Congruent Figures
Two figures are congruent when one can be moved to sit exactly on top of the other: same shape and same size. This NSW Year 8 maths subtopic goes beyond Stage 4, since the NSW syllabus places congruence in Stage 5, so it has no Stage 4 outcome of its own.
You will see that a slide, flip or turn keeps a figure congruent but an enlargement does not, write statements such as \(\triangle ABC \equiv \triangle PQR\) with the vertices in matching order, and read off matching sides and angles. You will also find pronumerals and perimeters, show that circles with equal radii are congruent, and explain why two floor tiles with the same area need not be. It is part of Year 8 Congruent Figures.
Every question with a fully worked solution.
- Congruent Figures - Video - What are Congruent Figures? Watch
Theory
Two figures are congruent when one can be moved to sit exactly on top of the other: the same shape and the same size. Write \(\triangle ABC \equiv \triangle PQR\) with the vertices in matching order, then read off the equal parts: \(AB = PQ\), \(\angle B = \angle Q\). This Year 8 maths page shows how.
A slide, a flip or a turn keeps a figure congruent. A translation (slide), a reflection (flip) and a rotation (turn) change where a figure sits, but not one length and not one angle. Enlarging or reducing it does not give a congruent figure.
The symbol is \(\equiv\). \(\triangle ABC \equiv \triangle PQR\) is read “triangle \(ABC\) is congruent to triangle \(PQR\)”.
Name the vertices in matching order. The first letter of one name matches the first letter of the other, the second the second, and so on: \(A\) with \(P\), \(B\) with \(Q\), \(C\) with \(R\).
Then read the equal parts off the statement: \(AB = PQ\), \(BC = QR\), \(CA = RP\), \(\angle A = \angle P\), \(\angle B = \angle Q\), \(\angle C = \angle R\). Matching parts are also called corresponding parts.
Two circles are congruent when their radii are equal. Change a diameter to a radius first.
Congruent figures have equal perimeters and equal areas. The reverse is not true: two figures with the same area need not be congruent.
No length changes, so a missing matching side can be found with Pythagoras’ theorem, and angle sums finish any angle chase.
Reading a congruence statement \(\triangle ABC \equiv \triangle PQR\):
| Position | Vertices | Equal parts |
|---|---|---|
| 1st | \(A \leftrightarrow P\) | \(\angle A = \angle P\) |
| 2nd | \(B \leftrightarrow Q\) | \(\angle B = \angle Q\) |
| 3rd | \(C \leftrightarrow R\) | \(\angle C = \angle R\) |
| 1st and 2nd | \(AB \leftrightarrow PQ\) | \(AB = PQ\) |
| 2nd and 3rd | \(BC \leftrightarrow QR\) | \(BC = QR\) |
| 3rd and 1st | \(CA \leftrightarrow RP\) | \(CA = RP\) |
Writing a congruence statement
- Find a part that stands out (a right angle, the shortest side, a marked angle) in both figures.
- Match the vertices around both figures in the same direction.
- Write the names with matching vertices in the same positions.
Using a congruence statement
- Match letters by position: 1st with 1st, 2nd with 2nd, and so on.
- Copy the known length or angle to its matching part, then solve any equation.
(i) Find \(KL\).
(ii) Find \(\angle L\).
(iii) Name the side equal to \(LM\).
Match the vertices: \(A \leftrightarrow J\), \(B \leftrightarrow K\), \(C \leftrightarrow L\), \(D \leftrightarrow M\).
(i) \(KL\) matches \(BC\): \(14\) cm
(ii) \(\angle L\) matches \(\angle C\): \(118^\circ\)
(iii) \(LM\) matches \(CD\)
The second trapezium is the first after a half turn. The short side is \(FG\) in the first and \(ZW\) in the second, and the long side is \(EH\) and \(XY\). Going round both the same way: \(E \leftrightarrow Y\), \(F \leftrightarrow Z\), \(G \leftrightarrow W\), \(H \leftrightarrow X\).
\(EFGH \equiv YZWX\), so \(FG = ZW\).
Both signs change at every vertex: \((2, 1) \to (-2, -1)\), \((2, 3) \to (-2, -3)\), \((4, 3) \to (-4, -3)\). That is a rotation of \(180^\circ\) about the origin.
Rotation of \(180^\circ\) about the origin; \(\triangle ABC \equiv \triangle DEF\); \(DE = AB = 2\) units.
(i) Find \(b\).
(ii) Find the perimeter of \(\triangle XYZ\).
| \(2b + 5\) | \(=\) | \(15\) |
| \(b\) | \(=\) | \(5\) |
| \(KM^2\) | \(=\) | \(15^2 + 8^2 = 289\) |
| \(KM\) | \(=\) | \(17~\text{cm}\) |
| \(15 + 8 + 17\) | \(=\) | \(40\) |
(i) \(b = 5\) (\(KL\) matches \(XY\))
(ii) \(40\) cm: congruent triangles have equal perimeters.
| \(\angle F\) | \(=\) | \(180^\circ - 52^\circ - 63^\circ = 65^\circ\) |
\(\angle T\) matches \(\angle F\), so \(\angle T = 65^\circ\).
(ii) A triangle with base \(12\) cm and height \(4\) cm and a triangle with base \(8\) cm and height \(6\) cm have equal areas. Are they congruent?
| \(\tfrac{1}{2} \times 12 \times 4\) | \(=\) | \(24\) |
| \(\tfrac{1}{2} \times 8 \times 6\) | \(=\) | \(24\) |
(i) Yes: the radius of the first is \(12 \div 2 = 6\) cm.
(ii) No: both areas are \(24~\text{cm}^2\), but the sides are different, so no turn or flip puts one on the other.
Common pitfalls
Frequently asked questions
What does congruent mean?
Exactly the same shape and size, so one figure fits exactly on top of the other.
What is the symbol for congruent?
\(\equiv\), as in \(\triangle ABC \equiv \triangle DEF\).
Why does the order of the letters matter?
Letters in the same position name matching vertices, so the order tells you which sides and angles are equal.
Do reflections give congruent figures?
Yes. Translations, reflections and rotations all keep every length and angle.
Are two circles with the same radius congruent?
Yes. Circles are congruent exactly when their radii are equal.