Every triangle has three angles that add to 180°. Add to that the facts for special triangles: an isosceles triangle has two equal base angles, an equilateral triangle has three angles of 60°, and an exterior angle equals the sum of the two opposite interior angles. These facts appear on their own and inside diagrams with parallel lines and circles.
This is the second lesson in angles and geometric reasoning. It builds on parallel-line angle relationships and leads to polygons.
Which triangle facts should you know?
| Fact | Wording to write |
|---|---|
| Angle sum | The angles in a triangle add to 180° |
| Isosceles | Base angles of an isosceles triangle are equal |
| Equilateral | Each angle of an equilateral triangle is 60° |
| Exterior angle | An exterior angle equals the sum of the two opposite interior angles |
In an isosceles triangle, the equal sides tell you which angles are equal. The equal angles sit opposite the equal sides. If AB = AC, then ∠B = ∠C.
How do you choose the right fact?
- List what is given: angles, equal-side marks, extended lines.
- Look for equal sides. They point to equal base angles.
- Look for a straight line at a vertex. It gives an exterior angle or a 180° link.
- Use the angle sum to finish the triangle.
- Write the reason for every angle you find, in words.
Worked example
In triangle ABC, AB = AC and ∠BAC = 40°. BC is extended to D beyond C. Find ∠ABC and ∠ACD, with reasons.
Step 1: The equal sides are AB and AC, so the base angles ∠ABC and ∠ACB are equal.
Step 2: The angles add to 180°, so ∠ABC + ∠ACB = 180° − 40° = 140°.
Step 3: Each base angle is 140° ÷ 2 = 70°, so ∠ABC = 70°.
Step 4: ∠ACD is an exterior angle, so it equals ∠BAC + ∠ABC = 40° + 70° = 110°.
Check: ∠ACB = 70°, and ∠ACB + ∠ACD = 70° + 110° = 180°, because BCD is a straight line.
The mistake to watch for
A common slip is to treat the given angle as a base angle.
Mistaken answer: The other base angle is 40°, so the apex angle is 180° − 40° − 40° = 100°.
The student assumed 40° sat at the base. The question says ∠BAC = 40°, and A is the apex, between the equal sides AB and AC.
The correction is to find the vertex where the equal sides meet, and only then decide which angle is the apex. A quick check: the mistaken answer gives ∠BAC = 100°, which contradicts the 40° in the question, so the working cannot be right.
Check yourself
Try these, then open each answer.
1. The angles of a triangle are x°, 2x° and (3x + 30)°. Find the three angles.
Show answer
x + 2x + 3x + 30 = 180, so 6x = 150 and x = 25. The angles are 25°, 50° and 3(25) + 30 = 105°. Check: 25 + 50 + 105 = 180. 25°, 50°, 105°
2. In triangle PQR, PQ = PR and ∠PQR = 54°. Find ∠QPR.
Show answer
Base angles are equal, so ∠PRQ = 54°. Then ∠QPR = 180° − 54° − 54° = 72°.
3. An exterior angle of a triangle is 125°. One of the two opposite interior angles is 50°. Find the other, and the interior angle next to the exterior angle.
Show answer
The exterior angle equals the sum of the opposite interior angles, so the other angle is 125° − 50° = 75°. The adjacent interior angle is 180° − 125° = 55°. Check: 50° + 75° + 55° = 180°.
Where this leads next
Triangles are the building block for polygons, so move on to interior and exterior polygon angles. The non-calculator working trainer helps with halving and subtracting without slips, and the mixed practice set mixes all the facts.
Some students know every triangle fact but pick the wrong one under time pressure. Our teachers work on that choice in online one-to-one Mathematics tuition.