TRIANGLES: Finding Unknown Angles

Properties of Triangles

Following are the important properties of triangles:

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

We will prove some of these properties in the questions below.

1. Angles of an isosceles triangle

Make an isosceles triangle as below.
angles in an isosceles triangle
Now, using a protractor, measure the 2 angles opposite its equal sides. What do you notice?

In the figure above,
∠a  =  70°
∠b  =  70°

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

2. Angles of an equilateral triangle

Make an equilateral triangle as below with equal sides.
angles in an equilateral triangle
Now, using a protractor, measure all the angles of the triangle. What do you notice?

In the figure above,
∠x  =  60°
∠y  =  60°
∠z  =  60°

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

3. Angles of a right-angled triangle

angles in 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°.

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

finding angles in a triangle
180°  −  80°  =  100°
100°  ÷  2  =  50°

∠a  =  50°
∠c  =  50°

The sum of angles of a triangle is 180°.
Since AB = BC, ΔABC is an isosceles triangle.
So, ∠a = ∠c.

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

finding angles in a triangle
∠p  =  ∠r
∠r  =  45°
∠q  =  180°  −  45°  −  45°  =  90°
Since, PQ = QR, ΔPQR is an isosceles triangle, and ∠p = ∠r.
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).