Congruent Figures
Congruent figures are exactly the same shape and size, so one can be moved onto the other by translations, reflections and rotations. This NSW Year 9 Advanced maths subtopic writes congruence statements with the symbol \(\equiv\) and matching vertices (NESA outcome MA5‑GEO‑P‑01).
You will write congruence statements, name matching sides and angles, and use them to find lengths and angles. You will name the transformation between two congruent triangles, decide whether two circles are congruent, find unknowns in congruent quadrilaterals, count tiles for a floor, and test claims about pentagons with equal sides and rectangles with the same perimeter and area. It is part of Year 9 Advanced Congruence and Similar Figures.
Every question with a fully worked solution.
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Theory
Congruent figures are exactly the same shape and the same size. One can be moved onto the other by sliding, turning or flipping it. This Year 9 Advanced maths page shows how to write a congruence statement, match sides and angles, and use algebra to find unknowns. It comes up if you are heading for Extension maths.
Two figures are congruent if one fits exactly on top of the other. They have the same shape and the same size.
A translation (slide), reflection (flip) or rotation (turn) moves a figure without changing any length or angle. So the image is always congruent to the original. An enlargement changes the size, so its image is not congruent (unless the scale factor is 1).
The symbol \(\equiv\) means "is congruent to". In a congruence statement the vertices are written in matching order: \(\triangle ABC \equiv \triangle PQR\) means \(A\) matches \(P\), \(B\) matches \(Q\) and \(C\) matches \(R\).
Corresponding (matching) sides are equal and corresponding angles are equal. Two circles are congruent when their radii are equal.
If \(\triangle ABC \equiv \triangle PQR\):
Transformations about the origin on the number plane:
How to match two congruent figures
- Find the odd one out: if several shapes are given, compare sizes first. A bigger or smaller copy is not congruent.
- Find a stand-out vertex: where the shortest and longest sides meet, the sharpest angle, a right angle or a dent. Match it first.
- Go round in order: walk round both shapes from the matched vertex. If one shape is flipped, walk round it the other way.
- Write the statement: list the vertices in matching order, e.g. \(\triangle ABC \equiv \triangle EDF\).
Finding unknowns
- Read the statement: the 1st letters match, the 2nd letters match, and so on. A side such as \(SP\) joins the 4th and 1st letters.
- Set matching parts equal: an expression equals its matching side or angle. Two expressions that match each other also make an equation.
- Use the angle sum or perimeter to find a part that is not given directly.
- Check: substitute back and make sure the matching parts really are equal.
Is it congruent?
- Transformations: compare coordinates. If every point follows one rule such as \((x, y) \to (-x, y)\), name the transformation; the image is congruent.
- Coordinates: find side lengths by counting squares or with Pythagoras' theorem, then match equal sides.
- Must they be congruent? Find all the measurements the facts fix. If anything can still change, such as an angle, give a counterexample.
(i) Find the length of \(LM\).
(ii) Find the size of \(\angle N\).
The order of the letters gives the matching vertices: \(E \leftrightarrow K\), \(F \leftrightarrow L\), \(G \leftrightarrow M\), \(H \leftrightarrow N\).
(i) \(LM\) joins the 2nd and 3rd letters, so it matches \(FG\): \(LM = \mathbf{8}\) cm.
(ii) \(\angle N\) matches \(\angle H\). \(\angle G\) matches \(\angle M\), so \(\angle G = 86^\circ\). The angles of a quadrilateral add to \(360^\circ\).
| \(\angle H\) | \(=\) | \(360^\circ - 72^\circ - 114^\circ - 86^\circ\) |
| \(\) | \(=\) | \(88^\circ\) |
| \(\angle N\) | \(=\) | \(\mathbf{88^\circ}\) |
(i) Find the coordinates of \(J'\), \(K'\) and \(L'\).
(ii) Show that \(KL = K'L'\) and write a congruence statement.
(i) A \(90^\circ\) anticlockwise turn about \(O\) sends \((x, y)\) to \((-y, x)\).
| \(J(1, 2)\) | \(\to\) | \(J'(-2, 1)\) |
| \(K(4, 2)\) | \(\to\) | \(K'(-2, 4)\) |
| \(L(1, 6)\) | \(\to\) | \(L'(-6, 1)\) |
(ii) Use the distance formula (or Pythagoras on the grid).
| \(KL\) | \(=\) | \(\sqrt{(1 - 4)^2 + (6 - 2)^2} = \sqrt{25} = 5\) |
| \(K'L'\) | \(=\) | \(\sqrt{(-6 + 2)^2 + (1 - 4)^2} = \sqrt{25} = 5\) |
A rotation keeps every length and angle, so \(\mathbf{\triangle JKL \equiv \triangle J'K'L'}\).
(i) Find \(a\) and \(b\).
(ii) Find the perimeter of \(WXYZ\).
(i) Match the sides from the statement: \(AB \leftrightarrow WX\) and \(BC \leftrightarrow XY\). That gives two equations.
| \(2a + b\) | \(=\) | \(11 \quad (1)\) |
| \(a + 2b\) | \(=\) | \(13 \quad (2)\) |
| \((1) \times 2 - (2):\ 3a\) | \(=\) | \(9\) |
| \(a\) | \(=\) | \(\mathbf{3}\) |
| \(b\) | \(=\) | \(11 - 2(3) = \mathbf{5}\) |
Check in (2): \(3 + 2(5) = 13\) \(\checkmark\)
(ii) Congruent figures have equal perimeters.
| \(P\) | \(=\) | \(11 + 13 + 9 + 7\) |
| \(\) | \(=\) | \(\mathbf{40}\text{ cm}\) |
(i) Two rectangles each have a diagonal of \(13\) cm and one side of \(5\) cm. Explain why they must be congruent.
(ii) Two parallelograms each have sides of \(12\) cm and \(5\) cm. Must they be congruent?
(i) The diagonal cuts a rectangle into two right-angled triangles. Find the other side with Pythagoras' theorem.
| \(x^2 + 5^2\) | \(=\) | \(13^2\) |
| \(x^2\) | \(=\) | \(169 - 25 = 144\) |
| \(x\) | \(=\) | \(12\text{ cm}\) |
Both rectangles are \(12\) cm by \(5\) cm with four right angles, so they are congruent.
(ii) No. A parallelogram's angles can change while its sides stay the same. A \(12\) cm by \(5\) cm rectangle and a parallelogram that leans at \(55^\circ\) have the same sides but different angles.
Equal sides alone are not enough for a quadrilateral: the angles must match too.
Common pitfalls
Frequently asked questions
What does congruent mean in maths?
Congruent figures are exactly the same shape and the same size. One can be placed exactly on top of the other after sliding, turning or flipping it.
What is the symbol for congruent?
The symbol is \(\equiv\). For example, \(\triangle ABC \equiv \triangle PQR\) means triangle \(ABC\) is congruent to triangle \(PQR\), with the vertices in matching order.
How do you write a congruence statement?
Find which vertex of one figure matches each vertex of the other. Then write the vertices of both figures in that matching order, joined by \(\equiv\).
Are reflected shapes congruent?
Yes. A reflection flips a shape over but does not change any length or angle, so the image is congruent to the original. Translations and rotations also give congruent images.
What is the difference between congruent and similar?
Congruent figures have the same shape and the same size. Similar figures have the same shape, but one may be an enlargement of the other.
When are two circles congruent?
Two circles are congruent when they have equal radii. Equal diameters, equal circumferences or equal areas all mean equal radii.