# TRIANGLES: Finding Unknown Angles

*Angles of an isosceles triangle*

*Angles of an isosceles triangle*

Make an isosceles triangle as below.

Now, using a protractor, measure the 2 angles opposite its equal sides. What do you notice?

In our case,

∠a = 70°

∠b = 70°

The angles opposite the equal sides of the isosceles triangle are equal.

*Angles of an equilateral triangle*

*Angles of an equilateral triangle*

Make an equilateral triangle as below with equal sides.

Now, using a protractor, measure all the angles of the triangle. What do you notice?

∠x = 60°

∠y = 60°

∠z = 60°

All 3 angles of an equilateral triangle are equal. Each angle is 60°.

*Angles of a right-angled triangle*

*Angles of a right-angled triangle*

Consider the right-angled triangle above.

∠p + ∠r + ∠q = 180°

∠p + ∠r + 90° = 180°

∠p + ∠r = 90°

The sum of the other two angles of a right-angled triangle is 90°.

**Summary:**

- The sum of angles of a triangle is 180°.
- Isosceles triangles have two sides that are equal.
- The angles opposite the equal sides in an isosceles triangle are equal.
- All sides of an equilateral triangle are equal.
- All angles of an equilateral triangle are equal. Each angle is 60°.
- A right-angled triangle has one angle that is equal to 90°.
- The sum of the other two angles of a right-angled triangle is 90°.

*Example 1*

*Example 1*

In the triangle below, AB = BC and ∠ABC = 80°. Find the other two angles.

The sum of angles of a triangle is 180°.

180° - 80° = 100°

Since AB = BC, ΔABC is an isosceles triangle, and ∠a = ∠c.100° ÷ 2 = 50°

∠a = 50°

∠c = 50°

*Example 2*

*Example 2*

In triangle PQR below, PQ = QR and ∠QPR = 45°. Is ΔPQR a right-angled triangle?

Since, PQ = QR, ΔPQR is an isosceles triangle, and ∠p = ∠r.∠p = ∠r

∠r = 45°

∠q = 180° - 45° - 45° = 90°

Since, one of the angles of triangle PQR is 90°, ΔPQR is a right-angled triangle.

It is, in fact, a right-angled isosceles triangle (as one of its angles is 90° and two of its sides are equal).