2.3 Triangle Angle Theorems

The Triangle Angle-Sum Theorem

The most-used fact about triangles is the Triangle Angle-Sum Theorem: the three interior angles of any triangle add to exactly 180°. New York standard G-CO.C.10 requires you to prove and apply this and related triangle theorems.

Worked example. A triangle has angles measuring 47°, 68°, and x°. Then 47 + 68 + x = 180, so 115 + x = 180 and x = 65°. If instead the angles are (2x)°, (3x)°, and (4x)°, then 9x = 180, so x = 20 and the angles are 40°, 60°, and 80°.

Why 180°? Standard G-CO.C.10 expects you to be able to justify the theorem, not just use it. Draw a line through one vertex parallel to the opposite side. The two "outer" angles formed there are alternate interior angles with the triangle's other two angles, so they are congruent to them. Those three angles lie along the straight auxiliary line, which measures 180° - proving the triangle's three interior angles must also total 180°. This parallel-line argument is the classic Regents justification for the angle-sum theorem.

The Exterior Angle Theorem

Extend one side of a triangle and you form an exterior angle. The Exterior Angle Theorem states that an exterior angle equals the sum of the two remote (non-adjacent) interior angles. It follows directly from the angle sum, because the exterior angle and its adjacent interior angle form a linear pair.

Worked example. Two remote interior angles measure 50° and 65°. The exterior angle at the third vertex is 50 + 65 = 115°. As a check, the adjacent interior angle is 180 minus 115 = 65°, and 50 + 65 + 65 = 180°. A frequent trap is adding all three interior angles or using the adjacent angle instead of the two remote ones.

The Third Angle Theorem

A quick corollary of the angle sum is the Third Angle Theorem: if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles is congruent too. You never have to prove the third pair separately - the 180° total forces it. This theorem underpins the AA criterion for similar triangles you will meet in Chapter 4, and it is a common one-line justification in Regents proofs.

Isosceles Triangle Base Angles

An isosceles triangle has two congruent sides called legs; the third side is the base. The Isosceles Triangle Theorem (Base Angles Theorem) says the two base angles - the angles opposite the congruent legs - are congruent. Its converse says that if two angles of a triangle are congruent, the sides opposite them are congruent, so the triangle is isosceles.

Worked example. An isosceles triangle has a vertex angle of 40°. The two base angles are equal, and together with the vertex they sum to 180°: 40 + 2b = 180, so 2b = 140 and each base angle is 70°.

Algebraic example. In isosceles triangle RST the legs RS and RT are congruent with RS = x + 5 and RT = 2x minus 3. Set x + 5 = 2x minus 3, giving 8 = x, so x = 8 and each leg measures 13.

An equilateral triangle is a special isosceles triangle: all three sides are congruent, so all three angles are congruent and each measures 180° / 3 = 60°.

Classifying Triangles

Regents questions expect you to name a triangle two ways at once - by its sides and by its angles.

The Triangle Inequality

Three lengths form a triangle only if each side is shorter than the sum of the other two. For sides 7 and 10, the third side t must satisfy 10 minus 7 < t < 10 + 7, that is 3 < t < 17. So 16 works, but 3, 17, and 18 do not.

Worked Angle-Chase

Suppose a diagram shows parallel lines with a triangle resting on them. One base angle equals an alternate interior angle of 55°, another interior angle is a vertical angle measuring 65°, and you need the third. Since a triangle's angles sum to 180°, the third angle is 180 minus 55 minus 65 = 60°. Angle-chasing rewards you for combining vertical angles, parallel-line pairs, and the angle sum in one problem - exactly what Part II items test.

A second angle-chase with algebra. A triangle has angles (x + 10)°, (2x)°, and (3x minus 10)°. Because they sum to 180°, write (x + 10) + 2x + (3x minus 10) = 180, so 6x = 180 and x = 30. The angles are 40°, 60°, and 80° - an acute, scalene triangle. Always finish by classifying and by checking that no angle came out negative or over 180°, which would signal an arithmetic slip. When a figure combines an exterior angle with the angle sum, decide first which theorem is shorter: often the Exterior Angle Theorem saves a step over computing the adjacent interior angle and subtracting.