Investigate equal angles and proportional sides in similar triangles

Ontario Grade 10 Mathematics

Trigonometry

How to identify matching parts and compare side lengths

Two triangles can have the same shape but different sizes. When triangles are similar, their corresponding angles are equal and their corresponding side lengths are proportional. Proportional means that matching side lengths have equal ratios. To compare triangles safely, first identify which parts correspond. Then compare side lengths in a consistent direction.

What you will learn

Recall what equal angles, ratios, and proportions mean.
Identify corresponding angles and sides from a similarity statement.
Explain how equal angles and proportional sides describe similar triangles.
Use a scale factor to find a missing side length.

1. Grade 9 bridge: angles, ratios, and matching parts

An angle measures a turn. Its measure is often written with a degree symbol. For example, angles measuring

50^\circ

are equal, even if the triangles containing them are different sizes.

A ratio compares two quantities by division. For example, the ratio of

6

to

8

can be written as CAD 6:8 or

\frac{6}{8}

. The order matters: CAD 6:8 compares the first quantity to the second. A proportion is a statement that two ratios are equal.

A triangle is named by its three vertices, or corner points. In triangle

ABC

, the vertices are

A

,

B

, and

C

. Its angles are at those vertices, and its sides are

AB

,

BC

, and

AC

. Corresponding parts are parts that match between two shapes. They can be vertices, angles, or sides.

To find a matching side, look at the vertices at its ends. If

A

matches

D

and

B

matches

E

, then side

AB

matches side

DE

. This matching process matters because comparing the wrong sides gives a misleading ratio.

\frac{a}{b}=\frac{c}{d}

Equal angles have the same measure.
A ratio compares quantities in a stated order; a proportion says two ratios are equal.
Corresponding parts are parts that match between two shapes.

2. What makes triangles similar?

Similar triangles have the same shape, although they may have different sizes. Their corresponding angles have equal measures. Their corresponding side lengths are proportional, meaning every matching pair has the same ratio when compared in the same direction.

The symbol

\sim

means “is similar to.” In the statement

\triangle ABC \sim \triangle DEF

, the order tells you which vertices match:

A

with

D

,

B

with

E

, and

C

with

F

. The triangle symbol indicates that the letters name a triangle.

These vertex matches give the matching angles and sides. The angle at

A

matches the angle at

D

, and the angle at

B

matches the angle at

E

. Side

AB

matches

DE

, side

BC

matches

EF

, and side

AC

matches DF.

A scale factor is the number that multiplies the side lengths of one triangle to give the matching side lengths of the other. The direction must be stated. If a matching side changes from

6

units to

9

units, the scale factor from the first triangle to the second is

\frac{9}{6}

. In the reverse direction, it is

\frac{6}{9}

.

The table shows how the order of the letters guides the matches. A sketch may not be drawn to scale, so do not rely only on how long a side looks in a picture.

\triangle ABC \sim \triangle DEF

Similar triangles have equal corresponding angles and proportional corresponding side lengths.
The letter order in a similarity statement identifies corresponding vertices.
A scale factor compares matching side lengths in a chosen direction.

3. Compare matching side lengths

After identifying corresponding sides, compare them in a consistent order. For example, if you compare lengths in triangle

DEF

to their matches in triangle

ABC

, put each

DEF

side over its matching

ABC

side. Similar triangles give the same ratio for each pair.

A missing side can be found with a scale factor. First use a pair of known corresponding sides to calculate the factor. Then multiply the other known side by that same factor. This works because all corresponding side lengths are proportional.

You can also write a proportion using two pairs of matching sides. Keep the corresponding sides in the same positions in both ratios. Once you find the unknown length, check whether it fits the direction of scaling. If the second triangle is larger, its matching sides should be longer.

Equal-angle matches help confirm that the side matches make sense. The vertices at the ends of corresponding sides must match. Do not reverse a side pair or compare sides just because they appear in a similar place on the page.

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

List corresponding side pairs before forming ratios.
Use the same direction for every ratio in a comparison.
Check that the answer fits the stated size relationship.

4. Independent practice

Try these questions before using the quick check. Start by using the order of the letters to match vertices, angles, and sides. For a scale factor, state which triangle is the starting triangle and which is the target triangle.

If

\triangle RST \sim \triangle UVW

, identify the angle corresponding to the angle at

S

and the side corresponding to side

ST

.

Two similar triangles have matching side lengths

5

and

15

, in that order. Find the scale factor from the first triangle to the second.

A side of length

7

in one triangle corresponds to a side of length

21

in a similar triangle. Another side in the first triangle is

4

. Find the length of its matching side in the second triangle.

For each question, explain how the vertex order or scale factor supports your answer. This helps you check that you are comparing matching parts, not just choosing numbers that seem to fit.

Use the similarity statement to identify corresponding parts.
Apply one consistent scale factor to every pair of corresponding sides.

Corresponding parts when $\triangle ABC \sim \triangle DEF$

Part in $\triangle ABC$	Matching part in $\triangle DEF$
Vertex $A$	Vertex $D$
Vertex $B$	Vertex $E$
Vertex $C$	Vertex $F$
Angle at $A$	Angle at $D$
Angle at $B$	Angle at $E$
Angle at $C$	Angle at $F$
Side $AB$	Side $DE$
Side $BC$	Side $EF$

Worked example

Finding a missing side in similar triangles

Triangles

ABC

and

DEF

are similar, with

\triangle ABC \sim \triangle DEF

. Side

AB

is

6

units, side

BC

is

8

units, and side

DE

is

9

units. Find the length of

EF

.

Match the parts
The order of the similarity statement pairs $A$ with $D$ , $B$ with $E$ , and $C$ with $F$ . Therefore, side $AB$ matches $DE$ , and side $BC$ matches $EF$ .
$AB\leftrightarrow DE,\quad BC\leftrightarrow EF$
Find the scale factor
Compare triangle $DEF$ to triangle $ABC$ . The known matching lengths change from $6$ to $9$ , so divide the length in $DEF$ by its match in $ABC$ .
$\frac{DE}{AB}=\frac{9}{6}=1.5$
Apply the same factor
Side $EF$ matches side $BC$ . Multiply $8$ by $1.5$ , because proportional sides use the same scale factor for every matching pair.
$EF=8(1.5)=12$

Answer: The length of

EF

is

12

units.

Check: The matching side in the second triangle should be

1.5

times as long. The check

\frac{12}{8}=1.5

confirms this. The result is reasonable because the second triangle is larger.

Common mistakes and how to avoid them

Matching sides because they look alike in a sketch.

Correction: Use the vertex order in the similarity statement. A sketch may not be drawn to scale.

Using the scale factor in opposite directions for different side pairs.

Correction: State which triangle you are scaling from and keep that direction for every pair.

Comparing sides that do not correspond.

Correction: List the matching vertices and sides before forming ratios.

Assuming similar triangles have equal side lengths.

Correction: Similar triangles have equal corresponding angles and proportional corresponding sides. Their sizes may differ.

Lesson summary

Similar triangles have equal corresponding angles.
Their corresponding side lengths are proportional.
The letter order in a similarity statement identifies matching vertices, angles, and sides.
A scale factor compares matching side lengths in a chosen direction.
Use matching sides to form ratios, then check that the answer fits the triangles’ relative sizes.

Check your understanding

Question 1

If

\triangle JKL \sim \triangle MNP

, which side corresponds to

KL

?

$MN$
$NP$
$MP$
$MK$

Show answer and explanation

NP

The letter order pairs

J

with

M

,

K

with

N

, and

L

with

P

. So side

KL

corresponds to side

NP

.

Question 2

A side of length

4

in a smaller triangle corresponds to a side of length

10

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

$\frac{2}{5}$
$2$
$\frac{5}{2}$
$6$

Show answer and explanation

\frac{5}{2}

Divide the larger matching length by the smaller one:

\frac{10}{4}=\frac{5}{2}

.

Question 3

A matching side pair in two similar triangles has lengths

6

and

9

, in that order. Another side in the triangle with the side of length

6

is

8

. What is its matching length in the other triangle?

$\frac{16}{3}$
$10$
$12$
$17$

Show answer and explanation

12

The scale factor from the first triangle to the second is

\frac{9}{6}=1.5

. Multiply the matching side length by that factor:

8(1.5)=12

.

Key terms

Corresponding parts: Parts of two shapes that match, such as vertices, angles, or sides.
Proportion: A statement that two ratios are equal.
Scale factor: The number that multiplies lengths in one figure to give matching lengths in another.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths.

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