Quadrilaterals · Lesson 11 of 13
Trapeziums
“Use parallel-side relationships to understand trapeziums and the special isosceles trapezium.”
• Define a trapezium using at least one parallel pair. • Find angles along its non-parallel sides. • Recognise an isosceles trapezium. • Explain why its base angles are equal.
A bridge support or a tapered container may have one short horizontal edge and one longer horizontal edge. Those two edges can be parallel even though the sides joining them slope in different directions. We no longer have the full parallel-side structure of a parallelogram, but one parallel pair still gives us useful angle information. The key is to follow the two sides that connect the parallel edges.
After finding those angle relationships, we will examine the balanced case where the two non-parallel sides have equal lengths. Its symmetry can be explained using congruent right triangles.
Trapezium
A parallelogram has both pairs of opposite sides parallel. Relax that requirement to at least one pair and you obtain the definition used here for a trapezium. The phrase “at least” matters: a parallelogram also passes this test.
For an ordinary trapezium with just one parallel pair, call those parallel sides the bases and the other sides the legs. In ABCD-style naming always state which pair is parallel before choosing an angle equation.
A quadrilateral with at least one pair of parallel opposite sides.
Let PQ ∥ SR. Side PS is a transversal, so ∠P + ∠S = 180°. Side QR is another transversal, giving ∠Q + ∠R = 180°. These are the pairs along the legs, not the two angles along one base.
You can therefore find the upper two angles after measuring the lower two. Do not assume that the two lower angles are equal unless an additional property makes that true.
Problem
PQ ∥ SR, ∠S = 135° and ∠R = 105°. Find P and Q.
A trapezium whose non-parallel sides have equal lengths. The two angles at either base are equal.
Construct UVWX with UV ∥ XW and equal legs UX = VW. Draw perpendiculars XY and WZ to the longer base UV, with Y and Z on that base. The middle quadrilateral XWZY has four right angles, so it is a rectangle. Consequently XY = WZ.
Now compare right triangles UXY and VWZ. Their hypotenuses UX and VW are equal, and the legs XY and WZ are equal. By the right-angle–hypotenuse–side congruence condition, the triangles are congruent. Hence ∠U = ∠V.
Each upper angle is supplementary to its lower neighbour. Since those lower angles are equal, the upper pair is equal as well. This is the precise meaning of equal base angles: the two endpoints of the same base have equal angles.
Problem
One upper angle is 100°. Find the other three angles.
Problem
In isosceles trapezium UVWX with UV ∥ XW, ∠U = 65°. Find V, W and X.
Here “trapezium” means at least one parallel pair. The isosceles case above has one parallel pair and two equal non-parallel legs; do not use the mere equality of opposite sides in a slanted parallelogram to claim equal base angles.
Quiz
A trapezium has:
If PQ ∥ SR and ∠P = 120°, then ∠S =
Which condition defines the isosceles trapezium studied here?
In an isosceles trapezium, matching base angles are:
If one base angle of an isosceles trapezium is 65°, its adjacent angle on the same leg is:
Practice Problems
- In trapezium PQRS, PQ ∥ SR and ∠P = 105°. Find S.
- An isosceles trapezium has lower base angles 85° and 85°. Find the upper angles.
- In PQRS, PQ ∥ SR, ∠P = 100° and ∠Q = 110°. Find R and S.
- Explain why an isosceles trapezium has equal base angles.
- Does every parallelogram satisfy our trapezium definition? Does every trapezium have to be a parallelogram?
1. P and S are same-side interior angles along PS. 2. S = 180° − 105° = 75°.
1. Each upper angle is supplementary to its lower neighbour. 2. Both are 180° − 85° = 95°.
1. Along QR, R = 180° − 110° = 70°. 2. Along PS, S = 180° − 100° = 80°. 3. The boundary order is P = 100°, Q = 110°, R = 70°, S = 80°.
1. Drop perpendiculars from the shorter base to the longer base. They are equal opposite sides of the middle rectangle. 2. The outer right triangles have equal hypotenuses, the equal legs of the trapezium, and equal perpendicular heights. 3. They are congruent by RHS, so the corresponding lower base angles are equal.
1. Yes to the first: two parallel pairs include at least one parallel pair. 2. No to the second: a trapezium can have exactly one parallel pair. 3. A drawing with unequal parallel bases supplies an example that is not a parallelogram.
Key Takeaways
• A trapezium has at least one pair of parallel opposite sides. • Angles at the ends of each leg total 180°. • An isosceles trapezium has equal non-parallel sides. • Its angles along the same base are equal. • Congruent right triangles explain that equality. • State the parallel pair and the definition convention before classifying.