Congruence and Similarity
Two shapes are congruent if they are exactly the same size and shape, so one can be mapped onto the other using only reflections, rotations and translations, without any resizing, while two shapes are similar if they are the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides all in the same ratio, called the scale factor. Congruent triangles can be shown using the conditions SSS, SAS, ASA or RHS, matching three sides, two sides and the included angle, two angles and the included side, or a right angle, hypotenuse and one other side.
Before you start
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Method
- To decide if two shapes are congruent, check whether every side length and every angle matches exactly, in the same order around the shape.
- For triangles, you only need to check one of the four conditions, SSS, SAS, ASA or RHS, rather than every side and angle.
- To decide if two shapes are similar, check that all corresponding angles are equal and that all corresponding sides are in the same ratio.
- Find the scale factor between similar shapes by dividing a length on the larger shape by the matching length on the smaller shape (or the reverse, to scale down).
- To find a missing length in a similar shape, multiply (or divide) the matching known length by the scale factor.
- Always match corresponding vertices carefully: in triangle ABC similar to triangle PQR, side AB corresponds to side PQ, so compare the sides in the same order.
Worked example
Triangle ABC is similar to triangle PQR. AB = 6 cm, BC = 9 cm, and AC = 7.5 cm. The side PQ, which corresponds to AB, is 18 cm. Find the lengths of QR and PR.
- Find the scale factor from triangle ABC to triangle PQR using the pair of corresponding sides you know both lengths for: scale factor = PQ / AB = 18 / 6 = 3.
- QR corresponds to BC, so QR = BC x scale factor = 9 x 3 = 27 cm.
- PR corresponds to AC, so PR = AC x scale factor = 7.5 x 3 = 22.5 cm.
Practice questions
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Q1Triangle DEF is congruent to triangle XYZ. DE = 8 cm and angle D = 55 degrees. State the length of XY and the size of angle X.Show answer
Answer: XY = 8 cm and angle X = 55 degrees, since congruent shapes have exactly matching sides and angles
Q2State which condition, SSS, SAS, ASA or RHS, proves two triangles are congruent when all three sides of each triangle match.Show answer
Answer: SSS (side, side, side)
Q3State which condition proves two right-angled triangles are congruent when the hypotenuse and one other side match in both triangles.Show answer
Answer: RHS (right angle, hypotenuse, side)
Q4Rectangle A measures 4 cm by 6 cm. Rectangle B is similar to rectangle A, with its shorter side 10 cm. Find the length of the longer side of rectangle B.Show answer
Answer: scale factor = 10/4 = 2.5, so the longer side = 6 x 2.5 = 15 cm
Q5Triangle GHI is similar to triangle JKL, with scale factor 4. Side GH = 3.2 cm. Find the length of side JK.Show answer
Answer: 3.2 x 4 = 12.8 cm
Q6Triangle MNO is similar to triangle PQR, with scale factor 4. Side PQ, which corresponds to MN, is 30 cm. Find the length of MN.Show answer
Answer: 30 / 4 = 7.5 cm
Q7Two similar quadrilaterals have a pair of corresponding sides of 5 cm and 8 cm. A second pair of corresponding sides has a length of 12 cm on the larger shape. Find the length of the matching side on the smaller shape.Show answer
Answer: scale factor = 8/5 = 1.6, so the smaller side = 12 / 1.6 = 7.5 cm
Q8A photo measuring 10 cm by 15 cm is enlarged so that it is similar to the original, with a new shorter side of 25 cm. Find the length of the new longer side.Show answer
Answer: scale factor = 25/10 = 2.5, so the longer side = 15 x 2.5 = 37.5 cm
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Triangle ABC has AB = 5 cm, BC = 7 cm and angle B = 90 degrees. Triangle PQR has PQ = 5 cm, QR = 7 cm and angle Q = 90 degrees. (a) State the condition that proves triangle ABC is congruent to triangle PQR. (b) Triangle ABC is then reflected to form triangle A'B'C'. State whether A'B'C' is congruent or similar to ABC, giving a reason.
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Two similar triangles are shown. Triangle 1 has sides of 6 cm, 10 cm and 12 cm. The shortest side of triangle 2, which corresponds to the shortest side of triangle 1, is 15 cm. (a) Find the scale factor from triangle 1 to triangle 2. (b) Find the lengths of the other two sides of triangle 2. (c) The perimeter of triangle 1 is 28 cm. Use this to find the perimeter of triangle 2, and state the connection between your two answers.
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Free printable worksheet
Want more practice on paper? Download the congruence and similarity worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 18 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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