Math

Lines, Angles, and Triangles

2 min readEasy5-question drill

Lines and triangles show up on nearly every math section, and they almost always come down to a handful of angle rules you can memorize in ten minutes. Know them cold and these become free points.

Where this lesson fits

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This topic is about the rules that govern angles (measured in degrees, °) when lines cross and when triangles are formed.

Straight lines and crossing lines:

Angles on a straight line add up to 180° (they form a straight angle).
Angles all the way around a point add up to 360°.
When two lines cross, the angles directly across from each other are vertical angles — they're always equal.

Parallel lines: When a line (called a transversal) cuts across two parallel lines (lines that never meet), it creates 8 angles, but only two different values. Equal pairs include corresponding angles (same position at each intersection) and alternate interior angles (the Z-shaped pair between the lines). Any two angles that aren't equal are supplementary — they add to 180°.

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Two parallel lines cut by a transversal create equal corresponding angles and equal alternate interior angles; any other pair sums to 180 degrees.

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Angle pairs formed when a transversal crosses parallel lines.

Triangle angle sum: The three interior angles of any triangle add to 180°. This single fact solves a huge fraction of these questions.

Exterior angle rule: Extend one side of a triangle and you get an exterior angle. It equals the sum of the two non-adjacent interior angles (the two angles it doesn't touch). Example: if those two interior angles are 70° and 60°, the exterior angle is 130°.

Triangle types you should recognize:

Equilateral — all sides equal, all angles 60°.
Isosceles — two sides equal, and the angles opposite those equal sides are also equal.
Right — one angle is 90°.

Triangle Inequality: The third side of a triangle must be bigger than the difference of the other two sides and smaller than their sum. If two sides are 5 and 11, the third side x must satisfy 11 − 5 < x < 11 + 5, i.e. 6 < x < 16.

Similar triangles: If two triangles have the same angles, their sides are proportional — you can set up a fraction to find a missing length.

Most questions just combine two or three of these facts. Find one angle, use it to find the next, and chain your way to the answer.

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Quick check

Check your understanding with a question from this topic:

In a triangle, two angles measure 55° and 80°. What is the measure of the third angle?

45°

35°

55°

65°

Worked examples

Example 1

In a triangle, two of the angles measure 42° and 73°. What is the measure of the third angle?

Example 2

An exterior angle of a triangle measures 125°. One of the two non-adjacent interior angles measures 48°. What is the measure of the other non-adjacent interior angle?

Example 3

In triangle ABC, side AB = side AC, and the angle at B measures 50°. What is the measure of the angle at A?

Common pitfalls

Confusing which sides give equal angles in isosceles

The equal angles sit opposite the equal sides, not between them. Mark the two equal sides, then look across from each to find the matching angles.

Forgetting the lower bound in the triangle inequality

Students remember the third side must be less than the sum but forget it must also be greater than the difference. Always write both: |a − b| < x < a + b.

Adding exterior angle to a wrong interior angle

The exterior angle equals the two non-adjacent interior angles only. The interior angle right next to it is its supplement (adds to 180°), not part of the sum.

Assuming a figure is to scale

Don't eyeball angle sizes from the picture — figures are often not drawn accurately. Trust the given numbers and the rules, not how it looks.

Key takeaways

Angles on a straight line sum to 180°; around a point sum to 360°; vertical angles are equal.
The three interior angles of any triangle sum to 180°.
An exterior angle equals the sum of the two non-adjacent interior angles.
Triangle Inequality: the third side lies strictly between the difference and the sum of the other two.
Isosceles triangles have equal angles opposite their equal sides; equilateral triangles are all 60°.

Watch & learn

Curated Khan Academy walkthroughs on Lines, Angles, and Triangles. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Lines, Angles, and Triangles, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Lines, Angles, and Triangles

You are here

→Right Triangles and Trigonometry →Area and Volume →Circles