Complementary angles are two angles whose measures add up to 90 degrees. Each angle is the complement of the other, so the complement of an angle x is 90° − x. They can share a side and form a right angle, or sit apart. For example, 35° and 55° are complementary.
Complementary is about the sum, not the position. Two angles that add to 90° are complementary whether they touch or not.
Formal definition and notation
Two angles ∠ A and ∠ B are complementary if and only if m∠ A + m∠ B = 90°. Each must be an acute angle, since both measures are positive. The angles need not be adjacent; when they are, their outer sides form a right angle. The word applies to a pair of angles, never to one angle alone.
Symbol
Read it as
Meaning
m∠ A + m∠ B = 90°
the measure of angle A plus the measure of angle B equals 90 degrees
the test for a complementary pair
Key facts
Memory hook: C comes before S in the alphabet, and 90 comes before 180. Complementary is 90, supplementary is 180.
The two acute angles of a right triangle are always complementary: if one is 28°, the other is 62°.
If two angles are complementary to the same angle, they are equal to each other.
The complement of 45° is 45°, the only angle that is its own complement.
In trigonometry, cosine means complement’s sine: cos 30° = sin 60°.
Complement formula
Complement formula
complement of x = 90° − x
x
the given angle, which must be between 0° and 90°
90°
one right angle, the total the pair must reach
Only acute angles have complements. A 90° or larger angle has no complement.
Try it: find the complement
Interactive
Drag the ray. The readout gives the complement for any angle under 90 degrees; go past 90 and see why no complement exists.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Complementary Angles examples, step by step
Example 1: Find a complement
Warm-up
Find the complement of a 23° angle.
Complements add to 90°.
90° − 23° = 67°.
Answer: The complement is 67°.
Example 2: Algebra with complements
Core
Two complementary angles measure (2x + 8)° and (3x + 7)°. Find both angles.
Two angles are complementary, and the larger is 4 times the smaller. What is the larger angle?
Let the smaller angle be x; the larger is 4x.
x + 4x = 90, so 5x = 90 and x = 18.
Larger angle = 4 × 18° = 72°.
Answer: The larger angle is 72° (the smaller is 18°).
Common mistakes
✗ MistakeComplement of 40°= 180° − 40° = 140°.
✓ InsteadComplement of 40°= 90° − 40° = 50°.
Subtracting from 180 gives the supplement. Complements total 90, supplements total 180.
✗ MistakeSaying a single 90° angle is complementary.
✓ InsteadA 90° angle is a right angle. Complementary describes a pair, such as 30° and 60°.
Complementary is a relationship between two angles, like being twins. One angle alone cannot be complementary.
✗ MistakeThinking 30° and 60° drawn far apart are not complementary.
✓ InsteadThey add to 90°, so they are complementary wherever they are drawn.
Only the sum matters. Adjacent complementary angles are just one special case.
Not to be confused with
Complementary Anglesvs.Supplementary angles
Supplementary angles add to 180°, not 90°. 30° has complement 60° but supplement 150°.
Complementary Anglesvs.Complimentary
Complimentary (with an i) means free or praising. The math word is complementary, with an e, from complete: the pair completes a right angle.
Quick check: 3 questions
Question 1 of 3
What is the complement of 71°?
Show answer and explanation
Answer: C.90° − 71° = 19°. 109° is the supplement (from 180), 29° is a subtraction slip, and 289° subtracts from 360.
Question 2 of 3
Which pair of angles is complementary?
Show answer and explanation
Answer: A.40 + 50 = 90. The pair 40°, 140° adds to 180 (supplementary), 45 + 55 = 100, and 90 + 90 = 180.
Question 3 of 3
Angle A and angle B are complementary. m∠ A = (x + 14)° and m∠ B = 2x°. What is x?
Show answer and explanation
Answer: C.x + 14 + 2x = 90 gives 3x = 76, so x = 2513. 5513 comes from using 180, 30 ignores the 14, and 3423 adds 14 to 90 instead of subtracting it.
Complementary Angles: frequently asked questions
Do complementary angles have to be adjacent?
No. Complementary angles only need measures that add to 90°. A 25° angle in one corner of a page and a 65° angle in another corner are complementary. When the two angles do share a vertex and a side, they fill a right angle, which is why diagrams often show them together.
How do you remember complementary vs supplementary?
Use the alphabet: C comes before S, and 90 comes before 180, so complementary means 90° and supplementary means 180°. Another hook: a right angle looks like the corner of a capital L, so complementary angles complete a corner, while supplementary angles make a straight line.
Can an obtuse angle have a complement?
No. An obtuse angle is already more than 90°, and a complement would need a negative measure to bring the total down to 90°. Only acute angles, those between 0° and 90°, have complements. A right angle has no complement either, since its partner would be 0°.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.G.B.5
Cite this entry
Effortless Math Team. “Complementary Angles.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/complementary-angles/
Flashcard
TermComplementary Anglestwo angles, total 90°
DefinitionTwo angles whose measures add to 90°. The complement of x is 90° − x: 25° and 65° are complementary.
Acute AngleAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2TrigonometryTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2, PrecalculusSineSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2, PrecalculusLike TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8Supplementary AnglesSupplementary angles are two angles whose measures add up to 180 degrees. Each one is the supplement of the other, so the supplement of an angle x is 180° − x. When they share a side, they form a straight line, called a linear pair. For example, 118° and 62° are supplementary.Geometry · Grade 7, Grade 8, GeometryObtuse AngleAn obtuse angle is an angle that measures more than 90 degrees and less than 180 degrees. It opens wider than a square corner but does not flatten into a straight line. Angles of 100, 135 and 179 degrees are obtuse. A triangle can have at most one obtuse angle.Geometry · Grade 4, Grade 5, Grade 7AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5CosineCosine is the trigonometric ratio that compares the side adjacent to an angle with the hypotenuse of a right triangle: cos θ = adjacenthypotenuse. On the unit circle, cos θ is the x-coordinate of the point at angle θ. The cosine of an angle equals the sine of its complement.Trigonometry · Geometry, Algebra 2, PrecalculusVertical AnglesVertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.Geometry · Grade 7, Grade 8, Geometry