Tests for Congruent Triangles
Every question with a fully worked solution.
- Tests for Congruent Triangles - Video - Congruent Triangle Rules Watch
Theory
Three of the right facts prove two triangles congruent. There are four tests: SSS (three sides), SAS (two sides and the angle between them), AAS (two angles and a side in the same position) and RHS (right angle, hypotenuse, side). Write the test after the statement: \(\triangle ABC \equiv \triangle DEF\) (SSS). This Year 8 maths page shows how.
SSS: the three sides of one triangle equal the three sides of the other.
SAS: two sides and the angle between them (the angle where those two sides meet).
AAS: two angles and one side in the same position in both triangles. Once two angles match, so does the third.
RHS: both triangles have a right angle, equal hypotenuses and one other equal side. The hypotenuse is the longest side, opposite the right angle.
Write the test in brackets after the statement: \(\triangle LMN \equiv \triangle RST\) (SAS).
Some equal parts are free. A side shared by both triangles is a common side. Crossing lines make equal vertically opposite angles, and parallel lines make equal alternate angles.
Three sides but no angle marked? Test for a right angle with Pythagoras’ theorem: if \(a^2 + b^2 = c^2\), RHS may be used as well as SSS.
The tests:
| Test | Equal parts needed |
|---|---|
| SSS | all three sides |
| SAS | two sides and the angle between them |
| AAS | two angles and a side in the same position |
| RHS | right angle, hypotenuse and one other side |
Choosing the test
- List the equal pairs: given, marked, common sides, vertically opposite or alternate angles.
- Check positions: is the angle between the two sides? Is the side in the same position?
- Name the test and write the statement with matching vertices in order.
Finding missing parts
- Prove the triangles congruent first.
- Copy matching sides and angles; finish with the angle sum or Pythagoras’ theorem.
In both triangles the equal angle is at the vertex where the \(11\) m and \(7\) m sides meet, so it is the angle between them.
SAS: \(\triangle PQR \equiv \triangle XYZ\).
(i) Name the test.
(ii) Find \(AB\).
\(AC\) and \(DF\) are opposite the right angles, so they are the hypotenuses.
| \(AB^2\) | \(=\) | \(26^2 - 10^2 = 576\) |
| \(AB\) | \(=\) | \(24~\text{cm}\) |
(i) RHS
(ii) \(AB = 24\) cm
(i) Name the test.
(ii) Find \(\angle Z\).
\(PR\) is opposite \(\angle Q\) and \(XZ\) is opposite \(\angle Y\), so the equal sides are in the same position.
| \(\angle R\) | \(=\) | \(180^\circ - 47^\circ - 68^\circ = 65^\circ\) |
(i) AAS
(ii) \(\angle Z\) matches \(\angle R\), so \(\angle Z = 65^\circ\).
The hypotenuses are opposite the right angles: \(KM\) and \(RT\).
\(KM = RT\). (The fact \(LM = ST\) would also prove congruence, but then the right angle is between the equal sides, so the test is SAS.)
\(AB = CB\) (given), \(AD = CD\) (given) and \(BD = BD\) (common side).
\(\triangle ABD \equiv \triangle CBD\) (SSS).
| \(5^2 + 12^2\) | \(=\) | \(25 + 144 = 169 = 13^2\) |
Yes. Triangle \(A\) is right-angled with hypotenuse \(13\) cm, and both have a \(12\) cm side, so they are congruent (RHS).
Common pitfalls
Frequently asked questions
What are the four tests for congruent triangles?
SSS, SAS, AAS and RHS.
What does SAS mean?
Two sides and the included angle, the angle between those two sides, are equal.
Why is SSA not a congruence test?
Two sides and an angle that is not between them can make two different triangles.
What does RHS stand for?
Right angle, hypotenuse, side.
What is a common side?
A side shared by both triangles; it is equal to itself, so it counts as one pair of equal sides.