Congruence

Congruent shapes are exactly the same shape and size, but one may be turned or flipped over.

Think about the difference between congruent shapes and enlargements.

Swap each letter of BC for the letter it matches.

Find which side of triangle ABC matches EF. Use the 6 cm sides and the 50° angles to help.

Angle Z matches angle R. Use the angles in triangle PQR.

Angle y is between the side with one dash and the side with two dashes. Find the angle in triangle ABC between those two sides.

1. In triangle DEF, angle y is between the one-dash side and the two-dash side.
2. In triangle ABC, the angle between the one-dash side AB and the two-dash side BC is angle B.
3. Matching angles in congruent triangles are equal, so y=64.

B is at the end of the longer side that meets the right angle. Find the vertex in the same place on triangle PQR.

1. A is the right angle and B is at the other end of the 4-unit side.
2. In triangle PQR, R is the right angle and P is at the other end of the 4-unit side.
3. So vertex B matches vertex P.

Congruent triangles have exactly the same side lengths, in any order.

Part (a)

1. 9 cm, 5 cm and 8 cm are the same three lengths as triangle T, just listed in a different order.
2. 10 cm, 16 cm and 18 cm is an enlargement: the same shape but a different size.
3. The congruent triangle has sides 9 cm, 5 cm and 8 cm.

Part (b)

1. Triangle U has the same three side lengths as triangle T: 5 cm, 8 cm and 9 cm.
2. 9 cm and 5 cm are already used, so the third side of triangle U is 8 cm.

In triangle ABC the 40° angle is at B, where the 6 cm side and the 9 cm side meet.

1. Angle ABC is the angle between AB and BC, so triangle ABC gives two sides and the angle between them.
2. Angle PQR is at Q, which is where PQ and QR meet.
3. So PQ = 6 cm, QR = 9 cm and angle PQR = 40° gives the same two sides and the angle between them.
4. In the other options the 40° angle is not between the two given sides, or a side is the wrong length.

Turning a triangle round does not change its size. Could the third sides be different lengths? Think about Pythagoras' theorem.

1. Turning a triangle round does not change its shape or size.
2. In each triangle the third side can be found with Pythagoras' theorem: \sqrt{13^2-5^2}=\sqrt{144}=12 cm.
3. So both triangles have sides 5 cm, 12 cm and 13 cm.
4. Yes, they are congruent: both have a right angle, a 13 cm side opposite it and a 5 cm side.

Swap each letter for the letter it matches. For example, DE matches BC.

Part (a)

1. D matches B and E matches C, so DE matches BC.
2. BC = 11 cm, so x=11.

Part (b)

1. E matches C and F matches A, so EF matches CA.
2. AC = 8 cm, so y=8.

Congruent triangles are the same size, so what is the perimeter of triangle XYZ?

1. Congruent triangles are exactly the same size, so triangle XYZ also has a perimeter of 30 cm.
2. XY + YZ = 8 + 12 = 20 cm.
3. XZ = 30 − 20 = 10 cm.

Think about an enlargement of a triangle. Which facts stay the same and which change?

1. An enlargement has the same three angles as the original triangle, but its sides are a different length.
2. So equal angles alone do not show the triangles are the same size.
3. Each of the other three fixes the size of the triangle as well as its shape, so each one does show that two triangles are congruent.

A side is between two angles. Find the side of triangle PQR between the same two angles.

Part (a)

1. AB is the side between the 35° angle and the 100° angle.
2. In triangle PQR, the 35° angle is at Q and the 100° angle is at R.
3. So AB matches QR.

Part (b)

1. AC is the side between the 35° angle and the 45° angle.
2. In triangle PQR, the 35° angle is at Q and the 45° angle is at P.
3. So AC matches PQ.

First find angle ACB in triangle ABC. Then find the angle that matches it in triangle ADC.

1. In triangle ABC, angle ACB =180-32-118=30°.
2. C matches C, A matches A and B matches D, so angle ACD matches angle ACB. Angle ACD = 30°.
3. Angle BCD = angle ACB + angle ACD = 30 + 30 = 60°.

DE matches AB. Use this to find AD first.

1. DE matches AB, so DE = 10 cm.
2. AD = AE − DE = 16 − 10 = 6 cm.
3. DB = AB − AD = 10 − 6 = 4 cm.

Angle ECD matches angle ABC. Then use angles on a straight line at C.

1. B matches C, so angle ECD matches angle ABC. Angle ECD = 70°.
2. Angles on a straight line add up to 180°.
3. At C: x=180-45-70=65.

Find all three sides of one triangle, using the matching side from the other triangle. Then change the total length to metres.

1. AC matches DF, so AC = 75 cm.
2. Perimeter of one triangle = 45 + 60 + 75 = 180 cm.
3. Two triangles need 2 × 180 = 360 cm = 3.6 m of wire.
4. Cost = 3.6 × £1.20 = £4.32.

AC is inside the parallelogram, so it is not part of its perimeter. Find AB + BC first.

1. AB + BC = 40 − 14 = 26 cm.
2. Triangle CDA matches triangle ABC, so CD = AB and DA = BC. So CD + DA = 26 cm.
3. The diagonal AC is inside the shape, so it is not part of the perimeter.
4. Perimeter of ABCD = 26 + 26 = 52 cm.

Use the angles in a triangle to find angle Q, then compare what the two triangles have in common.

Part (a)

1. The angles in a triangle add up to 180°.
2. Angle PQR =180-50-70=60°.

Part (b)

1. Angle P = angle A = 50° and angle Q = angle B = 60°.
2. PQ is between angles P and Q, and AB is between angles A and B. Both are 8 cm.
3. So two angles and the side between them are equal: the triangles are congruent.
4. Equal angles alone would not be enough, because an enlargement has the same angles.

Part (c)

1. QR is between the 60° angle and the 70° angle.
2. In triangle ABC, BC is between angle B (60°) and angle C, which is 180-50-60=70°.
3. So QR matches BC, and y=6.5.