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.
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%20Congruent%20Triangles%20(Angle-Side-Angle).webp)
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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?
Correct! 🎉 +10 pointsNot quite right
Since all corresponding sides are equal in length, the triangles are congruent by the Side-Side-Side (SSS) congruence rule.
Are these two triangles congruent?
Correct! 🎉 +10 pointsNot quite right
Since the two sides and the included angle are the same in both triangles, they are congruent by the Side-Angle-Side (SAS) rule.
Two right-angled triangles have the same hypotenuse of and one side of . Are these two triangles congruent?
Correct! 🎉 +20 pointsNot quite right
Since the hypotenuse and one side of both right-angled triangles are the same, the Right-Angle Hypotenuse-Side (RHS) rule applies, guaranteeing the triangles are congruent.
Are these two triangles congruent?
Correct! 🎉 +20 pointsNot quite right
The triangles are congruent because they have two equal angles and the side between them is also the same length. This follows the ASA (Angle-Side-Angle) rule.
Which condition cannot be used to determine congruent triangles?
Correct! 🎉 +20 pointsNot quite right
AAS is not enough to prove congruence because it does not always fix the size of the triangle. Different triangles can share the same angles but have different side lengths.
Two triangles each have one side of , another side of . The non-included angle in both triangles is . Are the triangles congruent?
Correct! 🎉 +30 pointsNot quite right
SSA (Side-Side-Angle) does not guarantee congruence. The non-included angle can result in two different triangles, so congruence is not guaranteed.
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Interactive Activity
Explore the 4 rules of triangle congruence: SSS, SAS, ASA, RHS
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Students Also Ask
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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.