Properties of triangles
Find unknown angles using the angle sum of a triangle, the exterior angle theorem, and properties of isosceles and equilateral triangles.
Worked examples
Finding a missing angle using the angle sum
Straightforward
Problem
A triangle has two angles measuring and . Find the size of the third angle.
1
State the angle sum property of a triangle.
The angles of any triangle add to .
2
Add the two known angles.
3
Subtract from to find the missing angle.
Answer
The third angle is .
Finding base angles in an isosceles triangle
Moderate
Problem
An isosceles triangle has an apex angle of . Find the size of each base angle.
1
Recall that the two base angles of an isosceles triangle are equal. Let each base angle be .
Base angles: and , apex angle:
2
Write an equation using the angle sum of a triangle.
3
Solve for .
, so
Answer
Each base angle is .
Finding angles using algebraic expressions
Challenging
Problem
A triangle has angles , , and . Find and state the size of each angle.
1
Write an equation using the angle sum of a triangle.
2
Collect like terms on the left side.
3
Solve for .
, so
4
Substitute to find each angle.
, ,
5
Check: the three angles should add to .
✓
Answer
; the angles are , , and .
Practise
Q1·Straightforward
A triangle has two angles measuring and . Find the size of the third angle.
Explanation
Third angle .
Q2·Straightforward
A right-angled triangle has one acute angle of . Find the size of the other acute angle.
Explanation
The three angles add to . The right angle is , so the missing angle .
Q3·Straightforward
An equilateral triangle has all three angles equal. What is the size of each angle, in degrees?
Explanation
Each angle .
Q4·Straightforward
A triangle has angles , , and . Which type of triangle is this?
Explanation
One of the angles is exactly , so the triangle is right-angled.
Q5·Moderate
An exterior angle of a triangle measures . One of the non-adjacent interior angles is . Find the other non-adjacent interior angle.
Explanation
By the exterior angle theorem, the exterior angle equals the sum of the two non-adjacent interior angles. So the missing angle .
Q6·Moderate
An isosceles triangle has an apex angle of . Find the size of each base angle.
Explanation
Let each base angle be . Then , so and .
Q7·Moderate
A triangle has angles , , and . Find the value of .
Explanation
, so and .
Q8·Moderate
An exterior angle of a triangle is formed at one vertex. The two non-adjacent interior angles are both equal to . Find the size of the exterior angle.
Explanation
Exterior angle .
Q9·Challenging
A triangle has angles , , and . Find the value of .
Explanation
\\ \\ \\
Q10·Challenging
In triangle , and . Find the size of .
Explanation
Since , we have . Using the angle sum: , so , , .
Q11·Challenging
An isosceles triangle has an exterior angle of at one of its base vertices. Find the size of the apex angle.
Explanation
The interior base angle . Since the triangle is isosceles, both base angles equal . Apex angle .
Q12·Challenging
In triangle , (so is the apex). The exterior angle at vertex is . Find the size of the apex angle at .
Explanation
Interior angle at . Since , angles and are the base angles and are equal, so . Apex angle .