Congruent Figures
Use the four congruence tests, SSS, SAS, AAS and RHS, to prove two triangles are identical, and write a congruence statement with the vertices in the order that actually matches.
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Think First
What does it mean for two shapes to be exactly the same? Do they need to face the same direction, or be in the same position? Could a shape that has been flipped still be "the same"?
Congruent Figures
Two figures are congruent (symbol $\cong$) when they are exactly the same shape and size. Position and orientation do not matter, a shape that has been moved, rotated, or reflected can still be congruent to the original. For triangles, we have four reliable tests that let us prove congruence without measuring every part.
What You'll Master
- Define congruence and use the symbol $\cong$ correctly
- State and apply the four congruence tests for triangles: SSS, SAS, AAS, RHS
- Write correct congruence statements matching corresponding vertices in order
- Use congruence to find unknown sides and angles
Words You Need
⚠ Spot the Trap
SSA is NOT a valid congruence test. Two sides and a non-included angle are not enough, two different triangles can be constructed from the same SSA information. The angle must be the included angle (between the two sides) to use SAS. Similarly, AAA proves triangles are similar, not necessarily congruent.
The Four Congruence Tests for Triangles
You only need to check enough information to guarantee the triangles are identical. The four valid tests are:
- SSS If all three sides of one triangle equal all three sides of the other, the triangles are congruent.
- SAS If two sides and the included angle (the angle between those two sides) are equal, the triangles are congruent.
- AAS If two angles and any corresponding side are equal, the triangles are congruent. (Two angles fix the shape; one side fixes the size.)
- RHS In right-angled triangles only: if the hypotenuse and one other side are equal, the triangles are congruent.
Worked Example, SSS Congruence
Triangle $ABC$ has sides $AB = 5$ cm, $BC = 8$ cm, $AC = 10$ cm.
Triangle $DEF$ has sides $DE = 5$ cm, $EF = 8$ cm, $DF = 10$ cm.
Test: Compare all three pairs:
- $AB = DE = 5$ cm ✓
- $BC = EF = 8$ cm ✓
- $AC = DF = 10$ cm ✓
Conclusion: All three pairs of sides are equal, so by SSS, $\triangle ABC \cong \triangle DEF$.
Note the vertex order: $A$ corresponds to $D$, $B$ to $E$, $C$ to $F$. This means $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$ automatically.
Writing Congruence Statements Correctly
When writing $\triangle ABC \cong \triangle DEF$, the order of vertices matters. The first vertex of each triangle must correspond, the second must correspond, and the third must correspond.
If $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$ then:
- Corresponding sides: $AB = DE$, $BC = EF$, $AC = DF$
- Corresponding angles: $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$
Writing vertices in the wrong order is a common error, always check which parts actually match before writing the statement.
Common Pitfalls
- Using SSA as a congruence test, it is not valid
- Writing vertices in the wrong order in the congruence statement
- In SAS, using a non-included angle (the angle must be between the two sides)
- Applying RHS to non-right-angled triangles
- Confusing congruence ($\cong$, same size) with similarity ($\sim$, same shape only)
Copy This Into Your Book
Congruence tests (triangles only):
SSS: 3 sides equal | SAS: 2 sides + included angle | AAS: 2 angles + corresponding side | RHS: right angle + hypotenuse + 1 side
Congruence statement: $\triangle ABC \cong \triangle DEF$ means $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$, vertex order matters!
SSA and AAA are NOT valid congruence tests.
Two triangles each have sides of 6 cm and 9 cm with an included angle of 50° between them. Which congruence test applies?
Two right-angled triangles each have a hypotenuse of 13 cm and a leg of 5 cm. Which test proves them congruent?
Triangles $PQR$ and $XYZ$ have $\angle P = \angle X$, $\angle Q = \angle Y$, and $PQ = XY$. Which test applies?
$\triangle ABC \cong \triangle DEF$. If $BC = 7$ cm, what is $EF$?
In two triangles, $\angle A = \angle P$, $\angle B = \angle Q$, and $AB = PQ$. Which is the correct congruence statement?
Q6. Two triangles have $\angle A = \angle X = 70°$, $\angle B = \angle Y = 55°$, and $AB = XY = 8$ cm. State which congruence test applies and write the congruence statement.
Q7. $\triangle PQR \cong \triangle XYZ$. Given $PR = 5$ cm and $PQ = 7$ cm, find the lengths of $XZ$ and $XY$. Explain which sides correspond.
Q8. In quadrilateral $ABCD$, the diagonals $AC$ and $BD$ bisect each other at point $M$ (i.e. $AM = MC$ and $BM = MD$). Prove that $\triangle ABM \cong \triangle CDM$, stating the congruence test used.
Show Answers
Q6
Two angles ($\angle A = \angle X$ and $\angle B = \angle Y$) and the included side $AB = XY$ are equal. Test: AAS.
Congruence statement: $\triangle ABC \cong \triangle XYZ$
Q7
In $\triangle PQR \cong \triangle XYZ$: P↔X, Q↔Y, R↔Z.
$PR$ corresponds to $XZ$, so $XZ = 5$ cm.
$PQ$ corresponds to $XY$, so $XY = 7$ cm.
Q8
In $\triangle ABM$ and $\triangle CDM$:
$AM = CM$ (diagonals bisect each other, given)
$BM = DM$ (diagonals bisect each other, given)
$\angle AMB = \angle CMD$ (vertically opposite angles)
Therefore $\triangle ABM \cong \triangle CDM$ by SAS.
Stretch Challenge
In quadrilateral $ABCD$, $AB = CD$ and $AB \parallel CD$. Using the diagonal $AC$ as a shared side, prove that $\triangle ABC \cong \triangle CDA$. State clearly which congruence test you use and explain what type of quadrilateral $ABCD$ must be.