38
Property of an exterior angle of a triangle
Textbook: pp. 100–101
GoalKnow the exterior angle of a triangle and its property.
New words
exterior angle · tashqi burchaknon-adjacent interior angles · qo‘shni bo‘lmagan ichki burchaklarcorollary · natijainterior angle · ichki burchak
Explanation
An angle adjacent to an interior angle of a triangle is an exterior angle. There are two exterior angles at each vertex and they are equal because they are vertical. Theorem: an exterior angle of a triangle equals the sum of the two interior angles not adjacent to it. Proof: for the exterior angle ∠4 and its adjacent interior angle ∠3, ∠3 + ∠4 = 180°, and by the angle sum ∠1 + ∠2 + ∠3 = 180°; hence ∠4 = ∠1 + ∠2. Corollary: an exterior angle is greater than each interior angle not adjacent to it. For example, if the other two angles are 40° and 65°, the exterior angle at the third vertex is 40° + 65° = 105°.
Worked examples
In △ABC with ∠A = 35° and ∠B = 50°, the exterior angle at C is 35° + 50° = 85°.
The exterior angle at A is 120° and ∠B = 45°, so ∠C = 120° − 45° = 75°.
Class activity
“Show the exterior angle”: colour both exterior angles at each vertex of a drawn triangle and check them by measuring.
Practice
1
In △ABC, ∠A = 28° and ∠C = 61°. Find the exterior angle at B (degrees).
89
2
An exterior angle is 130° and one non-adjacent interior angle is 80°. Find the other (degrees).
50
3
An interior angle of a triangle is 72°. Find its adjacent exterior angle (degrees).
108
4
Why is an exterior angle greater than a non-adjacent interior angle?
It equals the sum of the two non-adjacent angles, and the other one is positive.