Congruent triangles: worked examples and method

Same size and shape

Congruent shapes are identical in all side lengths and angles. A full criterion is required to prove congruence.

SAS criterion

Two sides and the included angle prove congruence. The angle must be situated between the two known sides.

ASA criterion

Two angles and the included side prove congruence. The side must connect the vertices of the two known angles.

AAS criterion

Two angles and a side that is not between them prove congruence because the third angle is fixed by the 180° angle sum.

SSS and RHS

SSS requires all three sides equal. RHS requires a Right angle, Hypotenuse and one other Side equal.

Invalid conditions

AAA proves similarity, not congruence. SSA (two sides and a non-included angle) is ambiguous.

What congruent means

If two triangles are congruent, every pair of corresponding sides is equal and every pair of corresponding angles is equal. The perimeter of one equals the perimeter of the other. A triangle with sides 7 cm, 8 cm and 9 cm and a second triangle with those same three sides are congruent by SSS. Both perimeters are 24 cm. You do not need to find an angle first.

The order of the vertices records the match. If triangle ABC is congruent to triangle DEF, then A matches D, B matches E and C matches F. Side AB matches side DE, and angle B matches angle E. Writing the letters in a scrambled order hides which side copies across. After the congruence is proved, you may copy a length or an angle onto the other triangle, naming the correspondence.

Method: 1. Match the vertices in the order written, then copy corresponding sides and angles. 2. Three matching sides give SSS, so sides 7cm, 8cm and 9cm make both perimeters 24cm. 3. If ABC matches DEF, side AB copies to side DE.

The four conditions

SSS is three pairs of equal sides. SAS is two pairs of equal sides and the equal angle between them. If the equal angle is not between the two sides, you do not have SAS. ASA is two pairs of equal angles and the equal side between those angles. If the side you know is opposite one of the angles instead, find the third angle using 180^°. The information is then ASA. That is why AAS is accepted: it is ASA after one subtraction.

RHS is for right-angled triangles only. You need the right angle, the hypotenuse, and one other side. Two triangles that share a side of 8 cm, each with a right angle and a hypotenuse of 10 cm, are congruent by RHS. The third side is √(10^2 - 8^2) = 6 cm in each triangle. The 6, 8, 10 triple is a check, not a new condition. Without the right angle, a hypotenuse is not defined, so you cannot quote RHS.

Method: 1. Use SSS for three sides, SAS for two sides with the angle between them, and ASA for two angles with the side between them. 2. Turn AAS into ASA by subtracting the known angles from 180^°. 3. For right-angled triangles with hypotenuse 10cm and a shared side of 8cm, quote RHS and find sqrt10^2-8^2=6cm.

What is not enough

AAA means the angles match, so the triangles are similar. The sides could be doubled, or at any other scale factor, so the triangles need not be congruent. Do not claim congruence from three angles. SSA means two sides and an angle that is not the included angle. Two different triangles can share that information, so SSA is not a congruence test. The exception is RHS, where the known angle is the right angle and one of the sides is the hypotenuse. The right angle removes the second possible triangle.

A common diagram looks like SAS until you notice the marked angle is next to only one of the marked sides. Move the angle in your mind: if it is not locked between the two sides, it is not included. Likewise, two right angles and one pair of equal legs is not RHS unless you also know the hypotenuses are equal, or you know another matching side in the right place. Quote the condition by name. A marker looks for SSS, SAS, ASA, AAS or RHS, not for the word congruent on its own.

Method: 1. AAA only matches the angles, so a scale factor of 2 still fits and the triangles need not be congruent. 2. SSA is not a test, unless the angle is 90^° and one side is the hypotenuse. 3. Before you write SAS, check the equal angle sits between the two equal sides.

Using congruence in a proof

List the equal pieces, name the condition, then state the triangles congruent in matching order. Only after that may you claim a further pair of equal angles or sides. In an isosceles triangle ABC with AB = AC, let M be the midpoint of BC. Then BM = MC, and AM is common. Triangles ABM and ACM have three pairs of equal sides, so they are congruent by SSS. Corresponding angles at B and C are equal.

The angles at M are also equal, and they sit on the straight line BC, so each is 90^°. That is why AM is a height. If BC = 10 cm and AB = 13 cm, then BM = 5 cm and AM = √(13^2 - 5^2) = 12 cm. The congruence did the angle work. Pythagoras then found the length, and it was valid only because the right angle had been earned. Do not drop a perpendicular and assume the right angle without a reason when the question is a proof.

Method: 1. List the equal pieces, name SSS, SAS, ASA or RHS, then state the triangles congruent. 2. In isosceles ABC with M the midpoint of BC, triangles ABM and ACM are congruent by SSS. 3. The right angle at M then allows AM=sqrt13^2-5^2=12cm.

Setting out a congruence proof

A "prove that triangles are congruent" question is marked line by line. Each statement about a pair of equal parts needs its own reason, and the last line names the condition. A good layout is three or four short lines such as: AB = DC (opposite sides of a parallelogram), BC is common, angle ABC = angle DCB (reason), so the triangles are congruent (SAS). Do not write a paragraph, because the reasons are easy to miss in one.

Name the triangles with matching letters. If the proof says triangle ABC is congruent to triangle DCB, then A matches D, B matches C and C matches B. Writing the vertices in the right order is part of the answer, and it also tells you which sides and angles are equal when you use the result later. Finish with the condition: SSS, SAS, ASA, AAS or RHS.

Method: 1. List the pairs of equal parts one per line, each with its reason. 2. Check that the equal angle is between the two equal sides for SAS. 3. Write the final line: "so triangle ... is congruent to triangle ... (condition)". 4. If the question says "hence", use the congruence to state the length or angle that follows.

Tip: Reasons earn marks. "AB = CD" on its own is a statement, but "AB = CD (opposite sides of a parallelogram)" is a mark.

Equal parts that the diagram does not show

Most proofs depend on equal parts that are not marked. A side shared by two triangles is equal to itself, so write "common side". Vertically opposite angles are equal. Alternate and corresponding angles are equal when lines are parallel. In a parallelogram opposite sides are equal and opposite angles are equal. In a rectangle every angle is 90^° and the diagonals are equal.

Circle questions give more. Two radii of the same circle are equal. A line from the centre to the midpoint of a chord makes a right angle with the chord. A midpoint splits a line into two equal halves, and an equilateral triangle has three equal sides and three 60^° angles. Before you write anything, tick every pair you can justify on a copy of the diagram, and then look for three that make a condition.

Method: 1. Scan the diagram for common sides and vertically opposite angles. 2. Look for parallel lines, midpoints and radii that give equal parts. 3. Mark each equal pair with matching ticks or arcs. 4. Choose three pairs that form SSS, SAS, ASA, AAS or RHS.

Using congruence to find lengths and angles

When two triangles are congruent, the matching parts are equal. If triangle ABC is congruent to triangle PQR then AB = PQ, BC = QR and AC = PR, and angle A = angle P, angle B = angle Q and angle C = angle R. The perimeters and the areas are equal too. Questions often give measurements for one triangle only and ask about the other, so read the letter order carefully.

The same idea powers many numerical questions. If a rectangle has diagonals AC and BD, triangles ABC and DCB are congruent, so AC = BD, and a Pythagoras calculation in one triangle gives the length of both. Use a short chain: show congruent, state the equal length, then calculate. The calculation marks are separate from the proof marks.

Method: 1. Write the pairing of the vertices from the order of the letters. 2. Copy across each given length or angle to the matching part. 3. Use Pythagoras or angles in a triangle if a value is still missing. 4. State perimeters or areas as equal when the triangles are congruent.

Which conditions work and which do not

Four conditions work. SSS needs three pairs of equal sides. SAS needs two pairs of equal sides and the equal angle between them. ASA and AAS need two angles and one side, in the same relative position. RHS applies only to right-angled triangles: the right angle, the hypotenuse and one other side. Three equal angles (AAA) show the triangles are similar, but they can be different sizes, so it is not enough.

A classic trap is "SSA". An angle that is not between the two sides does not fix the triangle, because two different triangles can often be drawn from the same information. "Explain why" questions are set on exactly this, for example "Mia says two triangles with angles 50^°, 60^°, 70^° are congruent. Is she right?" The reason is that one could be an enlargement of the other.

Method: 1. Count what you can prove: sides and angles, and where each sits. 2. Reject AAA and SSA as conditions on their own. 3. Name the condition at the end of the proof. 4. For an explain question, say what extra information would be needed.