TRIANGLES: Finding Unknown Angles
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
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
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°.
- 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°.
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°
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).