How to Find Missing Angles of Triangles

Find the third angle when two angles are known

An interior angle is an angle inside the triangle at one of its three vertices. In ordinary flat-plane geometry, the three interior angles total 180°, the same measure as a straight angle.

Missing angle = 180° − (first angle + second angle)

1. Add the two given interior angles.
2. Subtract that sum from 180°.
3. Check that all three angles add to 180° and that each is positive.

Example: Two angles measure 60° and 45°. Their sum is 105°, so the missing angle is 180° − 105° = 75°. Check: 60° + 45° + 75° = 180°.

If two supposed interior angles already add to 180° or more, they cannot belong to a triangle. Check whether one of the labels is an exterior angle instead.

Right, isosceles and equilateral triangles

Right triangle: subtract from 90°

A right triangle has one 90° angle, so its other two angles total 90°. If one acute angle is 32°, the other is 90° − 32° = 58°.

Isosceles triangle: use the equal base angles

An isosceles triangle has two equal sides, and the angles opposite those sides are equal. If the vertex angle between the equal sides is 40°, subtract it from 180° and divide the remainder equally: (180° − 40°) ÷ 2 = 70° for each base angle.

If a base angle is 65° instead, the other base angle is also 65°. The vertex angle is 180° − 65° − 65° = 50°. Identify which angle is given before choosing a calculation.

Equilateral triangle: every angle is 60°

All three sides and all three angles are equal. Divide 180° by 3 to get 60° at each vertex.

Use the stated triangle type or matching side marks. A drawing that looks symmetrical does not establish that two sides or angles are equal.

Use an exterior angle

An exterior angle is formed when one side of a triangle is extended past a vertex. It and the adjacent interior angle form a straight line, so they total 180°. The exterior angle also equals the sum of the two nonadjacent interior angles.

Example: An exterior angle is 125°, and one nonadjacent interior angle is 50°. The other nonadjacent interior angle is 125° − 50° = 75°. The adjacent interior angle is 180° − 125° = 55°. Check: 50° + 75° + 55° = 180°.

Solve when angles contain variables

Write an equation using all three interior angle expressions.

Example: The angle measures are x°, 2x° and 3x°. Then x + 2x + 3x = 180, so 6x = 180 and x = 30. The angles are 30°, 60° and 90°. Finding x is only the first step; substitute it into each expression to find the requested angle.

Match corresponding angles in similar triangles

Similar triangles have equal corresponding angles, even when one triangle is rotated, reflected or larger. Side-length ratios describe their scale; you do not need to calculate a side ratio to copy a known corresponding angle.

Example: If triangle ABC is similar to triangle DEF in that order, A corresponds to D, B to E, and C to F. If angle A = 30° and angle B = 60°, then C = 90°. Therefore D = 30°, E = 60° and F = 90°.

Use vertex names, angle marks or the stated correspondence. Knowing that one angle in each triangle is 30° does not, by itself, establish that the triangles are similar.

Try these missing-angle problems

1. Two interior angles are 48° and 67°. Find the third.
2. A right triangle has an acute angle of 27°. Find its other acute angle.
3. An isosceles triangle has a vertex angle of 36°. Find both base angles.
4. An exterior angle is 140°. One nonadjacent interior angle is 62°. Find the other nonadjacent interior angle.
5. A triangle has angles x°, (x + 20)° and 80°. Find x and all three angle measures.

When a task asks you to measure a printed angle rather than calculate from given information, review how to align and read a protractor. Do not measure a diagram that says it is not drawn to scale.

Continue your geometry study

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Teaching reference: Common Core 8.G.A.5 covers triangle angle sums, exterior angles and angle-angle similarity. The worked examples and practice above are original.