GCSE Maths: Congruent shapes and triangles (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Congruent shapes and triangles
Two shapes are congruent if they are exactly the same size and shape. You can place one exactly on top of the other, perhaps after turning it, sliding it or flipping it over. Reflections, rotations and translations all give congruent images. For congruent shapes, all matching (corresponding) sides are equal and all matching angles are equal.
For triangles we do not need to check all six facts (three sides and three angles). If three particular facts are equal, the rest are forced to be equal as well. There are four conditions to remember. For triangles ABC and PQR:
- SSS: all three sides are equal.
- SAS: two sides and the angle between them (the included angle) are equal.
- ASA: two angles and the side between them are equal. (If the equal side is not between the angles it is AAS, which also works because the third angle is then known too.)
- RHS: both are right-angled, the hypotenuses are equal and one other side is equal.
What does not work. Three equal angles (AAA) are not enough: triangles with the same angles can be different sizes. They are similar, not congruent (see section 21.8). Also, two sides and an angle that is not between them (SSA) is not enough. The picture below shows two different triangles that both have a 30° angle, a side of 10 cm next to it and a side of 6 cm opposite it.
Writing a proof. In a proof you must give a reason for every fact. Typical reasons are “given”, “common side” (shared by both triangles), “vertically opposite angles are equal”, “alternate angles are equal” (parallel lines) and “base angles of an isosceles triangle are equal”. Finish by naming the condition, for example “triangle ABC is congruent to triangle CDA (SAS)”, and write the vertices in matching order.
Key ideas
- Congruent shapes are exactly the same size and shape. Reflection, rotation and translation give congruent images.
- Triangles are congruent if: SSS, SAS (included angle), ASA (or AAS), or RHS.
- AAA is not enough (similar only). SSA is not enough (unless the angle is a right angle, which is RHS).
- In a proof, give a reason for each fact and name the condition. Write the vertices in matching order.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 6 cm, BC = 4 cm, CA = 5 cm
Triangle 2: PQ = 6 cm, QR = 4 cm, RP = 5 cm - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.AB = PQ = 6 cm (side)
BC = QR = 4 cm (side)
CA = RP = 5 cm (side) - Decide which condition is satisfied. SSS (side, side, side): all three sides of one triangle equal the three sides of the other triangle.The three equal parts make SSS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (SSS).
Check. Put the three sides in order: 4, 5, 6 in both triangles. If you cut one out and flip it over, it fits exactly. ✓
Watch out. Match the sides by length, not by where they appear on the page: the second triangle has been turned over.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 6 cm, AC = 8 cm, angle A = 40°
Triangle 2: PQ = 6 cm, PR = 8 cm, angle P = 40° - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.AB = PQ = 6 cm (side)
angle A = angle P = 40° (angle, between the two sides)
AC = PR = 8 cm (side) - Decide which condition is satisfied. SAS (side, angle, side): two sides and the angle between them (the included angle) are equal.The three equal parts make SAS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (SAS).
Check. Use the cosine rule to find BC and QR: both are √(62 + 82 − 2 × 6 × 8 × cos 40°) = 5.14 cm, so all three sides match. ✓
Watch out. The angle must be between the two sides. A 40° angle in another position would not be enough.
Key words. Two shapes are congruent if they are exactly the same size and shape. The included angle is the angle between two given sides.
Method. Check the known facts against the four conditions for congruent triangles. If none is satisfied, we cannot conclude that the triangles are congruent.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angles 40°, 60°, 80° (no sides given)
Triangle 2: angles 40°, 60°, 80° (no sides given) - Compare with the four conditions. To be sure, the facts must match SSS, SAS, ASA/AAS or RHS.Three equal angles is AAA, which is not one of the conditions: no side is known to be equal.
- State the conclusion. If none of the conditions is met, we cannot say that the triangles are congruent.The triangles are not necessarily congruent.
Check. Draw a triangle with angles 40°, 60° and 80° with a base of 5 cm, then another with a base of 10 cm: both have the same angles but different sizes. ✓
Watch out. AAA proves the triangles are similar (same shape) but not congruent (same size).
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: right angle at B, AC = 13 cm (hypotenuse), BC = 5 cm
Triangle 2: right angle at Q, PR = 13 cm (hypotenuse), QR = 5 cm - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.angle B = angle Q = 90° (right angle)
AC = PR = 13 cm (hypotenuse)
BC = QR = 5 cm (side) - Decide which condition is satisfied. RHS (right angle, hypotenuse, side): both triangles have a right angle, the hypotenuses are equal and one other pair of sides is equal.The three equal parts make RHS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (RHS).
Check. Pythagoras gives the missing side in each triangle: √(132 − 52) = √144 = 12 cm, so all three sides match. ✓
Watch out. The side must be a different side from the hypotenuse, and the equal angle must be the right angle.
Key words. A proof is a chain of reasons that shows a statement is always true. Each line must have a reason, such as “given”, “common side”, “alternate angles” or “vertically opposite angles”.
Method. To prove triangles are congruent, state three pairs of equal parts, each with a reason, name the condition (SSS, SAS, ASA or RHS), and then write the conclusion.
- AB = CD Opposite sides of a parallelogram are equal (marked with one tick).
- BC = DA Opposite sides of a parallelogram are equal (marked with two ticks).
- AC = CA AC is a common side, shared by both triangles.
- Conclusion. All the facts needed for SSS are shown.Triangle ABC is congruent to triangle CDA (SSS).
Check. The matching vertices are A ↔ C, B ↔ D, C ↔ A, and the pairs of sides match: AB = CD, BC = DA, AC = CA. ✓
Watch out. Write the vertices in matching order: triangle ABC is congruent to triangle CDA, not triangle ACD.
Watch out
Check that the equal parts are in the correct positions. For SAS the angle must be between the two equal sides. AAA and SSA do not prove congruence. In a proof, write the vertices in matching order and give a reason for every statement.Exercise 21G Congruent triangles
Non-calculatorKey words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 9 cm, BC = 7 cm, CA = 8 cm
Triangle 2: PQ = 9 cm, QR = 7 cm, RP = 8 cm - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.AB = PQ = 9 cm
BC = QR = 7 cm
CA = RP = 8 cm - Decide which condition is satisfied. SSS (side, side, side): all three sides of one triangle equal the three sides of the other triangle.The three equal parts make SSS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (SSS).
Check. The three sides are 7, 8, 9 in both triangles. ✓
Watch out. Match sides by length.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 5 cm, AC = 7 cm, angle A = 50°
Triangle 2: PQ = 5 cm, PR = 7 cm, angle P = 50° - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.AB = PQ = 5 cm
angle A = angle P = 50° (between the sides)
AC = PR = 7 cm - Decide which condition is satisfied. SAS (side, angle, side): two sides and the angle between them (the included angle) are equal.The three equal parts make SAS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (SAS).
Check. The included angle is the one between the two 5 cm and 7 cm sides. ✓
Watch out. Check that the angle is between the two sides.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angle B = 40°, angle C = 70°, BC = 6 cm (between B and C)
Triangle 2: angle Q = 40°, angle R = 70°, QR = 6 cm (between Q and R) - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.angle B = angle Q = 40°
BC = QR = 6 cm (side between the angles)
angle C = angle R = 70° - Decide which condition is satisfied. ASA (angle, side, angle): two angles and the side between them are equal.The three equal parts make ASA.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (ASA).
Check. The third angles are 180 − 40 − 70 = 70 and 180 − 40 − 70 = 70: they agree too. ✓
Watch out. The side must be between the two angles for ASA. If it is not, it is AAS, which also works.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angle B = 50°, angle C = 60°, CA = 7 cm (opposite B, not between B and C)
Triangle 2: angle Q = 50°, angle R = 60°, RP = 7 cm - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.angle B = angle Q = 50°
angle C = angle R = 60°
CA = RP = 7 cm (a side not between the two angles) - Decide which condition is satisfied. AAS (angle, angle, side): two angles and a side that is not between them are equal. This is a form of ASA, because the third angle is fixed by the angle sum.The three equal parts make AAS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (AAS).
Check. Third angles: 180 − 50 − 60 = 70 in both, so the side 7 cm is between the 70° and 60° angles in each: ASA. ✓
Watch out. Work out the third angle if the equal side is not between the two given angles.
Key words. Two shapes are congruent if they are exactly the same size and shape. The included angle is the angle between two given sides.
Method. Check the known facts against the four conditions for congruent triangles. If none is satisfied, we cannot conclude that the triangles are congruent.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angles 40°, 60°, 80° (no sides given)
Triangle 2: angles 40°, 60°, 80° (no sides given) - Compare with the four conditions. To be sure, the facts must match SSS, SAS, ASA/AAS or RHS.Three equal angles is AAA, which is not one of the conditions: no side is known to be equal.
- State the conclusion. If none of the conditions is met, we cannot say that the triangles are congruent.The triangles are not necessarily congruent.
Check. Draw a triangle with angles 40°, 60° and 80° with a base of 5 cm, then another with a base of 10 cm: both have the same angles but different sizes. ✓
Watch out. AAA proves the triangles are similar (same shape) but not congruent (same size).
Key words. Two shapes are congruent if they are exactly the same size and shape. The included angle is the angle between two given sides.
Method. Check the known facts against the four conditions for congruent triangles. If none is satisfied, we cannot conclude that the triangles are congruent.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 5 cm, AC = 7 cm, angle A = 40°
Triangle 2: PQ = 5 cm, PR = 7 cm, angle P = 60° - Compare with the four conditions. To be sure, the facts must match SSS, SAS, ASA/AAS or RHS.Two sides are equal but the angles between them are different (40° and 60°), so SAS is not satisfied. The third sides would be different lengths.
- State the conclusion. If none of the conditions is met, we cannot say that the triangles are congruent.The triangles are not necessarily congruent.
Check. Use the cosine rule: BC2 = 25 + 49 − 70 cos 40° ≈ 20.4 and QR2 = 74 − 70 cos 60° = 39, so the third sides differ. ✓
Watch out. Equal angles are needed too, and they must be the angles between the equal sides.
Key words. Two shapes are congruent if they are exactly the same size and shape. The included angle is the angle between two given sides.
Method. Check the known facts against the four conditions for congruent triangles. If none is satisfied, we cannot conclude that the triangles are congruent.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 5, BC = 6, CA = 7
Triangle 2: PQ = 5, QR = 6, RP = 8 - Compare with the four conditions. To be sure, the facts must match SSS, SAS, ASA/AAS or RHS.Two pairs of sides are equal (5 and 6) but the third pair is not (7 and 8), so SSS is not satisfied. No other condition applies.
- State the conclusion. If none of the conditions is met, we cannot say that the triangles are congruent.The triangles are not necessarily congruent.
Check. Check the sides in order: 5, 6, 7 and 5, 6, 8. The lists are not the same. ✓
Watch out. All three sides must match for SSS, not just two.
Method. Check what is known about each triangle against the four conditions.
- List the facts. Triangle ABC: three angles only. Triangle PQR: three angles and a side.No pair of equal sides is known.
- Compare with the conditions. All four conditions need at least one pair of equal sides. We have three equal angles (AAA) only.AAA does not prove congruence.
- Give an example. ABC could be an enlargement of PQR, with the same angles but all lengths doubled.Triangle ABC could be bigger than PQR.
Check. A triangle with angles 50°, 60° and 70° with a side of 10 cm fits the angle facts and is not congruent to one with PQ = 5 cm. ✓
Watch out. Equal angles give similar triangles. To get congruent triangles you need a pair of equal sides as well.
Method. Find the position of the angle in relation to the two sides.
- Find which sides the angle is between. Angle B is between sides BA and BC. The two given sides are AB and AC, which meet at A, not at B.The 40° angle is not between the two given sides.
- Match with the conditions. This is two sides and a non-included angle (SSA), which is not a condition for congruence.SAS is not satisfied.
- Conclude. Two different triangles can have the same SSA information.Not necessarily congruent.
Check. Compare with the picture in this section: two different triangles share a 30° angle, a 10 cm side and a 6 cm opposite side. ✓
Watch out. Always check that the angle is between the two sides before using SAS.
Key words. A proof is a chain of reasons that shows a statement is always true. Each line must have a reason, such as “given”, “common side”, “alternate angles” or “vertically opposite angles”.
Method. To prove triangles are congruent, state three pairs of equal parts, each with a reason, name the condition (SSS, SAS, ASA or RHS), and then write the conclusion.
- AB = AD Given (marked with one tick).
- CB = CD Given (marked with two ticks).
- AC = AC AC is a common side of both triangles.
- Conclusion. All the facts needed for SSS are shown.Triangle ABC is congruent to triangle ADC (SSS).
Check. Corresponding angles will then be equal, for example angle BAC = angle DAC, which shows that the diagonal AC bisects the angles at A and C. ✓
Watch out. Say “common side” for a side shared by both triangles; it counts as a pair of equal sides.
Key words. A proof is a chain of reasons that shows a statement is always true. Each line must have a reason, such as “given”, “common side”, “alternate angles” or “vertically opposite angles”.
Method. To prove triangles are congruent, state three pairs of equal parts, each with a reason, name the condition (SSS, SAS, ASA or RHS), and then write the conclusion.
- AB = AC Given: the triangle is isosceles.
- BM = CM M is the midpoint of BC.
- AM = AM Common side.
- Conclusion. All the facts needed for SSS are shown.Triangle ABM is congruent to triangle ACM (SSS).
Check. It follows that angle ABM = angle ACM, which proves that the base angles of an isosceles triangle are equal. ✓
Watch out. A midpoint splits a line into two equal parts. Use it as a reason.
Key words. A proof is a chain of reasons that shows a statement is always true. Each line must have a reason, such as “given”, “common side”, “alternate angles” or “vertically opposite angles”.
Method. To prove triangles are congruent, state three pairs of equal parts, each with a reason, name the condition (SSS, SAS, ASA or RHS), and then write the conclusion.
- AM = BM M is the midpoint of AB.
- angle AMC = angle BMD Vertically opposite angles are equal.
- CM = DM M is the midpoint of CD.
- Conclusion. All the facts needed for SAS are shown.Triangle AMC is congruent to triangle BMD (SAS).
Check. The equal angle is between the two pairs of equal sides: AM and CM make angle AMC, BM and DM make angle BMD. ✓
Watch out. The included angle must be between the two equal sides. Here the vertically opposite angles are between them.
Method. Find the area of one triangle, then double it, because two congruent triangles have equal areas.
- Check for a right angle. Test Pythagoras with the sides 6, 8, 10.62 + 82 = 36 + 64 = 100 = 102, so the angle at B is 90°.
- Find the area of triangle ABC. Area = ½ × base × height with the two shorter sides as base and height.½ × 6 × 8 = 24 cm2
- Find the area of the parallelogram. Congruent triangles have equal areas, and the two triangles fill the parallelogram.2 × 24 = 48 cm2
Check. The parallelogram has base 8 and perpendicular height 6, so area = 8 × 6 = 48 cm². ✓
Watch out. Congruent shapes always have the same area and perimeter.
Exercise 21H Congruence: more practice and proof
Non-calculatorKey words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 5, BC = 7, CA = 9
Triangle 2: PQ = 5, QR = 7, RP = 9 - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.AB = PQ = 5
BC = QR = 7
CA = RP = 9 - Decide which condition is satisfied. SSS (side, side, side): all three sides of one triangle equal the three sides of the other triangle.The three equal parts make SSS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (SSS).
Check. Put the sides in order: 5, 7, 9 in both triangles. ✓
Watch out. Match sides by length.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: AB = 6, angle B = 50°, BC = 8
Triangle 2: PQ = 6, angle Q = 50°, QR = 8 - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.AB = PQ = 6
angle B = angle Q = 50° (between AB and BC, and between PQ and QR)
BC = QR = 8 - Decide which condition is satisfied. SAS (side, angle, side): two sides and the angle between them (the included angle) are equal.The three equal parts make SAS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (SAS).
Check. In both triangles the 50° angle is the angle between the 6 and the 8. ✓
Watch out. The angle must be between the two sides.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angle A = 40°, angle B = 60°, AB = 7
Triangle 2: angle P = 40°, angle Q = 60°, PQ = 7 - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.angle A = angle P = 40°
AB = PQ = 7 (the side between angles A and B, and between angles P and Q)
angle B = angle Q = 60° - Decide which condition is satisfied. ASA (angle, side, angle): two angles and the side between them are equal.The three equal parts make ASA.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (ASA).
Check. The third angles are both 180 − 40 − 60 = 80°. ✓
Watch out. The side AB lies between angle A and angle B.
Key words. Two shapes are congruent if they are exactly the same size and shape. The included angle is the angle between two given sides.
Method. Check the known facts against the four conditions for congruent triangles. If none is satisfied, we cannot conclude that the triangles are congruent.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angle A = 30°, angle B = 70°, angle C = 80°
Triangle 2: angle P = 30°, angle Q = 70°, angle R = 80° - Compare with the four conditions. To be sure, the facts must match SSS, SAS, ASA/AAS or RHS.Three equal angles is AAA. No side is known to be equal, so none of SSS, SAS, ASA or RHS applies.
- State the conclusion. If none of the conditions is met, we cannot say that the triangles are congruent.The triangles are not necessarily congruent.
Check. A triangle with these angles can be enlarged to any size and keep its angles. ✓
Watch out. AAA gives similar triangles, not congruent ones.
Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.
Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.
- Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.Triangle 1: angle B = 90°, AC = 10, AB = 6
Triangle 2: angle Q = 90°, PR = 10, PQ = 6 - Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.angle B = angle Q = 90° (right angle)
AC = PR = 10 (hypotenuse, opposite the right angle)
AB = PQ = 6 - Decide which condition is satisfied. RHS (right angle, hypotenuse, side): both triangles have a right angle, the hypotenuses are equal and one other pair of sides is equal.The three equal parts make RHS.
- State the conclusion with its reason. In an exam, name the condition as your reason.The triangles are congruent (RHS).
Check. By Pythagoras the third sides are both √(100 − 36) = 8. ✓
Watch out. RHS needs the right angle, the hypotenuse and one other side.
Method. Check which of the four conditions Ella has used.
- List what Ella knows. Two angles in each triangle, and so the third angles are equal as well (70°). No side lengths are given.Angles only: 50°, 60°, 70°
- Compare with the conditions. Every condition needs at least one equal pair of sides.AAA is not a condition.
- Show that Ella is wrong. Two triangles with the same angles can be different sizes.A triangle with a 5 cm base and one with a 10 cm base can have the same angles but are not congruent.
Check. This is why we use the word similar for triangles with equal angles. ✓
Watch out. You need at least one side as well as angles.
Method. Congruent shapes must have all matching sides equal, not only the same perimeter.
- Check the perimeters. 2 × (6 + 2) and 2 × (5 + 3).16 cm and 16 cm: equal
- Check the sides. Congruent rectangles would need the same length and the same width.6 by 2 is different from 5 by 3.
- Conclude. The areas differ too: 12 cm2 and 15 cm2.The rectangles are not congruent.
Check. If two shapes are congruent, their areas must also be equal. Here 12 ≠ 15. ✓
Watch out. Equal perimeters are not enough to prove congruence.
Method. Think about what reflections and rotations do to a shape.
- First transformation. A reflection flips the shape but does not change any length or angle.A and B are congruent.
- Second transformation. A rotation turns the shape but does not change any length or angle.B and C are congruent.
- Link them together. If A fits exactly onto B and B fits exactly onto C, then A fits exactly onto C (maybe after being flipped).A and C are congruent.
Check. Only an enlargement would change the size of the shape. ✓
Watch out. Congruent shapes may be flipped or turned: they are still congruent.
Method. Congruent triangles have the same perimeter, because their sides are equal in pairs.
- State the perimeter of PQR. It is the same as the perimeter of ABC.Perimeter of PQR = 24 cm
- Add the two known sides. PQ + QR = 7 + 9.7 + 9 = 16 cm
- Take this away from the perimeter. What is left is PR.24 − 16 = 8 cm
Check. 7 + 9 + 8 = 24. ✓
Watch out. Congruent shapes have equal perimeters and equal areas.
Method. For squares, equal side lengths are enough, since all the angles are 90°.
- Work out the side of each square. A square has four equal sides, so side = perimeter ÷ 4.20 ÷ 4 = 5 cm
- Compare the angles. Every angle in a square is 90°.All angles are 90° in both squares.
- Conclude. The sides are equal and the angles are equal.The squares are congruent.
Check. Area = 5 × 5 = 25 cm2 for both. ✓
Watch out. All squares have the same angles. Only the side lengths decide whether two squares are congruent.
Key words. A proof is a chain of reasons that shows a statement is always true. Each line must have a reason, such as “given”, “common side”, “alternate angles” or “vertically opposite angles”.
Method. To prove triangles are congruent, state three pairs of equal parts, each with a reason, name the condition (SSS, SAS, ASA or RHS), and then write the conclusion.
- AB = AD All four sides of a rhombus are equal.
- BC = DC All four sides of a rhombus are equal.
- AC = AC AC is a common side of both triangles.
- Conclusion. All the facts needed for SSS are shown.Triangle ABC is congruent to triangle ADC (SSS).
Check. The matching angles are equal, so the diagonal AC bisects angle A and angle C. ✓
Watch out. State why the sides are equal (“all sides of a rhombus are equal”).
Key words. Corresponding sides of congruent triangles are the sides in the same position. They are equal.
Method. To prove the triangles congruent, find two pairs of equal sides and the equal angles between them (SAS). Then corresponding sides of congruent triangles are equal.
- AB = DC Opposite sides of a rectangle are equal.Equal sides: AB = DC
- Angle ABC = angle DCB Every angle of a rectangle is a right angle.Both are 90°, and each is between the equal sides AB, BC and DC, CB.
- BC = CB BC is a common side.Third pair of equal sides
- Conclude the congruence. Two sides and the angle between them are equal: SAS.Triangle ABC is congruent to triangle DCB (SAS).
- Deduce the diagonals. Corresponding sides of congruent triangles are equal. AC in the first triangle matches DB in the second.AC = DB, so the diagonals are equal.
Check. In a 6 by 8 rectangle, both diagonals are √(36 + 64) = 10. ✓
Watch out. “Hence” means: use the result you have just proved.
Key words. A proof is a chain of reasons that shows a statement is always true. Each line must have a reason, such as “given”, “common side”, “alternate angles” or “vertically opposite angles”.
Method. To prove triangles are congruent, state three pairs of equal parts, each with a reason, name the condition (SSS, SAS, ASA or RHS), and then write the conclusion.
- AB = CD Given.
- angle BAC = angle DCA Alternate angles are equal, because AB is parallel to DC and AC crosses both lines.
- AC = CA Common side.
- Conclusion. All the facts needed for SAS are shown.Triangle ABC is congruent to triangle CDA (SAS).
Check. The angle BAC is between AB and AC, and the angle DCA is between CD and CA, so it is the included angle in both triangles. ✓
Watch out. Alternate angles need parallel lines; give the reason.
Method. Find where the 30° angle is in relation to the two sides.
- Locate the angle. Angle A is between AC and AB. The two sides given are AC and BC.Angle A is not between AC and BC.
- Name the pattern. Two sides and an angle that is not between them is SSA, which is not a condition for congruence.SSA is not enough.
- Explain with the diagram. Swing the 6 cm side round from C: it can meet the line from A at two different points, B1 and B2. Both triangles have A = 30°, AC = 10 cm and a 6 cm side opposite A.Two different triangles fit the information, so they need not be congruent.
Check. In triangle AB2C the angle at B is obtuse, but in triangle AB1C it is acute, so the triangles are different. ✓
Watch out. For SAS, check that the angle is between the two sides.