().
Can vertical angles ever be supplementary to each other?
Only in one special case: when the two lines are perpendicular, all four angles measure , so each vertical pair adds to . In every other crossing, vertical angles are congruent but not supplementary.
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