Quadrilaterals · Ages 14–15
Rectangles and squares
Rectangle: equal diagonals, AE = BE = CE = DE; square: diagonals at 45° to the sides.
How to do it
Method
Rectangle: equal diagonals, AE = BE = CE = DE; square: diagonals at 45° to the sides
∠EAB = 35°: ∠EBA = 35°, ∠AEB = 180° − 70° = 110°
Another way
Use the right angle
∠EBC = 90° − 35° = 55°
In full
A rectangle is a parallelogram whose four angles are right angles, so it has every property of a parallelogram and one more: its diagonals are equal (prop. of rectangle). Equal diagonals that bisect each other give AE = BE = CE = DE, so △ABE, △BCE and the rest are isosceles. ∠EAB = 35° gives ∠EBA = 35° (base ∠s, isos. △) and ∠AEB = 180° − 2 × 35° = 110°. A square is a rectangle with four equal sides: its diagonals are equal, at right angles, bisect each other and bisect each right angle, so each makes 45° with a side (prop. of square). Another way: use the right angle. ∠ABC = 90°, so ∠EBC = 90° − 35° = 55°, the same result.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
ABCD is a rectangle whose diagonals cross at E, and AE = 8 cm. x =
Show the answer and working
Answer: 8 cm
A rectangle's diagonals are equal and bisect each other, so AE = BE = CE = DE: x = 8 (prop. of rectangle).
Example 2
The parallelogram ABCD has diagonals of 20 cm and 20 cm. ABCD is a rectangle.
True or false?
Show the answer and working
Answer: True
A parallelogram is a rectangle exactly when its diagonals are equal; 20 cm and 20 cm must match.
Example 3
The rectangle ABCD's diagonals cross at E, and ∠AEB = 94°. ∠ACB =
Show the answer and working
Answer: 47°
∠BEC = 180° − 94° = 86° (adj. ∠s on st. line); EB = EC (prop. of rectangle), so ∠ECB = (180° − 86°) ÷ 2 = 47°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.ABCD is a square. x =
Answer:
2.ABCD is a rectangle whose diagonals cross at E, and AE = 8 cm. x =
Answer:
3.The parallelogram ABCD has diagonals of 16 cm and 18 cm. ABCD is a rectangle.
True or false?
Answer:
4.ABCD is a rectangle whose diagonals cross at E. ∠EBA = °, ∠AEB = °, ∠EBC = °
Fill in each blank
Answer:
5.The rectangle ABCD's diagonals cross at E, with AE = (3y + 6) cm and BD = (8y + 6) cm. y =
Answer:
All the answers
- 1. 45°
- 2. 8 cm
- 3. False
- 4. 50, 80, 40
- 5. 3
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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