Investigate properties of similar triangles

Ontario Grade 10 Mathematics

Measurement and Trigonometry

How matching angles and side lengths show that triangles have the same shape

A small road sign and a larger version can have different side lengths but the same shape. Triangles can work the same way. In this lesson, you will compare angles and side lengths to investigate whether two triangles are similar. First, remember that an angle measures a turn and that the three interior angles of a triangle add to

180^\circ

. A ratio compares two quantities by division. These ideas help describe relationships between triangles.

What you will learn

Explain what it means for two triangles to be similar.
Identify corresponding angles and sides.
Use equal corresponding angles and proportional corresponding sides to investigate similarity.
Find an unknown side length using a scale factor.

1. Similar means same shape, not necessarily same size

Two triangles are similar when they have the same shape. Their sizes may differ. To investigate whether two triangles are similar, compare their angles and side lengths.

Angles in matching positions are called corresponding angles. Sides between the same pair of corresponding angles are called corresponding sides. For example, if angle

A

matches angle

D

, and angle

B

matches angle

E

, then side

AB

matches side

DE

.

For similar triangles, corresponding angles have equal measures. Corresponding side lengths have the same ratio. That ratio is called the scale factor when it describes how one triangle's side lengths compare with the other's. A scale factor greater than

1

means the compared triangle is larger. A positive scale factor less than

1

means it is smaller.

The order of the letters matters when naming triangles. In

\triangle ABC \sim \triangle DEF

, the first letters match, the second letters match, and the third letters match. So

A

matches

D

,

B

matches

E

, and

C

matches

F

. The symbol

\sim

means “is similar to.”

Similar triangles have equal corresponding angles.
Their corresponding side lengths are proportional, meaning matching side ratios are equal.
Match vertices in order before comparing side lengths.

2. Investigate the angle and side relationships

A practical way to investigate similarity is to mark angles that appear to match, then compare the matching sides. A drawing can suggest a match, but a sketch alone does not establish that the side lengths are proportional. Use given measurements or calculate them.

Suppose two triangles each have angle measures

40^\circ

,

60^\circ

, and

80^\circ

. Each angle in one triangle matches an equal angle in the other. In a triangle, knowing two angles also determines the third, because the three angle measures add to

180^\circ

.

If corresponding sides have lengths

3

,

4

, and

5

in one triangle, and

6

,

8

, and

10

in another, each side in the second triangle is twice its match. The ratios are equal, so the side lengths are proportional. The triangles have the same shape and differ by a scale factor of

2

.

Use matching parts to investigate a pair of triangles. Equal corresponding angles support a similarity match. Equal ratios for all three pairs of corresponding sides also support a match. In either case, match the parts correctly. Comparing sides that do not correspond can lead to a false conclusion.

\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}

The interior angles of a triangle add to $180^\circ$ .
Check angles and side ratios using correctly matched parts.
A common scale factor multiplies every side length by the same amount.

3. Use a scale factor to find a missing side

Once you know two triangles are similar and have matched their vertices, use a pair of corresponding sides to find the scale factor. Apply that same factor to the other matching sides. This works because the side ratios stay equal in similar triangles.

Keep the order of a ratio consistent. If you divide a side in the larger triangle by its match in the smaller triangle, use that same order for every pair. You can also write ratios in the opposite direction, but then every ratio must use that reversed order.

When solving for an unknown side, write a proportion. A proportion is a statement that two ratios are equal. Multiply or divide to isolate the unknown, then check that the result makes the corresponding side ratios agree. The answer should also make sense compared with the scale factor.

For example, if

k

compares the larger triangle to the smaller triangle, divide a larger side length by its matching smaller side length to find

k

.

k=\frac{\text{larger side}}{\text{matching smaller side}}

Find a scale factor from a known pair of corresponding sides.
Use the same direction for each ratio.
Check that the unknown side fits the established scale factor.

4. Guided example and independent practice

Use the worked example to see how correspondence, a scale factor, and a check fit together. The triangles are already stated to be similar, so the task is to use their matching sides rather than decide whether they are similar.

For independent practice, consider two similar triangles. A side of length

7

in the smaller triangle corresponds to a side of length

10.5

in the larger triangle. Another smaller side has length

6

. Find its matching side in the larger triangle. Set up a ratio in one consistent direction, solve, and check using the scale factor. Do not assume that sides match just because they look close in a sketch.

Similarity gives a relationship between corresponding sides.
A proportion helps find a missing side length.
Use the stated correspondence, not the apparent orientation of a drawing.

Matching parts in the similarity statement

Part in triangle $ABC$	Corresponding part in triangle $DEF$
Angle $A$	Angle $D$
Angle $B$	Angle $E$
Angle $C$	Angle $F$
Side $AB$	Side $DE$
Side $BC$	Side $EF$
Side $AC$	Side $DF$

Worked example

Find a missing corresponding side

Triangle

ABC

is similar to triangle

DEF

, in that order. Side

AB

is

6

cm and matches

DE

, which is

9

cm. Side

BC

is

8

cm and matches

EF

, which is

x

cm. Find

x

.

Match the sides
The order in the similarity statement shows that $AB$ matches $DE$ and $BC$ matches $EF$ . The larger-to-smaller ratio is consistent for both pairs.
Find the scale factor
Divide the known side in the second triangle by its matching side in the first triangle. The result tells how much the side lengths have been multiplied by.
$\frac{9}{6}=1.5$
Apply the same factor
The side matching $BC$ must also be multiplied by $1.5$ . This follows because corresponding side ratios are equal for similar triangles.
$x=8(1.5)=12$
Check the ratio
Compare the known and calculated pairs in the same direction. Both ratios equal $1.5$ , so the missing length is consistent with the similarity relationship.
$\frac{9}{6}=\frac{12}{8}=1.5$

Answer:

EF=12

cm.

Check: The scale factor from triangle

ABC

to triangle

DEF

is

1.5

. Multiplying

8

cm by

1.5

gives

12

cm.

Common mistakes and how to avoid them

Matching sides by their position on the page rather than by their endpoints.

Correction: Use the order in the similarity statement or identify which vertices correspond first. Then match the sides between those vertices.

Using different ratio directions for different side pairs.

Correction: Choose larger over smaller, or smaller over larger, and keep that order throughout the proportion.

Assuming that triangles are similar because they look alike in a drawing.

Correction: Use angle measures or side lengths to check the relationships. A sketch may not be drawn accurately.

Multiplying by the scale factor in the wrong direction.

Correction: Check which triangle is larger. A factor of

1.5

from smaller to larger means multiply the smaller side by

1.5

.

Lesson summary

Similar triangles have the same shape, though they may have different sizes.
Corresponding angles are equal, and corresponding side lengths are proportional.
The order of the triangle names identifies matching vertices and sides.
Use one consistent ratio direction to find or check a missing side.

Check your understanding

Question 1

In

\triangle PQR \sim \triangle XYZ

, which side corresponds to

QR

?

$XY$
$YZ$
$XZ$
$PQ$

Show answer and explanation

YZ

The second vertex

Q

matches

Y

, and the third vertex

R

matches

Z

. Therefore, side

QR

matches side

YZ

.

Question 2

A side of length

4

in a smaller triangle matches a side of length

10

in a similar larger triangle. What is the scale factor from smaller to larger?

$0.4$
$2.5$
$6$
$14$

Show answer and explanation

2.5

Divide the larger side by the smaller matching side:

10 \div 4=2.5

.

Question 3

A smaller triangle has a side of length

9

. Its matching side in a similar larger triangle is found using a scale factor of

2

. What is the larger side length?

$4.5$
$11$
$18$
$20$

Show answer and explanation

18

From smaller to larger, multiply by the scale factor:

9(2)=18

.

Key terms

Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths.
Corresponding parts: Angles or sides that match because they occupy matching positions in two figures.
Proportion: A statement that two ratios are equal.
Scale factor: The number used to multiply side lengths in one figure to get the corresponding side lengths in a similar figure.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M2. It is a study resource, not an official curriculum publication.

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