Congruent Triangles
Congruent triangles are the same size and shape, so their corresponding sides and angles are equal. One triangle can be slid, turned, or flipped to fit onto the other. You can prove this using SSS, SAS, ASA, and RHS.
Video Lesson
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Flashcards
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What Are Congruent Triangles?
- Two triangles are congruent if they have the same size and shape.
- This means all corresponding sides and angles are equal.
How Can You Recognise Congruent Triangles?
- Congruent triangles can be moved to overlap perfectly.
- This can be done by translating, reflecting, or rotating.
SSS (Side-Side-Side)
- Two triangles are congruent if all three corresponding sides are equal.
- The angles do not need to be given.
SAS (Side-Angle-Side)
- Two triangles are congruent if two corresponding sides and the included angle are equal.
- The angle must be the included angle.
ASA (Angle-Side-Angle)
- Two triangles are congruent if two corresponding angles and the included side are equal.
- The side must be between the two angles.
RHS (Right-Angle-Hypotenuse-Side)
- Applies only to right-angled triangles.
- Triangles are congruent if they have the same hypotenuse and one equal side.
Common Pitfall
- SSA does NOT guarantee congruence.
- Two equal sides and a non-included angle are not enough to prove congruence.
Practice Questions
Test your understanding
Are these two triangles congruent?
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Compare the three pairs of sides: both triangles have sides of , and .
All three pairs of corresponding sides are equal, so the triangles are congruent by the Side-Side-Side (SSS) rule.
The green triangle has been turned round, but rotating or reflecting a shape does not change any of its side lengths, so it is still congruent to the purple one.
Are these two triangles congruent?
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Both triangles have a side and a side with the angle between them, so they match by the Side-Angle-Side (SAS) rule.
The pink triangle is drawn turned round compared with the blue one, but rotating a triangle does not change any of its side lengths or angles, so the two triangles are still congruent.
Two right-angled triangles have the same hypotenuse of and one side of . Are these two triangles congruent?
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Both triangles have a right angle, the same hypotenuse and the same side, so the Right angle-Hypotenuse-Side (RHS) rule applies and they are congruent.
You do not need all six measurements. Pythagoras fixes the third side at cm, so these measurements can only make one triangle.
Are these two triangles congruent?
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Both triangles have angles of and with the side between those two angles, so they match by the Angle-Side-Angle (ASA) rule.
The green triangle is drawn turned round compared with the orange one, but rotating a triangle leaves every side and angle unchanged, so the two triangles are congruent.
Which condition cannot be used to determine congruent triangles?
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AAA is not enough to prove congruence, because three angles fix only the shape of a triangle, not its size. Two triangles can have exactly the same three angles but different side lengths - they are similar, not necessarily congruent.
ASA, SAS and SSS each include at least one side, which fixes the size as well as the shape. SSS still proves congruence when one triangle is a rotation or mirror image of the other, and in SAS the angle used must be the one between the two given sides.
Two triangles each have one side of , another side of . In both triangles, an angle of is opposite the side. Are the triangles congruent?
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The angle is opposite the side, not between the two given sides, so this is SSA.
By the sine rule, , which gives or . Both work, because is still less than , so two genuinely different triangles fit the same three measurements. SSA does not guarantee congruence.
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Interactive Activity
Explore the 4 rules of triangle congruence: SSS, SAS, ASA, RHS
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Congruent Triangles frequently asked questions
The questions students bump into most on this topic
The four congruence criteria are SSS, SAS, ASA, and RHS. SSS uses all three pairs of equal sides. SAS uses two pairs of sides plus the included angle. ASA uses two pairs of angles plus the included side. RHS applies to right-angled triangles with equal hypotenuse and one matching side.
In maths, congruent means having the same size and shape. Two triangles are congruent when corresponding sides are equal in length and corresponding angles are equal in measure. You can move one triangle onto the other through translation, rotation, or reflection without changing its dimensions.
SSS works because three given side lengths fix a triangle uniquely. There is only one way to construct a triangle from three given side lengths. Any two triangles with the same three side lengths must therefore have the same shape and size.
RHS works because Pythagoras's theorem fixes the third side. You only need the hypotenuse and one other side in a right-angled triangle. The third side must be equal in both triangles, reducing the situation to SSS, which guarantees congruence. RHS would not work without the right angle.
No. If you know two sides and an angle outside them, you can sometimes construct two different triangles that fit. They both match the given information. This ambiguity means the triangles are not guaranteed to be congruent, so none of the four criteria covers this case.
SAS uses two sides and the angle between them (the included angle) to prove congruence. ASA uses two angles and the side between them (the included side). Both depend on the equal element sitting between the matching pair, and swapping the position invalidates the criterion.