Compare similarity and congruence

Ontario Grade 10 Mathematics

Trigonometry

How to tell whether figures have the same shape, the same size, or both

A small photo and a larger copy can show the same scene without being the same size. They have the same shape, but one has been enlarged. Two identical floor tiles have the same shape and the same size. These are different relationships: similarity and congruence. To compare figures, match their corresponding angles and sides, then look at how their measurements relate.

What you will learn

Explain the difference between similar figures and congruent figures.
Match corresponding sides and angles in two figures.
Use angle measures and side lengths to decide whether two triangles are similar or congruent.
Describe the scale factor between similar figures.

1. Grade 9 bridge: match the parts

A polygon is a closed figure made from straight sides. Triangles and rectangles are examples. Before comparing two figures, identify which parts occupy matching positions. These matching parts are called corresponding parts.

Suppose we compare triangle

ABC

with triangle

DEF

. If

A

matches

D

,

B

matches

E

, and

C

matches

F

, then side

AB

corresponds to side

DE

. Angle

B

corresponds to angle

E

. The symbol

\angle

means angle.

An angle measure tells how wide an angle opens. It is usually measured in degrees. A side length tells how long a side is. Use the same units when comparing lengths. For example, convert centimetres to metres first if the measurements use different units.

Matching parts carefully is important. A side that looks close to another side on the page is not necessarily its corresponding side. Use the named vertices or the position of the parts in the figures to match them.

Corresponding parts occupy matching positions in two figures.
Match vertices first, then compare the sides and angles connected to them.
Use consistent units for side lengths.

2. Similar figures: same shape, possibly different size

Two figures are similar when they have the same shape, even if their sizes differ. For similar polygons, corresponding angles have equal measures, and all corresponding side lengths have one common ratio. A ratio compares two quantities by division.

For triangles, there are two useful ways to establish similarity. If all three pairs of corresponding angles are equal, the triangles are similar. Or, if all three pairs of corresponding sides have the same ratio, the triangles are similar. These tests are for triangles; they are not general tests for other polygons.

Imagine one triangle with side lengths

3

,

4

, and

5

units and a second triangle with corresponding side lengths

6

,

8

, and

10

units. Each side in the second triangle is twice its matching side in the first. Their matching angles are equal, so the triangles have the same shape but different sizes.

The common number that relates corresponding lengths is called the scale factor. To find it, divide a length in the figure you are moving to by its corresponding length in the starting figure. State the direction because reversing the comparison changes the scale factor.

From the smaller triangle to the larger one, the scale factor is

2

. From the larger triangle to the smaller one, it is

\frac{1}{2}

. A scale factor greater than

1

increases lengths. A positive scale factor less than

1

reduces them. In symbols, the scale factor is the corresponding length in the new figure divided by the corresponding length in the starting figure, as shown below.

k=\frac{\text{new length}}{\text{starting length}}

Similar polygons have equal corresponding angles and corresponding side lengths in one common ratio.
For triangles, equal measures for all three corresponding angles establish similarity.
For triangles, one common ratio for all three pairs of corresponding sides also establishes similarity.
Keep the direction of the scale factor consistent.

3. Congruent figures: same shape and same size

Two figures are congruent when they have the same shape and the same size. Their corresponding angles have equal measures, and their corresponding side lengths are equal. A figure can be turned or flipped and still be congruent to another figure. Its position does not change its shape or size.

Congruent figures are also similar. Their corresponding side lengths have a common ratio of

1

, because the lengths do not change. For example, two triangles with matching side lengths

5

,

7

, and

9

units, and matching corresponding angles, are congruent.

The reverse is not always true. Similar figures can have different sizes, so they do not have to be congruent. Similarity allows lengths to change by a common scale factor. Congruence requires corresponding lengths to stay equal.

For polygons with more than three sides, compare every corresponding angle and every corresponding side. Similarity requires all corresponding angles to match and all side pairs to have one common ratio. Congruence requires all corresponding angles and side lengths to match. Do not decide that two such polygons are similar by checking only a few parts.

\text{congruent} \Rightarrow \text{similar}

Congruent means same shape and same size.
Congruent figures have equal corresponding angles and equal corresponding side lengths.
Congruent figures are similar with scale factor $1$ .

4. Compare carefully, then practise

Use a steady process. First, write down which vertices correspond. Next, compare corresponding angles. If any matching angles differ, the figures are neither similar nor congruent.

For triangles, matching all three angles establishes similarity. You can also compare all three side ratios. If they agree, the triangles are similar. To decide whether similar triangles are congruent, check whether the corresponding side lengths are equal. A scale factor of

1

means the lengths are equal.

For other polygons, use the full definitions. Check every corresponding angle and every corresponding side. Similarity requires equal corresponding angles and one common ratio for all side pairs. Congruence requires equal corresponding angles and equal side lengths.

Independent practice: Two quadrilaterals have equal corresponding angles. The first has side lengths

4

,

6

,

8

, and

10

units. The corresponding sides of the second have lengths

6

,

9

,

12

, and

15

units. Each matching side in the second figure is

\frac{3}{2}

times its partner in the first. The quadrilaterals are similar, but not congruent, because the scale factor is not

1

.

Now suppose the last side in the second quadrilateral were

16

units instead of

15

. The first three side ratios would be

\frac{3}{2}

, but the last would be

\frac{16}{10}

. Since these ratios are not all equal, the figures would not be similar, even though their angles match.

Match parts before comparing measurements.
Use triangle similarity tests only for triangles.
For other polygons, check every corresponding angle and side ratio.
Equal angles alone do not show that figures are congruent.

Compare the relationships

Relationship	Corresponding angles	Corresponding side lengths	Size
Similar	Equal	All in one common ratio	May differ
Congruent	Equal	Equal	Same

Worked example

Compare two triangles

Triangle

ABC

has side lengths

4

,

6

, and

8

units. Triangle

DEF

has side lengths

6

,

9

, and

12

units. Their corresponding angles are equal, with

A

matching

D

,

B

matching

E

, and

C

matching

F

. Are the triangles similar, congruent, both, or neither?

Match the sides
Use the stated vertex matches to pair the sides. The side of length $4$ matches $6$ , the side of length $6$ matches $9$ , and the side of length $8$ matches $12$ .
Compare the ratios
Divide each length in triangle $DEF$ by its matching length in triangle $ABC$ . Each calculation gives the same scale factor, so all three side pairs have one common ratio.
$\frac{6}{4}=\frac{9}{6}=\frac{12}{8}=\frac{3}{2}$
Decide which relationship applies
The corresponding angles are equal, and all three side ratios agree. The triangles are similar. Their scale factor is not $1$ , so their corresponding side lengths are not equal and they are not congruent.

Answer: The triangles are similar but not congruent. The scale factor from triangle

ABC

to triangle

DEF

is

\frac{3}{2}

.

Check: Multiplying each side of triangle

ABC

by

\frac{3}{2}

gives

6

,

9

, and

12

, the side lengths of triangle

DEF

.

Common mistakes and how to avoid them

Calling figures congruent just because their angles match.

Correction: Matching angles alone do not show that figures are the same size. Check corresponding side lengths as well.

Using a triangle similarity test for a quadrilateral.

Correction: Triangle tests apply to triangles. For other polygons, check every corresponding angle and every corresponding side ratio.

Checking only one pair of corresponding sides.

Correction: For similarity, all corresponding side lengths must have one common ratio. Compare every pair.

Dividing side lengths in different directions.

Correction: Keep the order consistent. For example, divide every length in the second figure by its matching length in the first.

Thinking a rotated or flipped figure cannot be congruent.

Correction: Turning or flipping a figure does not change its size or shape. Match corresponding parts instead of relying on their position on the page.

Lesson summary

Similar figures have equal corresponding angles and side lengths in one common ratio.
For triangles, matching all three corresponding angles or having one common ratio for all three side pairs establishes similarity.
For other polygons, check every corresponding angle and side ratio.
Congruent figures have equal corresponding angles and equal corresponding side lengths.
Congruent figures are similar with scale factor $1$ .

Check your understanding

Question 1

Two triangles have equal corresponding angles. Each side of the second triangle is

3

times its matching side in the first triangle. What is true?

They are congruent but not similar.
They are similar but not congruent.
They are both similar and congruent.
They are neither similar nor congruent.

Show answer and explanation

They are similar but not congruent.

Equal corresponding angles show that the triangles are similar. The scale factor is

3

, not

1

, so their corresponding side lengths are not equal and they are not congruent.

Question 2

Two polygons have equal corresponding angles, and every corresponding side has equal length. How should they be described?

Similar but not congruent.
Congruent and similar.
Congruent but not similar.
Neither similar nor congruent.

Show answer and explanation

Congruent and similar.

All corresponding angles and side lengths match, so the polygons are congruent. Their side ratio is

1

, so they are also similar.

Question 3

Three pairs of corresponding sides of two triangles have ratios

2

,

2

, and

3

. What can you conclude?

The triangles are similar.
The triangles are congruent.
The triangles are not similar because the ratios do not agree.
The triangles must have equal corresponding angles.

Show answer and explanation

The triangles are not similar because the ratios do not agree.

Similar triangles must have corresponding side lengths in one common ratio. These ratios differ, so the triangles are not similar.

Key terms

Corresponding parts: Sides or angles that match in position when two figures are compared.
Ratio: A comparison of two quantities by division.
Scale factor: The common number used to multiply corresponding lengths when changing from one similar figure to another.
Similar figures: Figures with equal corresponding angles and corresponding side lengths in one common ratio.
Congruent figures: Figures with the same shape and size, with equal corresponding angles and side lengths.

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