35
Problem solving
Textbook: pp. 90–91
GoalMaster methods for solving problems on parallel lines.
New words
setting up an equation · tenglama tuzishpairs of angles · burchaklar juftlariangles with parallel sides · mos tomonlari parallel burchaklarchecking · tekshirish
Explanation
In a problem on parallel lines, first decide whether a ∥ b is given or must be proved. If given, alternate interior and corresponding angles are equal and same-side interior angles add up to 180°; if it must be proved, use a test for parallel lines. Call the unknown angle x and set up an equation. Two angles with respectively parallel sides are either equal (both acute or both obtuse) or add up to 180°. After solving, check the answer: angles must lie between 0° and 180°, and adjacent pairs must add to 180°.
Worked examples
a ∥ b; alternate interior angles 3x + 10° and 5x − 30°: 3x + 10 = 5x − 30, 2x = 40, x = 20°; the angles are 70° and 70°.
∠A and ∠B have respectively parallel sides and ∠A = 40°. If both are acute then ∠B = 40°; if ∠B is obtuse then ∠B = 140°.
Class activity
“Find the mistake”: the teacher writes a wrong solution (e.g. treating same-side interior angles as equal) and pupils find and fix the mistake.
Practice
1
a ∥ b. Alternate interior angles are 4x + 5° and 6x − 25°. Find x.
15
2
a ∥ b. Same-side interior angles are x + 20° and 2x + 10°. Find x.
50
3
Two angles have respectively parallel sides; one is 65° (acute), the other obtuse. Find the obtuse one (degrees).
115
4
Why should we check the solution?
The equation may be set up wrongly or give an impossible angle.