Tests for Congruent Triangles
Two triangles are congruent if they match by one of four tests: SSS, SAS, AAS or RHS. This NSW Year 9 Advanced maths subtopic uses the tests to show triangles are congruent and then find unknown sides and angles (NESA outcome MA5‑GEO‑P‑01).
You will name the test for a pair of triangles, use congruence to find pronumerals, and decide whether equal angles alone, or two sides and a non-included angle, are enough. You will find the extra fact a test needs, measure a river's width with congruent triangles, find the value of \(x\) that makes two triangles congruent, and work with a kite frame. It is part of Year 9 Advanced Congruence and Similar Figures.
Every question with a fully worked solution.
- Tests for Congruent Triangles - Video - Congruent Triangle Rules Watch
Theory
Two triangles are congruent when one of four congruence tests is met: SSS, SAS, AAS or RHS. This Year 9 Advanced maths page shows how to pick the test and use it to find unknowns. It comes up if you are heading for Extension maths.
Six measurements describe a triangle: three sides and three angles. A congruence test is a set of three of them that fixes the triangle completely. If two triangles share one of these sets, they are congruent.
SSS: three pairs of sides equal. SAS: two pairs of sides and the included angle (the angle between those sides) equal. AAS: two pairs of angles and one pair of matching sides equal. RHS: both triangles have a right angle, equal hypotenuses and one other pair of sides equal.
Useful facts for spotting equal parts: a side shared by both triangles is a common side; vertically opposite angles are equal. Once two triangles are congruent, all their matching sides and angles are equal.
Two combinations are not tests. Three equal angles (AAA) gives the same shape but not always the same size. Two sides and a non-included angle can give two different triangles.
The four tests:
For RHS, the third side is fixed by Pythagoras' theorem, which is why the test works:
How to decide if two triangles are congruent
- Mark what is equal: given sides and angles, a common side, vertically opposite angles, and the third angle from the angle sum.
- Match the vertices: equal angles match; a side matches the side opposite the equal angle.
- Pick the test: three sides (SSS); two sides with the angle between them (SAS); two angles and a matching side (AAS); right angle, hypotenuse and side (RHS).
- Write the statement with the vertices in matching order and the test in brackets.
Using congruent triangles
- Find unknowns: set each unknown equal to its matching part, then solve. Two expressions that match also make an equation.
- What extra fact is needed? Work out which test is nearly complete and name the missing part, e.g. the included angle for SAS.
- Use Pythagoras' theorem to find a missing side in a right-angled triangle before comparing.
- Real problems: build a pair of congruent triangles, prove them congruent, then measure the side you can reach to find the one you cannot.
(i) Two angles are equal, and the \(7\) cm side is opposite the \(80^\circ\) angle in both: AAS.
(ii) Right angle, equal hypotenuses (\(15\) cm, opposite the right angle) and one more equal side: RHS.
(iii) Two equal sides with the \(48^\circ\) angle between them: SAS.
In (i), check the side is in the same position. A \(7\) cm side opposite \(35^\circ\) in one triangle and opposite \(80^\circ\) in the other would prove nothing.
(i) Prove that \(\triangle ABM \equiv \triangle ACM\).
(ii) Hence find \(BM\) and the height \(AM\).
(i) In \(\triangle ABM\) and \(\triangle ACM\):
\(\angle AMB = \angle AMC = 90^\circ\) (given)
\(AB = AC\) (given, the hypotenuses)
\(AM\) is common
\(\therefore\ \triangle ABM \equiv \triangle ACM\) (RHS)
(ii) Matching sides are equal, so \(BM = MC\) and \(M\) is the midpoint of \(BC\).
| \(BM\) | \(=\) | \(18 \div 2 = \mathbf{9}\text{ cm}\) |
| \(AM^2\) | \(=\) | \(15^2 - 9^2 = 144\) |
| \(AM\) | \(=\) | \(\mathbf{12}\text{ cm}\) |
(i) Write a congruence statement.
(ii) Find \(x\) and \(y\).
(i) Find the third angles: \(\angle C = 180^\circ - 40^\circ - 75^\circ = 65^\circ\) and \(\angle D = 180^\circ - 65^\circ - 40^\circ = 75^\circ\). Match equal angles: \(A \leftrightarrow F\), \(B \leftrightarrow D\), \(C \leftrightarrow E\), so \(\mathbf{\triangle ABC \equiv \triangle FDE}\).
(ii) Matching sides sit opposite equal angles. \(BC\) is opposite \(40^\circ\), like \(DE\). \(AC\) is opposite \(75^\circ\), like \(EF\).
| \(3x - 2\) | \(=\) | \(x + 8\) |
| \(2x\) | \(=\) | \(10\) |
| \(x\) | \(=\) | \(\mathbf{5}\) |
| \(4y + 1\) | \(=\) | \(17\) |
| \(y\) | \(=\) | \(\mathbf{4}\) |
Check: \(BC = 13 = DE\) \(\checkmark\)
(i) \(\triangle KLM\) has \(KL = 9\) cm, \(LM = 12\) cm and \(\angle L = 90^\circ\). \(\triangle PQR\) has \(\angle Q = 90^\circ\), \(QR = 9\) cm and hypotenuse \(PR = 15\) cm. Are they congruent?
(ii) \(\triangle ABC\) has \(AB = 7\), \(BC = 10\), \(\angle A = 50^\circ\). \(\triangle DEF\) has \(DE = 7\), \(EF = 10\), \(\angle E = 50^\circ\). Do these facts prove them congruent?
(i) Find the hypotenuse of \(\triangle KLM\).
| \(KM^2\) | \(=\) | \(9^2 + 12^2 = 225\) |
| \(KM\) | \(=\) | \(15\text{ cm}\) |
Right angles, hypotenuses \(15\) cm and a side of \(9\) cm in each: yes, RHS, with \(\triangle KLM \equiv \triangle RQP\).
(ii) In \(\triangle DEF\) the \(50^\circ\) angle is between the \(7\) and \(10\) sides. In \(\triangle ABC\) it is at \(A\), which is not between \(AB\) and \(BC\).
The facts do not line up with any one test, so they do not prove congruence.
Common pitfalls
Frequently asked questions
What are the 4 congruence tests for triangles?
The four tests are SSS (three sides), SAS (two sides and the included angle), AAS (two angles and a matching side) and RHS (right angle, hypotenuse and one other side).
What is the difference between SAS and AAS?
SAS uses two sides and the angle between them. AAS uses two angles and one side, which must be opposite the same angle in both triangles.
Why is AAA not a congruence test?
Triangles with the same three angles have the same shape, but one can be bigger than the other. They are similar, not necessarily congruent.
Why is SSA not a congruence test?
With two sides and an angle that is not between them, the third side can often swing into two different positions, giving two different triangles. The facts do not fix one triangle.
Is ASA the same as AAS?
Yes, they work the same way. If two angles are equal the third angles are equal too, so a side between two angles or opposite one of them both fix the triangle. NSW uses the name AAS.
How do you set out a congruence proof?
Name the two triangles. List three pairs of equal parts, each with a reason such as given, common side or vertically opposite angles. Then write the congruence statement with the test in brackets.