Apply angle relationships (vertical, supplementary, parallel-line angles) and triangle properties (angle sum, exterior angle, isosceles and the triangle inequality) to find unknowns (Geometry).

What this topic is asking

This topic covers the angle and triangle relationships that let you find unknown angles and sides without measuring. The ACT tests vertical and supplementary angles, the angles made when a transversal crosses parallel lines, the triangle angle sum and exterior-angle rule, isosceles properties, and the triangle inequality.

Angle relationships

A few facts cover most angle questions.

Parallel lines and a transversal

When a line crosses two parallel lines, eight angles form with predictable relationships.

So once you know one angle, you can find all eight: equal to it (corresponding or alternate) or supplementary to it (same-side or linear pair).

Triangle angle sum and exterior angle

The interior angles of any triangle sum to $180°$.

Isosceles and equilateral triangles

An isosceles triangle has two equal sides, and the base angles opposite those sides are equal. So if a triangle has two sides equal and one base angle is $50°$, the other base angle is also $50°$, and the apex angle is $180° - 50° - 50° = 80°$. An equilateral triangle has all three sides equal and all three angles equal to $60°$. These equal-angle facts often unlock an otherwise underspecified figure.

The triangle inequality

The triangle inequality states that the length of any side must be less than the sum of the other two (and greater than their difference). So sides of 3 and 8 can form a triangle with a third side only if that side is between $8 - 3 = 5$ and $8 + 3 = 11$. This rule answers "which of these could be the side lengths of a triangle?" questions: test whether the two shorter sides sum to more than the longest.

Why these facts work together

ACT angle questions usually chain two or three of these facts: a vertical angle gives one measure, parallel lines transfer it, and the triangle sum finishes the unknown. Labelling every angle you can deduce on the figure, then using the $180°$ sum, resolves most problems. The triangle inequality and isosceles base-angle fact handle the side-length and equal-angle cases that pure angle-chasing does not.

Try this

Q1. Two angles of a triangle are $35°$ and $95°$. Find the third. [1 point]

• Cue. $180° - 35° - 95° = 50°$.

Q2. Can sides of length 4, 6 and 11 form a triangle? [1 point]

• Cue. $4 + 6 = 10 < 11$, so no; the two shorter sides do not reach.