Secondary 2 · Mathematics · Space
Congruence and quadrilaterals
Identify the conditions that show two triangles are congruent, and use them to explore the properties of parallelograms, rhombuses, rectangles and squares.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026. How we review
These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Understand what it means for two shapes to be congruent and match corresponding parts.
- Identify the four tests for congruent triangles: SSS, SAS, AAS and RHS.
- Use congruent triangles to explain properties of parallelograms, rhombuses, rectangles and squares.
Success looks like
- I can name the matching sides and angles of two congruent triangles.
- I can choose the correct congruence test from the information given, and explain why AAA is not a test.
- I can state the side, angle and diagonal properties of special quadrilaterals.
The big idea
Two shapes are congruent if they are exactly the same shape and size. We write △ABC ≡ △XYZ, and the order of the letters shows which vertices match: A with X, B with Y and C with Z. So side AB matches XY, BC matches YZ, and ∠B matches ∠Y.
We don't need to check all six sides and angles. Any one of these four tests is enough to prove two triangles are congruent:
- SSS: three pairs of equal sides.
- SAS: two pairs of equal sides and the included angle (the angle between those sides) is equal.
- AAS: two pairs of equal angles and one pair of matching sides are equal.
- RHS: both have a right angle, the hypotenuses are equal and one other pair of sides is equal.
Some combinations are not tests. AAA shows the triangles are the same shape but they could be different sizes. SSA (with the angle not between the sides) can produce two different triangles.
Congruent triangles explain the properties of special quadrilaterals. Draw a diagonal in a parallelogram: it splits the shape into two congruent triangles (using alternate angles on the parallel sides and the shared diagonal). This proves that opposite sides are equal and opposite angles are equal in a parallelogram.
Key quadrilateral properties to know:
- Parallelogram: opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.
- Rhombus: a parallelogram with all sides equal; diagonals bisect each other at right angles and bisect the angles.
- Rectangle: a parallelogram with four right angles; diagonals are equal in length and bisect each other.
- Square: has all the properties of a rhombus and a rectangle.
Worked examples
Choosing a test
In △ABC, AB = 5 cm, BC = 7 cm and ∠B = 40°. In △PQR, PQ = 5 cm, QR = 7 cm and ∠Q = 40°. Are the triangles congruent? If so, which side matches AC?
- Two pairs of sides are equal: AB = PQ and BC = QR.
- The angle between them is equal: ∠B = ∠Q. This is the included angle.
- So the triangles are congruent by SAS: △ABC ≡ △PQR.
- AC is opposite ∠B, so it matches PR, the side opposite ∠Q.
Answer: Yes, congruent by SAS, and AC = PR.
Why AAA is not enough
Two triangles both have angles of 50°, 60° and 70°. Must they be congruent?
- Draw a triangle with these angles and a 4 cm base, and another with an 8 cm base.
- Both have the same angles, but the second is twice as big.
Answer: No. They are the same shape (similar) but not necessarily the same size, so AAA is not a congruence test.
Opposite sides of a parallelogram
ABCD is a parallelogram with diagonal AC. Explain why AB = CD.
- AB ∥ DC, so ∠BAC = ∠DCA (alternate angles, parallel lines).
- AD ∥ BC, so ∠BCA = ∠DAC (alternate angles, parallel lines).
- AC is a side of both triangles (a common side).
- So △ABC ≡ △CDA by AAS (two angles and a matching side).
- AB and CD are matching sides of congruent triangles, so they are equal.
Answer: AB = CD because △ABC ≡ △CDA (AAS).
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Which of these is NOT a test for congruent triangles?
AAA — AAA only shows the triangles are the same shape. They could be different sizes.
Two triangles have all three pairs of matching sides equal. Which test shows they are congruent?
SSS — Three pairs of equal sides is the side–side–side (SSS) test.
Two right-angled triangles have equal hypotenuses and one other pair of equal sides. Which test applies?
RHS — Right angle, hypotenuse and one side is the RHS test.
Which of these quadrilaterals always has diagonals of equal length?
Rectangle — A rectangle's diagonals are always equal. A parallelogram, rhombus or kite usually has unequal diagonals.
In which quadrilateral do the diagonals always bisect each other at right angles?
Rhombus — A rhombus's diagonals bisect each other at 90°. A rectangle's diagonals only do this if it is a square.
One angle of a parallelogram is 70°. What is the size of the angle opposite it, in degrees?
70° — Opposite angles of a parallelogram are equal.
One angle of a parallelogram is 70°. What is the size of an angle next to it (an adjacent angle), in degrees?
110° — Adjacent angles in a parallelogram are co-interior angles between parallel sides, so they add to 180°: 180 − 70 = 110°.
△ABC ≡ △XYZ. AB = 6 cm, BC = 8 cm and AC = 9 cm. How long is YZ?
8 cm — The letter order shows B matches Y and C matches Z, so YZ matches BC = 8 cm.
Watch out for
- Matching sides by position on the page instead of by the letter order. Use the congruence statement (△ABC ≡ △XYZ) to pair the letters.
- Using SAS when the angle is not between the two sides. Have the learner trace the two sides with a finger — the angle must be where they meet.
- Thinking a square is not a rectangle or a rhombus. A square meets every requirement of both, so it is a special case of each.
Tips for parents
- Cut out two copies of a triangle and flip or turn one to show that congruent shapes can face different ways.
- Build triangles from straws of fixed lengths: with three set lengths only one triangle is possible (SSS), which is why triangles make strong structures.
- Make a table of quadrilateral properties together and tick which shapes have each property.
Go further
- Try to build two different triangles from the SSA information AB = 6 cm, BC = 4 cm, ∠A = 30° using a ruler and compasses.
- Prove that the diagonals of a parallelogram bisect each other, using a pair of congruent triangles.
- Look for congruent shapes in building frames, bridges, tiles and fabric patterns, and explain why triangles are used for strength.
About 50 minutes
You'll need
- Ruler, protractor and compasses
- Paper and scissors
- Straws or strips of card cut to set lengths (optional)
Key words
- Congruent
- Exactly the same shape and size. One shape could be placed on top of the other to match perfectly (it may need to be flipped or turned).
- Corresponding parts
- Sides or angles in the same position in two congruent shapes.
- SSS
- Side–side–side: all three pairs of sides are equal.
- SAS
- Side–angle–side: two pairs of sides and the angle between them are equal.
- AAS
- Angle–angle–side: two pairs of angles and a pair of matching sides are equal (also called ASA when the side is between the angles).
- RHS
- Right angle–hypotenuse–side: both triangles have a right angle, equal hypotenuses and one other pair of equal sides.
Still tricky?
For parents
Want an outside view of where your child is at?
Book a free, no-obligation English and maths assessment at a Success Tutoring centre and get a clear starting point.
Book a free assessmentRelated space lessons
Build up to it (Secondary 1)
Next steps (Secondary 3)