Triangles: Congruence Theorems

The One Thing

Triangle congruence proves that two triangles have exactly the same shape and size. The criteria SSS, SAS, ASA, and RHS establish congruence from sufficient corresponding measurements; CPCT then permits conclusions about the remaining corresponding sides and angles.

Definitions and Results

• Congruent figures: Figures that have exactly the same shape and size and can coincide completely when superimposed. Two triangles are congruent when one can be placed exactly over the other so that all corresponding vertices, sides, and angles coincide.
• Corresponding parts: Matching sides or angles in two figures, usually identified by the order of their vertices.
• Triangle rigidity: A triangle cannot change its shape when its three side lengths are fixed, unlike a quadrilateral, which may change shape.
• SSS congruence: Two triangles are congruent if their three corresponding sides are equal. Specifically, if $AB=PQ$, $BC=QR$, and $CA=RP$, then $△ABC\cong △PQR$.
• SAS congruence: Two triangles are congruent if two corresponding sides and the included angle between them are equal. The included angle is the angle formed by the two given sides.
• ASA congruence: Two triangles are congruent if two corresponding angles and the included side between them are equal. The included side is the side lying between the two given angles.
• RHS congruence: Two right-angled triangles are congruent if their hypotenuse and one corresponding side are equal. The right-angle condition is required.
• AAA: Not a congruence criterion because equal angles guarantee the same shape but not necessarily the same size.
• SSA: Generally not a valid congruence criterion because the given information may allow triangles of different shapes or sizes.
• CPCT: Corresponding Parts of Congruent Triangles are equal. CPCT is used after congruence has been established to show that corresponding sides and angles are equal.
• Isosceles triangle: A triangle having at least two equal sides.
• Base angles: The two angles opposite the equal sides of an isosceles triangle.
• Isosceles theorem: In an isosceles triangle, angles opposite equal sides are equal.
• Converse of the isosceles theorem: If two angles of a triangle are equal, then the sides opposite those angles are equal.
• Side–angle relationship: The side opposite the larger angle is longer, and the angle opposite the longer side is larger.
• Vertex correspondence: When naming congruent triangles, the order of the letters must show the correct correspondence between vertices.

Worked Methods

Proving congruence using SSS

• Identify the three pairs of corresponding sides.
• Verify that all three pairs are equal.
• Match the vertices in the correct order.
• State the congruence using SSS.

For example, if $$AB=PQ,BC=QR,CA=RP,$$ then $$△ABC\cong △PQR$$ by SSS.

Proving congruence using SAS

• Identify two pairs of corresponding equal sides.
• Confirm that the equal angle is the angle included between those two sides.
• Match the corresponding vertices correctly.
• State that the triangles are congruent by SAS.

The angle must be the included angle; an unrelated angle does not provide SAS information.

Proving congruence using ASA

• Identify two pairs of corresponding equal angles.
• Confirm that the equal side lies between those two angles.
• Match the corresponding vertices correctly.
• State that the triangles are congruent by ASA.

The side must be the included side between the two given angles.

Proving congruence using RHS

• Confirm that both triangles are right-angled.
• Identify the hypotenuse in each triangle.
• Show that the hypotenuses are equal.
• Show that one corresponding side is equal.
• State that the triangles are congruent by RHS.

Equality of the hypotenuse and one corresponding side is sufficient only because the triangles are right-angled.

Using CPCT after congruence

• Prove that the two triangles are congruent using SSS, SAS, ASA, or RHS.
• Write the congruence statement with the vertices in corresponding order.
• Identify the required corresponding sides or angles.
• Conclude that they are equal by CPCT.

For example, once $△ABC\cong △PQR$, the correspondence is $$A\leftrightarrow P,B\leftrightarrow Q,C\leftrightarrow R.$$ Therefore, corresponding sides and angles, such as $AB$ and $PQ$, or $\angle B$ and $\angle Q$, are equal by CPCT.

Applying the isosceles theorem

• Identify the two equal sides of the isosceles triangle.
• Identify the angles opposite those sides.
• Conclude that the two opposite angles, the base angles, are equal.
• Where required, construct or compare suitable smaller triangles to establish the result through congruence.

Thus, equal sides in an isosceles triangle produce equal base angles.

Applying the converse of the isosceles theorem

• Identify two equal angles in the triangle.
• Identify the sides opposite those angles.
• Use triangle congruence or the relationship between opposite sides and angles.
• Conclude that the two opposite sides are equal.

Thus, if two angles of a triangle are equal, the sides opposite them are equal.

Using the side–angle relationship

• Compare two angles and identify the larger one.
• Identify the sides opposite those angles.
• Conclude that the side opposite the larger angle is longer.
• Alternatively, compare two sides and conclude that the angle opposite the longer side is larger.

Where It Goes Wrong

• Treating AAA as a congruence test: equal angles establish the same shape, but not necessarily the same size.
• Applying SSA as though it were generally valid: the information may allow triangles of different shapes or sizes.
• Using SAS when the given angle is not the included angle between the two given sides.
• Using ASA when the given side is not the included side between the two given angles.
• Applying RHS without first confirming that both triangles are right-angled.
• Naming congruent triangles in an incorrect vertex order, so that the stated correspondence between sides and angles is wrong.

What Gets Asked

• Prove that two triangles are congruent using SSS, SAS, ASA, or RHS.
• Use the specific data $AB=PQ$, $BC=QR$, and $CA=RP$ to establish $△ABC\cong △PQR$ by SSS.
• Identify the included angle or included side in a proposed congruence proof.
• Explain why AAA and SSA do not generally prove congruence.
• State corresponding equal sides or angles using CPCT after congruence has been proved.
• Determine the correct order in which congruent triangles should be named.
• Prove that the base angles of an isosceles triangle are equal.
• Use the converse of the isosceles theorem to prove that two sides are equal when two angles are equal.
• Compare the sizes of sides and angles using the relationship between opposite sides and angles.