The angle sum and the exterior angle of a triangle
Mathematics Prep 1 Term 1, Unit 4, Lessons 2–3
GoalUse the angle sum and the exterior angle to find missing angles of a triangle.
New words
angle sum · the three angles of any triangle add up to 180°exterior angle · the angle between one side and the extension of another sideisosceles triangle · a triangle with two equal sides, so two equal base anglesequilateral triangle · a triangle with three equal sides and three angles of 60°
Explanation
The three angles of every triangle add up to 180°. To see why, draw a line through one vertex parallel to the opposite side: the alternate angles equal the other two angles of the triangle, and together with the third angle they make a straight angle. If you extend one side, the angle outside the triangle is an exterior angle. It is supplementary to the interior angle next to it, so it equals the sum of the two interior angles that are not next to it. In a right triangle the two acute angles add up to 90°. In an isosceles triangle the angles opposite the equal sides are equal.
★Remember
Angle sum of a triangle = 180°. Exterior angle = sum of the two remote interior angles.
Worked examples
Two angles known. A triangle has angles 50° and 65°. The third angle is 180° − (50° + 65°) = 180° − 115° = 65°.
Isosceles. The angle between the equal sides is 40°. The two base angles are equal and add up to 180° − 40° = 140°, so each is 70°.
Exterior angle. Side BC is extended to D. The remote interior angles are 60° and 45°, so the exterior angle x = 60° + 45° = 105°. Check: the interior angle at C is 180° − 60° − 45° = 75° and 180° − 75° = 105°.
With an equation. The angles are (x + 10)°, 2x and (3x − 10)°. Their sum is 180: x + 10 + 2x + 3x − 10 = 180, so 6x = 180 and x = 30. The angles are 40°, 60° and 80°. Check: 40 + 60 + 80 = 180.
Class activity
Tear the corners. With a grown-up, cut a paper triangle, tear off its three corners and place them side by side with the vertices together. They form a straight line. Which angle fact does this show?
Practice
1
A triangle has angles 72° and 38°. Find the third angle.
180° − (72° + 38°) = 180° − 110° = 70°.
2
In a right triangle one acute angle is 35°. Find the other.
90° − 35° = 55°.
3
An isosceles triangle has a base angle of 50°. Find the angle at the top.
The two base angles are 50° + 50° = 100°, so the top angle is 180° − 100° = 80°.
4
An exterior angle of a triangle is 120° and one remote interior angle is 45°. Find the other remote interior angle.
120° − 45° = 75°.
5
Hesham says a triangle can have angles 90°, 60° and 50°. Is he right?
No. 90 + 60 + 50 = 200, but the angles of a triangle add up to 180°.
