37
Theorem on the sum of the interior angles of a triangle
Textbook: pp. 98–99
GoalKnow and apply the theorem on the sum of interior angles of a triangle.
New words
interior angles · ichki burchaklarangle sum · burchaklar yig‘indisiisosceles triangle · teng yonli uchburchakratio · nisbat
Explanation
Theorem: the sum of the interior angles of a triangle equals 180°, that is ∠A + ∠B + ∠C = 180°. Proof: through A draw a line a parallel to BC. Since a ∥ BC, the alternate interior angle formed by a and AB equals ∠B, and the one formed by a and AC equals ∠C. At A the three angles equal to ∠B, ∠A and ∠C together form a straight angle, so ∠A + ∠B + ∠C = 180°. From this, every triangle has at most one right or obtuse angle. In problems, denote unknown angles by x or by parts of a ratio and set their sum equal to 180°.
Worked examples
In △ABC, if ∠A = 50° and ∠B = 65°, then ∠C = 180° − 50° − 65° = 65°.
The angles of a triangle are in ratio 3 : 4 : 5: 3x + 4x + 5x = 180°, x = 15°; the angles are 45°, 60°, 75°.
Class activity
“Torn corners”: tear the corners off a paper triangle (no scissors) and put them side by side at one point: they form a straight line.
Practice
1
In △ABC, ∠A = 72° and ∠B = 38°. Find ∠C (degrees).
70
2
The apex angle of an isosceles triangle is 40°. Find a base angle (degrees).
70
3
The angles of a triangle are in ratio 1 : 2 : 3. Find the largest (degrees).
90
4
Why can a triangle not have two obtuse angles?
Two obtuse angles already sum to more than 180°, and the third angle is positive.