Congruent Triangles
Every question with a fully worked solution.
- Congruent Triangles - Video - Congruent Triangle Rules Watch
Theory
Two triangles are congruent if they are exactly the same shape and size. This Year 10A maths page covers the four congruence tests (SSS, SAS, AAS and RHS), writing a congruence statement in matching order, and using congruent triangles to find unknown sides and angles and to prove that lengths or angles are equal.
Two figures are congruent if one can be moved exactly on top of the other by translating, reflecting or rotating it. All matching sides and all matching angles are equal. The symbol is \(\equiv\).
To show two triangles are congruent you do not need all six facts. Any one of the four congruence tests is enough:
SSS — three pairs of equal sides. SAS — two pairs of equal sides and the included angle (the angle between them). AAS — two pairs of equal angles and a pair of matching sides. RHS — a right angle, the hypotenuse and one other side.
In a congruence statement such as \(\triangle PQR \equiv \triangle LMN\), the vertices are written in matching order: \(P \leftrightarrow L\), \(Q \leftrightarrow M\), \(R \leftrightarrow N\). Then \(PQ = LM\), \(\angle Q = \angle M\), and so on.
Three equal angles (AAA) is not a test: the triangles have the same shape but may be different sizes. If no test fits the information, the triangles cannot be shown to be congruent.
| Test | What must be equal | Watch for |
|---|---|---|
| SSS | all three pairs of sides | a common side counts as one pair |
| SAS | two pairs of sides and the angle between them | the angle must be between the two sides |
| AAS | two pairs of angles and one pair of matching sides | the sides must be in the same position (opposite the same angle) |
| RHS | a right angle, the hypotenuse and one other side | only for right-angled triangles |
How to prove two triangles are congruent
- Name the two triangles you are comparing.
- List three pairs of equal parts, one per line, each with a reason: given, common, vertically opposite, alternate angles, radii, midpoint.
- Name the test (SSS, SAS, AAS or RHS) that the three facts make.
- Write the congruence statement with the vertices in matching order.
- Conclude what you were asked: a side, an angle or a parallel line, using "matching parts of congruent triangles".
What a congruence proof can show
- Two lengths are equal, such as a point being a midpoint, or two sides of a triangle being equal (so the triangle is isosceles).
- Two angles are equal, such as a line bisecting an angle.
- A right angle: two matching angles that are equal and also make a straight line are each \(90^\circ\).
- Parallel lines: equal alternate (or corresponding) angles show the lines are parallel.
Finding an unknown
- Match the vertices from the statement or from the equal marks.
- Set matching sides or angles equal. If they are written with a pronumeral, solve the equation.
- Finish with another rule if needed: the angle sum of a triangle, Pythagoras' theorem, or adding sides for a perimeter.
The \(72^\circ\) angle sits between the \(6\) cm and \(10\) cm sides in both triangles.
| \(PR\) | \(=\) | \(LN = 6\text{ cm}\) |
| \(\angle PRQ\) | \(=\) | \(\angle LNM = 72^\circ\) |
| \(RQ\) | \(=\) | \(NM = 10\text{ cm}\) |
Two sides and the included angle: SAS. Match the vertices: \(R \leftrightarrow N\) (the angle), \(P \leftrightarrow L\) (end of the \(6\) cm side), \(Q \leftrightarrow M\).
| \(\triangle PQR\) | \(\equiv\) | \(\boldsymbol{\triangle LMN}\ \text{(SAS)}\) |
(i) \(x\) and \(y\)
(ii) \(\angle L\)
(iii) \(\angle R\)
Read the statement in order: \(K \leftrightarrow R\), \(L \leftrightarrow S\), \(M \leftrightarrow T\). So \(KL\) matches \(RS\) and \(LM\) matches \(ST\).
(i) Two pairs of matching sides give two equations. Solve them simultaneously.
| \(x + 3y\) | \(=\) | \(17\) |
| \(3x - y\) | \(=\) | \(11\) |
From the second equation, \(y = 3x - 11\). Substitute into the first.
| \(x + 3(3x - 11)\) | \(=\) | \(17\) |
| \(10x - 33\) | \(=\) | \(17\) |
| \(x\) | \(=\) | \(\mathbf{5}\) |
| \(y\) | \(=\) | \(3 \times 5 - 11 = \mathbf{4}\) |
Check: \(KL = 5 + 12 = 17\) and \(LM = 15 - 4 = 11\).
(ii) \(\angle L\) matches \(\angle S\).
| \(\angle L\) | \(=\) | \(\mathbf{74^\circ}\) |
(iii) \(\angle R\) matches \(\angle K\). Use the angle sum of \(\triangle KLM\).
| \(\angle K\) | \(=\) | \(180^\circ - 74^\circ - 38^\circ = 68^\circ\) |
| \(\angle R\) | \(=\) | \(\mathbf{68^\circ}\) |
Write one fact per line, each with its reason.
| \(\angle APX\) | \(=\) | \(\angle BQX\ \text{(alternate, } PA \parallel QB)\) |
| \(\angle PXA\) | \(=\) | \(\angle QXB\ \text{(vertically opposite)}\) |
| \(PX\) | \(=\) | \(QX\ \text{(}X\text{ is the midpoint)}\) |
Two angles and a matching side: \(\therefore\ \triangle PXA \equiv \triangle QXB\) (AAS).
| \(AX\) | \(=\) | \(BX\) (matching sides of congruent triangles) |
So \(X\) is the midpoint of \(AB\). The side \(PX\) is opposite the angle at \(A\) in one triangle and \(QX\) is opposite the angle at \(B\) in the other, so the sides really do match.
| \(\angle TMA\) | \(=\) | \(\angle TMB = 90^\circ\ \text{(given)}\) |
| \(TA\) | \(=\) | \(TB\ \text{(hypotenuses, given)}\) |
| \(TM\) | \(\) | \(\text{is common}\) |
\(\therefore\ \triangle TMA \equiv \triangle TMB\) (RHS), so \(AM = BM\) (matching sides) and \(M\) is the midpoint.
| \(AM^2\) | \(=\) | \(3.4^2 - 3^2\) |
| \(\) | \(=\) | \(11.56 - 9 = 2.56\) |
| \(AM\) | \(=\) | \(\sqrt{2.56} = 1.6\text{ m}\) |
| \(AB\) | \(=\) | \(2 \times 1.6 = \mathbf{3.2}\text{ m}\) |
Common pitfalls
Frequently asked questions
What are the 4 congruence tests for triangles?
SSS (three sides), SAS (two sides and the included angle), AAS (two angles and a matching side) and RHS (right angle, hypotenuse and one side).
Why is AAA not a congruence test?
Triangles with the same three angles have the same shape but can be different sizes, like a small and a large equilateral triangle. AAA proves the triangles are similar, not congruent.
What is the difference between SAS and SSA?
In SAS the equal angle is between the two equal sides, and it proves congruence. In SSA the angle is not between them, and two different triangles can be made, so it is not a test.
How do you write a congruence statement?
Write the two triangle names with the vertices in matching order, joined by \(\equiv\). If \(P\), \(Q\), \(R\) match \(L\), \(M\), \(N\), write \(\triangle PQR \equiv \triangle LMN\).
What does a common side mean in a congruence proof?
It is a side shared by both triangles, such as a diagonal of a kite. It is equal to itself, so it gives one pair of equal sides with the reason "common".