Isosceles trapezium
Equal leaning sides give equal angles.
Equal leaning sides
In a trapezium PQRS, PQ ∥ SR. The leaning sides PS and QR can be any lengths. Now make the two leaning sides equal: PS = QR.
What do you think happens to ∠P and ∠Q, the angles at the two ends of the side PQ?
What this lesson covers
The idea
A trapezium whose non-parallel sides are equal is an isosceles trapezium, and in it the angles opposite to the equal sides are equal.
Equal leaning sides
In a trapezium PQRS, PQ ∥ SR. The leaning sides PS and QR can be any lengths. Now make the two leaning sides equal: PS = QR.
What do you think happens to ∠P and ∠Q, the angles at the two ends of the side PQ?
- They become equal
- They add up to 180°
- They can still be anything
Make the sides equal
PQ and SR lie on parallel rails, and Q stays put. Drag R and S until the two leaning sides PS and QR are equal. Then look at the angles. Do it 3 times.
Equal sides, equal angles
A trapezium whose non-parallel sides are equal is an isosceles trapezium, and in it the angles opposite to the equal sides are equal.
Whenever the leaning sides were equal and not parallel, the angles at the two ends of each parallel side matched: ∠P = ∠Q and ∠S = ∠R.
Equal leaning sides that were also parallel made a parallelogram, and there ∠P ≠ ∠Q. That is why the rule asks for the non-parallel sides to be equal.
Why? Drop the perpendiculars SY and RZ to PQ. Then SRZY is a rectangle, so SY = RZ.
The right triangles PYS and QZR have equal hypotenuses (PS = QR) and SY = RZ. So they are congruent (RHS), and ∠P = ∠Q.
The angles at the ends of each leaning side add up to 180°, so ∠S = 180° − ∠P and ∠R = 180° − ∠Q. Hence ∠S = ∠R too.
Notes
A trapezium whose non-parallel sides are equal is an isosceles trapezium, and in it the angles opposite to the equal sides are equal.
Check yourself
In isosceles trapezium PQRS, PQ ∥ SR and PS = QR. ∠P = 65°. How large is ∠Q?
Answer: 65 °
PS = QR, so the angles at the two ends of PQ are equal: ∠Q = ∠P = 65°.
In isosceles trapezium PQRS (PQ ∥ SR, PS = QR), ∠S = 110°. How large is ∠Q?
Answer: 70 °
∠R = ∠S = 110°. The angles at the ends of QR add up to 180°, so ∠Q = 180° − 110° = 70°.
In quadrilateral PQRS, PQ ∥ SR and PS = QR. Which is true?
In isosceles trapezium PQRS (PQ ∥ SR, PS = QR), ∠P + ∠Q = 130°. How large is ∠S?
Answer: 115 °
∠P = ∠Q, so each is 130° ÷ 2 = 65°. The angles at the ends of PS add up to 180°, so ∠S = 180° − 65° = 115°.
- If PS is not parallel to QR, it is an isosceles trapezium — correct. Yes! The leaning sides must be equal and not parallel. If they are parallel too, the shape is a parallelogram.
- It is always an isosceles trapezium, even if PS ∥ QR. If PS ∥ QR as well, the shape is a parallelogram, and its angles ∠P and ∠Q are not equal in general.
- It can never be an isosceles trapezium. It can: when PS and QR are equal and not parallel, it is an isosceles trapezium.