1
Triangles and quadrilaterals
Textbook: pp. 6–8
GoalRecall the main properties of triangles and quadrilaterals (angle sums, exterior angle, kinds of parallelogram, midline) and solve angle-finding problems.
New words
exterior angle · tashqi burchakmidline · o‘rta chiziqangle bisector · bissektrisaparallelogram · parallelogramm
Explanation
The interior angles of a triangle add up to 180°, and those of a convex quadrilateral add up to 360°. An exterior angle of a triangle equals the sum of the two interior angles not adjacent to it. In an isosceles triangle the base angles are equal. In a parallelogram opposite sides and angles are equal, adjacent angles add up to 180°, and the diagonals bisect each other. The midline of a triangle is parallel to the third side and half as long. In angle problems it helps to draw a figure first, mark the known angles and decide which property is needed.
Worked examples
In triangle ABC, ∠A = 62° and ∠B = 71°. Then ∠C = 180° − (62° + 71°) = 47°. The exterior angle at C is 180° − 47° = 133°, which agrees with the sum of ∠A and ∠B (62° + 71° = 133°).
If one angle of a parallelogram is 70°, the adjacent angle is 180° − 70° = 110°. Opposite angles are equal, so the angles are 70°, 110°, 70°, 110°.
Class activity
“Angle chain”: in pairs, one student gives one angle in a figure; the partner finds the next angle and names the property used. Then swap roles.
Practice
1
Two angles of a triangle are 48° and 77°. What is the third angle (degrees)?
55
2
Three angles of a quadrilateral are 95°, 80° and 110°. What is the fourth angle (degrees)?
75
3
The apex angle of an isosceles triangle is 40°. What is a base angle (degrees)?
70
4
Why do adjacent angles of a parallelogram add up to 180°?
Opposite sides are parallel, and adjacent angles are co-interior angles formed by two parallel lines and a transversal.