Quadrilaterals · Ages 14–15
Congruent quadrilaterals
Four equal sides aren't enough; four sides and a diagonal are.
How to do it
Method
Four equal sides aren't enough; four sides and a diagonal are
A 5 cm square and a 5 cm rhombus: equal sides, not congruent
Another way
Two triangles along a diagonal
AB, BC, AC equal and CD, DA, AC equal: SSS twice
In full
Two quadrilaterals are congruent when all four sides and all four angles match, so one fits exactly on the other; writing ABCD ≅ PQRS puts matching vertices in order, so AB = PQ and ∠C = ∠R. Three equal sides fix a triangle (SSS), but four don't fix a quadrilateral: a square and a rhombus with 5 cm sides have equal sides and different shapes. Exploring: four sides and a diagonal are enough, since the diagonal cuts each into two triangles, each SSS; four sides and an angle are enough too (two rhombuses with the same side and the same angle); four angles alone aren't (rectangles can be long or short). The method: cut each along a diagonal into two triangles and check those.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A square with 3 cm sides and a square with 5 cm sides are congruent.
True or false?
Show the answer and working
Answer: False
A square's angles are all 90°; two squares with the same side fit exactly, with different sides they don't.
Example 2
Two rhombuses both have 9 cm sides. One of them has an angle of 40°. For them to be congruent, the other's largest angle =
Show the answer and working
Answer: 140°
The first rhombus has angles 40° and 180° − 40° (int. ∠s); congruent means the same angles, the largest 140°.
Example 3
ABCD ≅ PQRS, with ∠P = 80°, ∠B = 90° and ∠R = 65°. ∠D =
Show the answer and working
Answer: 125°
∠A = ∠P = 80° and ∠C = ∠R = 65° (matching angles); ∠D = 360° − 80° − 90° − 65° = 125° (∠ sum of polygon).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A square with 8 cm sides and a square with 8 cm sides are congruent.
True or false?
Answer:
2.Which pair of quadrilaterals must be congruent?
- A. two squares with 6 cm sides
- B. two rhombuses with 6 cm sides
- C. two rectangles both 6 cm long
- D. two rectangles each with a perimeter of 18 cm
Choose the right answer
Answer:
3.Two rhombuses both have 3 cm sides. One of them has an angle of 135°. For them to be congruent, the other's largest angle =
Answer:
4.The parallelogram ABCD ≅ the parallelogram PQRS, and ∠A = 75°. ∠R =
Answer:
5.ABCD ≅ PQRS, with ∠P = 75°, ∠B = 100° and ∠R = 85°. ∠D =
Answer:
All the answers
- 1. True
- 2. A. two squares with 6 cm sides
- 3. 135°
- 4. 75°
- 5. 100°
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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