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You will learn: Use triangle angle sums, isosceles triangles and exterior angles with clear reasons.

Before you start: Angles in degrees, a straight angle of 180°, and the meaning of equal sides.

Follow what is given, not how a sketch looks

In Euclidean geometry, the three interior angles of a triangle sum to 180°. Angles on a straight line also sum to 180°. These two facts let us find many missing angles, provided we identify exactly which angles are involved.

The notation ∠ABC means the angle with vertex B, formed by BA and BC. The middle letter tells you where the angle is. An isosceles triangle has two equal sides, and the angles opposite those sides are equal.

Worked example 1: an isosceles triangle

In triangle ABC, AB = AC and ∠BAC = 40°. Find ∠ABC and ∠ACB. Since AB = AC, the opposite angles at C and B are equal. Let each be x degrees. Then x + x + 40 = 180, so 2x = 140 and x = 70. Each base angle is 70°.

Isosceles triangle with a 40 degree vertex angleA is above the midpoint of BC. AB equals AC, marked by single ticks. The angle at A is 40 degrees. The base angles at B and C are equal and labelled x.ABC40°xx
The matching ticks state AB = AC. The conclusion comes from equal base angles and the 180° sum.

Exterior angles

Extend side BC beyond C to a point D. The interior angle ∠ACB and the exterior angle ∠ACD form a straight angle, so their sum is 180°. The triangle sum also gives ∠BAC + ∠ABC + ∠ACB = 180°. Comparing the equations yields ∠ACD = ∠BAC + ∠ABC: an exterior angle equals the two non-adjacent interior angles added together.

Triangle with side BC extended beyond CB, C and D lie in that order on one straight line. A is above the line. The angle ACD is outside triangle ABC at C.ABCDexterior
D is beyond C. Changing the order of the points changes which angle is exterior.

Worked example 2: work back from an exterior angle

Suppose ∠BAC = 48° and ∠ACD = 125° in that configuration. The exterior-angle relation gives 125 = 48 + ∠ABC, so ∠ABC = 77°. Then ∠ACB = 180 − 125 = 55°. Check the triangle: 48 + 77 + 55 = 180.

What your diagram cannot tell you

A triangle that looks symmetric need not be isosceles. You need an equal-side statement, equal-side marks, or a proved result. Similarly, never measure a sketch to justify an exact answer. Write the reason beside each step: straight line, triangle sum, equal base angles, or another established fact.

Practise at your next step

A six-question session chooses from nine questions. Two correct answers in a row without hints move you up a level; an incorrect answer brings a simpler next question where one is available. Hints keep you at the same level. This is a practice suggestion, not an exam score or proof of mastery.

Interactive practice loads here. You can also use the complete question set below.

Write a proof of your own

In triangle ABC, AB = AC. Extend BC beyond C to D. Prove that ∠ACD = 90° + half of ∠BAC.

Write your reasoning on paper before comparing. The practice checker does not grade a written proof.

Compare your proof with a full solution

Let ∠BAC = a degrees. The two equal base angles have sum 180 − a, so each is (180 − a)/2 = 90 − a/2 degrees. The exterior angle ∠ACD is supplementary to the angle at C, so it equals 180 − (90 − a/2) = 90 + a/2 degrees. This is the required relation.

Check: did you state the assumptions, explain the key step, and reach the requested conclusion?

All nine practice questions, hints and solutions

This complete set works without the interactive practice. Hide each solution until you have made an attempt.

1. Two angles of a triangle are 50° and 60°. What is the third?

Foundation

  1. 60°
  2. 70°
  3. 80°
  4. 110°
Hint

Subtract the known angles from 180°.

Answer and explanation

70°. The third angle is 180° − 50° − 60° = 70° because the interior angles of a triangle sum to 180°.

2. Two adjacent angles form a straight line. One is 65°. What is the other?

Foundation

  1. 25°
  2. 65°
  3. 115°
  4. 125°
Hint

A straight angle measures 180°.

Answer and explanation

115°. The two angles sum to 180°, so the missing angle is 180° − 65° = 115°.

3. Which point is the vertex of ∠PQR?

Foundation

  1. P
  2. Q
  3. R
  4. All three
Hint

Read the middle letter.

Answer and explanation

Q. The middle letter Q identifies the vertex. The sides of the angle are QP and QR.

4. In triangle ABC, AB = AC and ∠BAC = 36°. What is ∠ABC?

Core

  1. 36°
  2. 54°
  3. 72°
  4. 144°
Hint

The two base angles are equal.

Answer and explanation

72°. The base angles at B and C sum to 180° − 36° = 144°. They are equal because AB = AC, so each is 144°/2 = 72°.

5. In triangle ABC, BC is extended beyond C to D. If ∠BAC = 45° and ∠ABC = 68°, what is ∠ACD?

Core

  1. 67°
  2. 90°
  3. 113°
  4. 135°
Hint

The exterior angle is the sum of the two non-adjacent interior angles.

Answer and explanation

113°. ∠ACD = ∠BAC + ∠ABC = 45° + 68° = 113°. Equivalently, the interior angle at C is 67° and its supplementary angle is 113°.

6. The three angles of a triangle have ratio 2 : 3 : 4. What is the largest?

Core

  1. 40°
  2. 60°
  3. 80°
  4. 90°
Hint

There are 2 + 3 + 4 equal parts in a total of 180°.

Answer and explanation

80°. There are 9 parts, so each part is 180°/9 = 20°. The largest angle has 4 parts and is 4 × 20° = 80°.

7. In triangle ABC, AB = AC and ∠BAC = 44°. AD bisects ∠BAC, with D on BC. What is ∠ADB?

Stretch

Triangle ABC and the bisector ADAB and AC are equal. D lies on BC, and AD bisects the 44 degree angle at A into two 22 degree angles. The angle ADB is to be found.ABCD22°22°?
Use this diagram for practice question 7. Perpendicularity is not assumed.
  1. 44°
  2. 68°
  3. 90°
  4. 112°
Hint

First find ∠ABD, then ∠BAD, and use triangle ABD.

Answer and explanation

90°. The base angle ∠ABC is (180° − 44°)/2 = 68°. Since D is on BC, ∠ABD = 68°. The bisector gives ∠BAD = 22°. Therefore ∠ADB = 180° − 68° − 22° = 90°.

8. In triangle ABC, BC is extended beyond C to D. If ∠ACD = 132° and ∠BAC = 57°, what is ∠ABC?

Stretch

  1. 48°
  2. 57°
  3. 75°
  4. 105°
Hint

Use the exterior-angle equation and subtract the known interior angle.

Answer and explanation

75°. 132° = ∠BAC + ∠ABC = 57° + ∠ABC, so ∠ABC = 75°. The angle at C is 48°, and 57° + 75° + 48° = 180° checks the result.

9. An isosceles triangle has an angle of 40°, but the question does not say which angle. Which list gives all possible largest angles?

Stretch

  1. Only 70°
  2. Only 100°
  3. 70° or 100°
  4. 40° or 140°
Hint

Consider 40° as the vertex angle and as a base angle.

Answer and explanation

70° or 100°. If 40° is the angle between the equal sides, the base angles are 70° each, so the largest is 70°. If 40° is a base angle, the other base angle is also 40° and the vertex angle is 100°. These cover both roles for the given angle.

Where to go next

Move on to the mixed proof-practice plan · Explore the wider Olympiad topic map

Teaching examples prepared for this guide, not attributed to a contest paper. Familiar elementary problems may appear in other learning materials. Report an unclear step or an error.