Triangles
Classify triangles by their sides and angles, and find missing angles using the angle sum property.
Worked examples
Classifying a triangle by its sides
Straightforward
Problem
A triangle has sides of length 7 cm, 7 cm, and 10 cm. Classify this triangle by its sides.
1
Count how many sides are equal.
Two sides are 7 cm. The third side is 10 cm.
2
Recall the classification rules for sides.
Equilateral: all three sides equal. Isosceles: exactly two sides equal. Scalene: no sides equal.
3
Apply the rule.
Exactly two sides are equal, so the triangle is isosceles.
Answer
Isosceles
Finding a missing angle in a triangle
Moderate
Problem
Two angles of a triangle are and . Find the third angle.
1
Recall the angle sum of a triangle.
The angles of any triangle add to .
2
Add the two known angles.
3
Subtract from .
Answer
Finding angles from a ratio
Challenging
Problem
The three angles of a triangle are in the ratio . Find the size of each angle.
1
Let the angles be , , and .
These represent the three angles in the given ratio.
2
Use the angle sum of a triangle.
3
Solve for .
, so
4
Find each angle.
, ,
Answer
, , and
Practise
Q1·Straightforward
A triangle has all three sides equal in length. What type of triangle is it?
Explanation
When all three sides are equal, the triangle is equilateral. It also has three equal angles of each.
Q2·Straightforward
A triangle has exactly two equal sides. What type of triangle is it?
Explanation
A triangle with exactly two equal sides is isosceles. The angles opposite the equal sides are also equal.
Q3·Straightforward
A triangle contains one angle of exactly . How is this triangle classified by its angles?
Explanation
Any triangle that contains a angle is called a right-angled triangle. The side opposite the right angle is the hypotenuse.
Q4·Straightforward
A triangle has angles of , , and . How is this triangle classified by its angles?
Explanation
All three angles (, , ) are less than , so the triangle is acute.
Q5·Moderate
Two angles of a triangle are and . Find the third angle in degrees.
Explanation
Q6·Moderate
Two angles of a triangle are and . Find the third angle in degrees.
Explanation
Q7·Moderate
Two angles of a triangle are and . Find the third angle in degrees.
Explanation
Q8·Moderate
An isosceles triangle has two equal base angles of each. Find the apex angle in degrees.
Explanation
Q9·Challenging
The angles of a triangle are in the ratio . Find the largest angle in degrees.
Explanation
The largest angle is .
Q10·Challenging
An isosceles triangle has an apex angle of . Find the size of each base angle in degrees.
Explanation
Let each base angle be .
Q11·Challenging
In a triangle, one angle is twice another angle. The third angle is . Find the smallest angle in degrees.
Explanation
The smallest angle is .
Q12·Challenging
A right-angled isosceles triangle has one angle of and the other two angles are equal. Find the size of each equal angle in degrees.
Explanation
Each equal angle is .