39
Problem solving
Textbook: pp. 102–103
GoalSolve problems with the angle sum and exterior angle.
New words
quadrilateral · to‘rtburchaksum 360° · yig‘indi 360°ratio · nisbatequation · tenglama
Explanation
In problems we use the figure to choose which triangle to work on. In an isosceles triangle the base angles are equal, so one known angle gives the others. Vertical and adjacent angles help. The angles of a quadrilateral add up to 360°: a diagonal splits it into two triangles, giving 180° + 180° = 360°. If angles are given as a ratio, write them as kx and equate the sum. At each step, note which theorem you used.
Worked examples
If three angles of a quadrilateral are 80°, 95° and 100°, the fourth is 360° − 275° = 85°.
In isosceles △ABC (AB = BC) with ∠B = 100°, ∠A = ∠C = (180° − 100°) : 2 = 40°.
Class activity
“Diagonal”: draw a quadrilateral, split it with a diagonal and compare the sum of its angles with 360°.
Practice
1
A quadrilateral has angles 70°, 110°, 90° and x. Find x (degrees).
90
2
The angles of a quadrilateral are in ratio 1 : 2 : 3 : 4. Find the largest (degrees).
144
3
Two angles of a triangle are 55° and 75°. Find the exterior angle adjacent to the third angle (degrees).
130
4
Why do the angles of a quadrilateral add up to 360°?
A diagonal splits it into two triangles with 180° each.