Triangles • Topic 1 of 6

Angle Sum & Exterior Angle

The three interior angles of any triangle add to 180° — the one fact you will use in nearly every triangle question. From it comes the exterior angle theorem: an exterior angle of a triangle equals the sum of the two interior angles it is not next to (the two "remote" interior angles). That shortcut saves a step whenever an angle sticks out beyond a side. When angles are given as a ratio, split 180° across the total number of parts. Because GRE figures are not drawn to scale, always assign the angles from the equations and never assume, from the look of the picture, that a triangle is equilateral or right-angled.

A triangle with its three interior angles and one exterior angle at a vertexAngle sum & exterior anglea°b°c°e°Exterior e = a + c · a + b + c = 180°

✅ Solved examples

1. Two angles of a triangle are 50° and 65°. Find the third.
Angles sum to 180°: third = 180 − (50 + 65) = 180 − 115 = 65°.
2. An exterior angle of a triangle is 120°, and one remote interior angle is 45°. Find the other remote interior angle.
An exterior angle equals the sum of the two remote interior angles: 120 = 45 + x → x = 75°.
3. The angles of a triangle are in the ratio 2 : 3 : 4. Find each angle.
Total parts = 9, and 180° / 9 = 20° per part. Angles = 40°, 60°, 80°.
4. Quantitative Comparison. A triangle has angles x°, x°, and (x + 30)°. Quantity A: x. Quantity B: 50. Which is greater?
Sum: x + x + (x + 30) = 180 → 3x + 30 = 180 → 3x = 150 → x = 50. Quantity A = 50 = Quantity B, so the two are equal.

✏️ Practice — try these, take hints as needed

1. Two angles of a triangle are 37° and 53°. Find the third and classify the triangle.
Sum to 180°.
180 − (37 + 53).
A 90° angle means right triangle.
90°; right triangle
2. An exterior angle of a triangle is 110° and the two remote interior angles are equal. Find each remote interior angle.
Exterior = sum of remote interiors.
110 = a + a.
2a = 110.
55° each
3. The angles of a triangle are in the ratio 1 : 2 : 3. Find the largest angle.
Total parts = 6.
180 / 6 = 30° per part.
Largest = 3 parts.
90°
4. One angle of a triangle is 90°. A second angle is twice the third. Find the two remaining angles.
The other two sum to 90°.
Let them be 2x and x.
3x = 90.
60° and 30°
5. An exterior angle of a triangle measures 130° and one remote interior angle is 80°. Find the other remote interior angle.
Exterior = sum of remote interiors.
130 = 80 + x.
Subtract 80.
50°

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
Types of Triangles →