42
Problem solving
Textbook: pp. 108–109
GoalSolve problems on triangle angles and congruence.
New words
problem solving · masala yechishintersection of bisectors · bissektrisalar kesishishiproving · isbotlashassuming the opposite · teskarisini faraz qilish
Explanation
In problem solving, it is important to draw the figure correctly, mark the signs and decide which theorem applies. The angle sum of 180° solves many calculation problems: for example, if bisectors from A and B meet at D, then ∠ADB = 180° − ∠A : 2 − ∠B : 2. If one angle of an isosceles triangle is given, consider two cases: it may be a base angle or the apex angle. In proofs, find equal triangles and conclude that corresponding sides or angles are equal. Sometimes assuming the opposite is convenient.
Worked examples
In △ABC with ∠A = 60° and ∠B = 80°, the bisectors from A and B meet at D: ∠ADB = 180° − 30° − 40° = 110°.
One angle of an isosceles triangle is 80°: either at the apex (base angles 50° and 50°) or at the base (80°, 80° and apex 20°).
Class activity
“Two cases”: one angle of an isosceles triangle is given; groups find all possible cases.
Practice
1
In △ABC, ∠A = 40°, ∠B = 100°; bisectors from A and B meet at D. Find ∠ADB (degrees).
110
2
One angle of an isosceles triangle is 100°. Find each of the other two (degrees).
40
3
An exterior angle of a triangle is 150° and the non-adjacent interior angles are equal. Find each (degrees).
75
4
Why must a 100° angle of an isosceles triangle be the apex angle?
Two equal base angles of 100° would sum to 200°, more than 180°.