Complementary Angles
In short: Complementary angles describe two angles with measures summing to exactly 90 degrees, so together they would fill a square corner, even when they sit far apart.
7th GradeGeometry7.G.B.5
What is Complementary Angles?
Two angles form complementary angles when their measures total \(90^{\circ}\). Placed side by side they build the corner of a square, and each angle is named the complement of the other.
If one angle measures \(x\), its partner measures \(90^{\circ} – x\). Both must be below \(90^{\circ}\), so each one is acute. The pair need not share a vertex: a \(37^{\circ}\) angle in one drawing and a \(53^{\circ}\) angle in another still qualify, although the match is easiest to see when they touch.
How to spot it
- A diagram marks a right angle with a small square and a ray cuts it into two pieces.
- The wording says the angles add to 90 degrees, or that one is the complement of the other.
- Two algebraic expressions stand for angle sizes and their total is a square corner.
- An acute angle is given and you are asked what is missing to reach a right angle.
The method
- Write an equation saying the two measures sum to \(90^{\circ}\).
- If one angle is known, subtract it from \(90^{\circ}\) to find the other.
- If both are expressions, combine like terms and solve for the variable.
- Put the value back to get each angle and confirm that they total \(90^{\circ}\).
Complementary Angles: solved example
Problem
A carpenter cuts a square corner into two pieces measuring \((2x + 5)^{\circ}\) and \((x + 10)^{\circ}\). Solve for \(x\) and give both angles.
Solution
A square corner is 90, so \(2x + 5 + x + 10 = 90\). That simplifies to \(3x + 15 = 90\) and \(x = 25\).
The pieces are \(2(25) + 5 = 55\) and \(25 + 10 = 35\), and \(55 + 35 = 90\) checks out. They measure \(55^{\circ}\) and \(35^{\circ}\).
Try it
Question 1
What angle completes \(68.5^{\circ}\) to a right angle?
Show the answer
\(90 – 68.5 = 21.5\), so \(21.5^{\circ}\).
Question 2
Complementary angles measure \(4x\) and \(5x\) degrees. Find both.
Show the answer
\(9x = 90\) gives \(x = 10\), so the angles are \(40^{\circ}\) and \(50^{\circ}\).
Complementary Angles vs. Supplementary Angles
Complementary pairs total \(90^{\circ}\) and fill a square corner. Supplementary pairs total \(180^{\circ}\) and fill a straight line. Handy memory aid: C precedes S in the alphabet, and 90 precedes 180. A complement is always acute, whereas a supplement may be obtuse.
Keep going
Lessons
Worksheets and quizzes
Related words
Whole dictionary: A–Z · 7th Grade · Geometry