Week 13 · Quadrilaterals and polygons: angle sums, special quadrilaterals, interior angles of polygons · 3 lessons
1
Angles of a quadrilateral and the trapezium
Mathematics Prep 1 Term 1, Unit 4, Lessons 4–5
GoalUse the angle sum of a quadrilateral and the properties of a trapezium to find angles.
New words
quadrilateral · a polygon with four sides and four anglestrapezium · a quadrilateral with exactly one pair of parallel sidesbases and legs · the parallel sides are the bases; the other two sides are the legsisosceles trapezium · a trapezium whose legs are equal; its base angles are equal
Explanation
The four angles of any quadrilateral add up to 360°. To see why, draw one diagonal: it splits the quadrilateral into two triangles and each has an angle sum of 180°, so 2 × 180° = 360°. A trapezium has exactly one pair of parallel sides, called the bases; the other two sides are the legs. The two angles on one leg are same-side interior angles between parallel lines, so they add up to 180°. A trapezium with equal legs is isosceles: its base angles are equal in pairs. A trapezium with a leg perpendicular to the bases is a right trapezium and has two right angles.
Worked examples
Angle sum 360°. Three angles of a quadrilateral are 85°, 100° and 70°. They add up to 255°, so x = 360° − 255° = 105°.
Isosceles trapezium. AB is parallel to DC and the legs are equal. The base angles at D and C are 70°. The angles at A and B are 180° − 70° = 110° each. Check: 70 + 70 + 110 + 110 = 360.
Right trapezium. AB is parallel to DC and angles A and D are 90°. Angle B = 115°, so angle C = 180° − 115° = 65°, because B and C are on the same leg. Check: 90 + 90 + 115 + 65 = 360.
With x. In trapezium ABCD, AB is parallel to DC, angle A = (2x + 10)° and angle D = (x + 20)°. They are on the same leg, so 2x + 10 + x + 20 = 180, 3x = 150 and x = 50. Angle A = 110° and angle D = 70°.
Class activity
Cut a quadrilateral. With a grown-up, draw any quadrilateral on paper, colour its four corners and cut them off. Put the four corners together round one point. They fill the whole turn. Which angle sum does this show?
Practice
1
Three angles of a quadrilateral are 90°, 75° and 110°. Find the fourth.
90 + 75 + 110 = 275, so the fourth is 360° − 275° = 85°.
2
In trapezium ABCD, AB is parallel to DC and angle D = 65°. Find angle A.
A and D are on the same leg: 180° − 65° = 115°.
3
One angle of an isosceles trapezium is 80° at a base. Find all four angles.
Two angles are 80° (the same base) and two are 180° − 80° = 100°. Check: 80 + 80 + 100 + 100 = 360.
4
The angles of a quadrilateral are x, 2x, 3x and 4x. Find them.
10x = 360, so x = 36. The angles are 36°, 72°, 108° and 144°.
5
Salma says a trapezium has two pairs of parallel sides. What is wrong?
A trapezium has exactly one pair of parallel sides. A shape with two pairs is a parallelogram.
