Similar Triangles

Introduction to Similar Triangles

In geometry, two figures are similar when they have the same shape but are not necessarily the same size.

Similar figures have equal corresponding angles and proportional corresponding side lengths.

For example, two triangles may have exactly the same angles, but one triangle may be twice as large as the other. The triangles are still similar because their shapes are identical.

Similarity is an important mathematical concept because it allows us to calculate unknown lengths, compare geometric figures, and solve problems involving objects that cannot be measured directly.

Similar triangles are particularly useful in:

• Architecture and construction.

• Maps and scale drawings.

• Photography and image enlargement.

• Surveying and measuring heights.

• Engineering and design.

• Shadows and indirect measurement.

• Mathematical modelling.

Understanding similarity involves recognising corresponding parts, applying similarity criteria, calculating scale factors, and using proportional reasoning.

Understanding Similar Figures

Two figures are similar if their corresponding angles are equal and their corresponding sides are proportional.

Corresponding Angles

Corresponding angles occupy matching positions in similar figures.

If triangle ABC is similar to triangle DEF, then:

∠A = ∠D

∠B = ∠E

∠C = ∠F

Corresponding Sides

Corresponding sides are sides that occupy matching positions in the two figures.

For similar triangles:

AB/DE = BC/EF = AC/DF

The ratios of corresponding side lengths are equal.

Similarity Notation

The symbol ∼ means "is similar to."

We write:

△ABC ∼ △DEF

The order of the letters is important because it identifies the corresponding vertices.

For example:

A ↔ D

B ↔ E

C ↔ F

This correspondence determines which sides should be compared.

Congruence Versus Similarity

Congruent figures have the same shape and the same size.

Similar figures have the same shape but may have different sizes.

All congruent figures are similar.

However, not all similar figures are congruent.

Example

Triangle ABC has sides:

3 cm, 4 cm, 5 cm

Triangle DEF has sides:

6 cm, 8 cm, 10 cm

The side ratios are:

6/3 = 2

8/4 = 2

10/5 = 2

All corresponding sides have been multiplied by 2.

Therefore, the triangles are similar but not congruent.

Scale Factors

A scale factor describes how much a figure has been enlarged or reduced.

For corresponding sides:

Scale factor = new length / original length

If the scale factor is:

• Greater than 1, the figure is enlarged.

• Between 0 and 1, the figure is reduced.

• Equal to 1, the figure remains the same size.

Example: Enlargement

A triangle has a side measuring 5 cm.

A similar triangle has the corresponding side measuring 15 cm.

Scale factor = 15/5

Scale factor = 3

The second triangle is three times as large in its linear dimensions.

Example: Reduction

A triangle has a side measuring 20 cm.

A smaller similar triangle has the corresponding side measuring 8 cm.

Scale factor = 8/20

Scale factor = 0.4

The second triangle is a reduction of the first.

Exploring the Scale Factor

The following comparison illustrates how changing the scale factor changes the side lengths of a triangle while preserving its angles.

Important Observation

A scale factor affects all corresponding side lengths equally.

It does not change the angles.

For example, if a triangle has angles:

40°, 60°, and 80°

then every similar triangle has those same three angles, regardless of its size.

Criteria for Similar Triangles

Two triangles can be proven similar using three main criteria:

• AA: Angle–Angle similarity.

• SSS: Side–Side–Side similarity.

• SAS: Side–Angle–Side similarity.

These criteria allow us to establish similarity without measuring every side and angle.

AA Similarity

AA stands for Angle–Angle.

Two triangles are similar if two angles of one triangle are equal to two corresponding angles of another triangle.

Why Two Angles Are Enough

The interior angles of every triangle add to 180°.

If two angles are equal, the third angles must also be equal.

For example:

Triangle ABC has:

∠A = 40°

∠B = 65°

Therefore:

∠C = 180° − 40° − 65°

∠C = 75°

Triangle DEF has:

∠D = 40°

∠E = 65°

Therefore:

∠F = 75°

All three pairs of corresponding angles are equal.

Thus:

△ABC ∼ △DEF by AA similarity.

Worked Example

Triangle PQR has angles:

35°, 65°, and 80°.

Triangle XYZ has angles:

35°, 65°, and 80°.

Are the triangles similar?

Step 1: Compare Angles

Both triangles contain corresponding angles of 35° and 65°.

Step 2: Apply AA

Two corresponding angles are equal.

Conclusion

The triangles are similar by AA.

No side lengths are needed to establish similarity.

SSS Similarity

SSS stands for Side–Side–Side.

Two triangles are similar if all three pairs of corresponding sides are proportional.

Unlike SSS congruence, the side lengths do not need to be equal.

They must have the same ratio.

Worked Example

Triangle ABC has sides:

4 cm, 6 cm, 8 cm

Triangle DEF has sides:

10 cm, 15 cm, 20 cm

Determine whether the triangles are similar.

Step 1: Compare Corresponding Sides

10/4 = 2.5

15/6 = 2.5

20/8 = 2.5

Step 2: Identify the Common Ratio

All three ratios are equal.

The scale factor is:

k = 2.5

Step 3: Apply SSS Similarity

△ABC ∼ △DEF by SSS similarity, with the sides matched in the order listed.

Important Difference

For SSS congruence, all three corresponding side lengths must be equal.

For SSS similarity, the corresponding side lengths must be proportional.

SAS Similarity

SAS stands for Side–Angle–Side.

Two triangles are similar if:

• Two pairs of corresponding sides are proportional.

• The included angles between those sides are equal.

Worked Example

Triangle ABC has:

AB = 6 cm

AC = 9 cm

∠A = 50°

Triangle DEF has:

DE = 10 cm

DF = 15 cm

∠D = 50°

Determine whether the triangles are similar.

Step 1: Compare Corresponding Side Ratios

DE/AB = 10/6 = 5/3

DF/AC = 15/9 = 5/3

The ratios are equal.

Step 2: Compare Included Angles

∠A = ∠D = 50°

Step 3: Apply SAS Similarity

The two side pairs are proportional, and the included angles are equal.

Therefore:

△ABC ∼ △DEF by SAS similarity.

Why the Included Angle Matters

If the equal angle is not between the proportional side pairs, the information may be insufficient to prove similarity.

The position of the angle is important.

Comparing Similarity Criteria

Important Observation

AAA also establishes triangle similarity, but it is unnecessary to check all three angles because AA is sufficient.

There is no general SSA similarity criterion.

Finding Missing Side Lengths

One of the most important applications of similarity is calculating unknown lengths.

If two triangles are similar, their corresponding sides are proportional.

Suppose:

△ABC ∼ △DEF

Then:

AB/DE = BC/EF = AC/DF

We can use these proportions to find missing side lengths.

Worked Example: Basic Proportions

Two similar triangles have corresponding sides:

Small triangle: 4 cm and 7 cm.

Large triangle: 12 cm and x cm.

Find x.

Step 1: Identify the Scale Factor

Scale factor = 12/4

k = 3

Step 2: Apply the Scale Factor

x = 7 × 3

Final Answer

x = 21 cm

Verification

12/4 = 3

21/7 = 3

The corresponding side ratios are equal.

Worked Example: Cross-Multiplication

Suppose two similar triangles have:

AB = 5 cm

DE = 15 cm

BC = 8 cm

EF = x cm

Find x.

Step 1: Write a Proportion

AB/DE = BC/EF

5/15 = 8/x

Step 2: Cross-Multiply

5x = 15 × 8

5x = 120

Step 3: Solve

x = 24 cm

Step 4: Verify

5/15 = 1/3

8/24 = 1/3

The proportions are equal.

Worked Example: Finding a Side in the Smaller Triangle

Two similar triangles have:

Small corresponding side = x

Large corresponding side = 18 cm

Another small side = 5 cm

Corresponding large side = 15 cm

Find x.

Step 1: Write a Proportion

x/18 = 5/15

Step 2: Simplify

x/18 = 1/3

Step 3: Solve

x = 18/3

x = 6 cm

Reasonableness Check

The larger triangle has a scale factor of 3 relative to the smaller triangle.

Therefore, the missing small side must be one-third of 18 cm.

The answer is reasonable.

Solving Algebraic Problems Using Similarity

Sometimes side lengths are given as algebraic expressions.

Similarity allows us to construct equations and solve for unknown variables.

Worked Example

Two similar triangles have corresponding sides:

Small triangle: 3x + 2 and 10 cm.

Large triangle: 20 cm and 25 cm.

The first pair is 3x + 2 and 20, and the second pair is 10 and 25.

Find x.

Step 1: Write a Proportion

(3x + 2)/20 = 10/25

Step 2: Cross-Multiply

25(3x + 2) = 200

Step 3: Expand

75x + 50 = 200

Step 4: Solve

75x = 150

x = 2

Step 5: Verify

3x + 2 = 3(2) + 2 = 8

8/20 = 0.4

10/25 = 0.4

The proportions match.

Final Answer

x = 2

Worked Example: Two Unknown Sides

Two similar triangles have side lengths:

Small triangle: 6 cm, 8 cm, and 10 cm.

Large triangle: 9 cm, x cm, and y cm.

The corresponding sides are listed in the same order.

Step 1: Calculate the Scale Factor

k = 9/6

k = 1.5

Step 2: Find x

x = 8 × 1.5

x = 12 cm

Step 3: Find y

y = 10 × 1.5

y = 15 cm

Final Answer

The larger triangle has side lengths:

9 cm, 12 cm, and 15 cm

Similar Triangles Formed by Parallel Lines

Parallel lines frequently create similar triangles.

Consider triangle ABC with points D on AB and E on AC.

If:

DE ∥ BC

then:

∠ADE = ∠ABC

and:

∠AED = ∠ACB

These are corresponding angles formed by parallel lines.

Therefore:

△ADE ∼ △ABC by AA similarity.

Proportional Relationships

Since:

△ADE ∼ △ABC

we have:

AD/AB = AE/AC = DE/BC

This relationship is extremely useful when a line parallel to one side divides the other two sides of a triangle.

Worked Example: Parallel-Line Similarity

In triangle ABC:

DE ∥ BC

AD = 4 cm

AB = 10 cm

AE = 6 cm

Find AC.

Step 1: Establish Similarity

△ADE ∼ △ABC by AA.

Step 2: Write the Proportion

AD/AB = AE/AC

4/10 = 6/AC

Step 3: Cross-Multiply

4AC = 60

Step 4: Solve

AC = 15 cm

Step 5: Verify

4/10 = 0.4

6/15 = 0.4

The ratios match.

Interpretation

The smaller triangle has a scale factor of 0.4 relative to the larger triangle.

Similar Triangles and the Pythagorean Theorem

Similarity and the Pythagorean theorem can be combined to solve right-triangle problems.

Worked Example

A right triangle has legs of 6 cm and 8 cm.

A similar right triangle has a corresponding leg of 15 cm instead of 6 cm.

Find the other two side lengths.

Step 1: Find the Original Hypotenuse

Using the Pythagorean theorem:

c² = 6² + 8²

c² = 36 + 64

c² = 100

c = 10 cm

Step 2: Calculate the Scale Factor

k = 15/6

k = 2.5

Step 3: Find the Other Leg

8 × 2.5 = 20 cm

Step 4: Find the Hypotenuse

10 × 2.5 = 25 cm

Final Answer

The larger triangle has sides:

15 cm, 20 cm, and 25 cm

Verification

15² + 20² = 225 + 400 = 625

25² = 625

The Pythagorean theorem is satisfied.

Similarity and Perimeter

When two figures are similar, their perimeters have the same ratio as their corresponding side lengths.

If the scale factor is k:

New perimeter = k × original perimeter

Worked Example

A triangle has side lengths:

5 cm, 7 cm, and 8 cm.

A similar triangle has a scale factor of 3.

Find the perimeter of the larger triangle.

Step 1: Calculate the Original Perimeter

P = 5 + 7 + 8

P = 20 cm

Step 2: Apply the Scale Factor

New perimeter = 3 × 20

Final Answer

60 cm

Important Principle

A scale factor of 3 makes the perimeter three times as large.

It does not make the perimeter nine times as large.

Similarity and Area

Area changes differently from side length.

If the scale factor is k:

New area = k² × original area

This happens because area involves two dimensions.

Worked Example

A triangle has an area of 24 cm².

A similar triangle has a scale factor of 2.5.

Find the area of the larger triangle.

Step 1: Square the Scale Factor

k² = 2.5²

k² = 6.25

Step 2: Calculate the New Area

New area = 24 × 6.25

Final Answer

150 cm²

Comparing Length, Perimeter, and Area

Why Area Changes by the Square

The area of a triangle is:

A = 1/2 × base × height

If both base and height are multiplied by k:

New area = 1/2 × (k × base) × (k × height)

New area = k² × (1/2 × base × height)

Therefore:

New area = k² × original area

This relationship applies to all similar two-dimensional figures.

Real-World Application: Measuring Heights Using Shadows

Similar triangles can be used to measure the heights of tall objects without climbing them.

When sunlight strikes objects at the same angle, their heights and shadow lengths can form similar right triangles.

Worked Example

A student who is 1.6 m tall casts a shadow measuring 2 m.

At the same time, a tree casts a shadow measuring 12 m.

Find the height of the tree.

Step 1: Establish Similarity

Both objects are vertical to level ground.

The sunlight creates the same angle with the ground.

Therefore, the height-and-shadow triangles are similar by AA.

Step 2: Write a Proportion

Tree height / tree shadow = student height / student shadow

h/12 = 1.6/2

Step 3: Cross-Multiply

2h = 12 × 1.6

2h = 19.2

Step 4: Solve

h = 9.6 m

Interpretation

The tree is approximately 9.6 m tall.

Reasonableness Check

The tree's shadow is six times longer than the student's shadow.

Therefore, its height should also be six times greater.

1.6 × 6 = 9.6 m

The answer is reasonable.

Real-World Application: Maps and Scale Drawings

Maps use similarity to represent large distances using smaller measurements.

Worked Example

A map uses the scale:

1 cm : 5 km

Two towns are 7.5 cm apart on the map.

Find their actual distance.

Step 1: Identify the Scale

1 cm represents 5 km.

Step 2: Apply Proportional Reasoning

Actual distance = 7.5 × 5

Final Answer

37.5 km

Interpretation

The towns are approximately 37.5 km apart according to the map scale.

This is a straight-line or map-measured distance, not necessarily the distance travelled along roads.

Real-World Application: Scale Models

Architects and engineers often use scale models to represent larger structures.

Worked Example

A building model is constructed at a scale of:

1 : 200

The model building is 18 cm tall.

Find the actual height of the building.

Step 1: Interpret the Scale

Every 1 cm on the model represents 200 cm in reality.

Step 2: Multiply

Actual height = 18 × 200

Actual height = 3600 cm

Step 3: Convert Units

3600 cm = 36 m

Final Answer

36 m

Interpretation

The actual building is 36 m tall.

Important Observation

When using scale ratios, all measurements must initially be expressed in compatible units.

Real-World Application: Photography

Images can be enlarged or reduced while maintaining their proportions.

If an image is enlarged uniformly, the original and enlarged shapes are similar.

Worked Example

A photograph measures:

12 cm wide

8 cm high

It is enlarged to a width of 30 cm.

Find the new height.

Step 1: Calculate the Scale Factor

k = 30/12

k = 2.5

Step 2: Calculate the New Height

New height = 8 × 2.5

Final Answer

20 cm

Interpretation

The enlarged photograph should measure:

30 cm × 20 cm

This preserves the original proportions without distortion.

Real-World Application: Indirect Measurement

Similarity can be used to determine distances that are difficult or unsafe to measure directly.

For example, a surveyor may construct two similar triangles to estimate the width of a river.

Worked Example

A surveying arrangement creates two similar triangles.

The smaller triangle has:

• A horizontal side of 4 m.

• A corresponding perpendicular side of 6 m.

The larger triangle has a corresponding horizontal side of 30 m.

Find the larger perpendicular side.

Step 1: Write a Proportion

6/4 = x/30

Step 2: Cross-Multiply

4x = 180

Step 3: Solve

x = 45 m

Interpretation

The unknown perpendicular distance is 45 m.

The validity of this result depends on the surveyor having established that the two triangles are similar.

Real-World Application: Engineering and Design

Engineers use similar triangles to calculate dimensions in scaled structures and mechanical systems.

Worked Example

A triangular support has:

Base = 2.4 m

Height = 1.8 m

A larger similar support must have a base of 4 m.

Find its height.

Step 1: Calculate the Scale Factor

k = 4/2.4

k = 5/3

Step 2: Calculate the New Height

New height = 1.8 × 5/3

Final Answer

3 m

Interpretation

The larger support must have a height of 3 m to preserve the original triangular shape.

Multi-Step Problems with Similar Triangles

Some problems require more than one mathematical technique.

They may involve:

• Angle calculations.

• Proportional relationships.

• Algebraic equations.

• Perimeter calculations.

• Area ratios.

• Pythagorean theorem.

Worked Example: Finding a Missing Side and Area

Two triangles are similar.

The smaller triangle has:

Base = 8 cm

Height = 6 cm

The larger triangle has a corresponding base of 20 cm.

Find:

1. The larger triangle's height.

2. The smaller triangle's area.

3. The larger triangle's area.

Step 1: Calculate the Scale Factor

k = 20/8

k = 2.5

Step 2: Find the Larger Height

Height = 6 × 2.5

Height = 15 cm

Step 3: Find the Smaller Area

A = 1/2 × 8 × 6

A = 24 cm²

Step 4: Find the Larger Area

Using the area scale factor:

Larger area = 24 × 2.5²

Larger area = 24 × 6.25

Larger area = 150 cm²

Step 5: Verify Directly

A = 1/2 × 20 × 15

A = 150 cm²

Both methods agree.

Worked Example: Combining Angles and Proportions

Triangle ABC is similar to triangle DEF.

Given:

∠A = 50°

∠B = 65°

AB = 8 cm

DE = 12 cm

BC = 10 cm

Find:

1. ∠F.

2. EF.

Step 1: Find the Third Angle

∠C = 180° − 50° − 65°

∠C = 65°

Since C corresponds to F:

∠F = 65°

Step 2: Find the Scale Factor

k = DE/AB

k = 12/8

k = 1.5

Step 3: Find EF

EF = BC × 1.5

EF = 10 × 1.5

EF = 15 cm

Final Answers

∠F = 65°

EF = 15 cm

Writing Similarity Proofs

A similarity proof explains why two triangles must have the same shape.

The proof should identify the information given and apply a valid similarity criterion.

Example: AA Similarity Proof

Given:

∠A = ∠D

∠B = ∠E

Prove:

△ABC ∼ △DEF

Paragraph Proof

The corresponding angles ∠A and ∠D are equal, and the corresponding angles ∠B and ∠E are equal. Therefore, the triangles have two pairs of equal corresponding angles. Hence, △ABC ∼ △DEF by AA similarity.

Why Mathematical Justification Matters

A diagram may appear to contain similar triangles, but visual appearance alone is not sufficient.

A mathematical proof must establish the required relationships.

For example, two triangles may look similar even when their side ratios differ slightly.

Accurate reasoning is more reliable than visual estimation.

A Systematic Method for Similarity Problems

Identify the Triangles

Determine which figures are being compared.

Match Corresponding Parts

Identify corresponding vertices, sides, and angles.

Establish Similarity

Use AA, SSS, or SAS similarity.

Determine the Scale Factor

Compare a known pair of corresponding sides.

Write a Correct Proportion

Keep the order of corresponding sides consistent.

Solve the Equation

Use multiplication, division, or cross-multiplication.

Check Units

Ensure that measurements use compatible units.

Verify the Result

Check that the corresponding side ratios match.

Interpret the Answer

Explain what the result means in the original geometric or practical situation.

Common Mistakes

Confusing similarity with congruence

Similar figures do not need to have equal side lengths.

Matching incorrect corresponding sides

Side ratios must compare matching sides.

Writing proportions inconsistently

All ratios must use the same triangle order.

Assuming equal angles imply congruence

Equal angles establish similarity, not necessarily congruence.

Using an invalid similarity criterion

AA, SSS, and SAS are the standard triangle similarity criteria.

Forgetting the included angle in SAS

The equal angle must lie between the proportional sides.

Calculating the scale factor backwards

Always identify which figure is the original and which is the scaled figure.

Using different measurement units without converting

For example, centimetres and metres must be converted appropriately.

Applying the scale factor incorrectly

Side lengths change by k, while areas change by k².

Assuming perimeter and area change by the same factor

They scale differently.

Failing to establish similarity before using proportions

A proportion based on similarity is justified only when the figures are known to be similar.

Ignoring the real-world context

Practical answers should include units and reasonable interpretations.

Key Terms

• Similar Figures: Figures with the same shape and proportional corresponding side lengths.

• Similar Triangles: Triangles with equal corresponding angles and proportional corresponding sides.

• Similarity Symbol (∼): A symbol meaning "is similar to."

• Corresponding Vertices: Vertices occupying matching positions in similar figures.

• Corresponding Sides: Sides that match in position and are proportional.

• Corresponding Angles: Angles occupying matching positions and having equal measures.

• Proportion: An equation stating that two ratios are equal.

• Ratio: A comparison of two quantities using division.

• Scale Factor: The ratio of a corresponding new length to an original length.

• Enlargement: A transformation that increases a figure's dimensions proportionally.

• Reduction: A transformation that decreases a figure's dimensions proportionally.

• Dilation: A transformation that changes a figure's size while preserving its shape.

• Congruence: A relationship between figures having exactly the same shape and size.

• AA Similarity: A criterion based on two pairs of equal corresponding angles.

• SSS Similarity: A criterion based on three pairs of proportional corresponding sides.

• SAS Similarity: A criterion based on two proportional side pairs and an equal included angle.

• Included Angle: The angle formed by two specified sides.

• Triangle Angle-Sum Theorem: The rule that the interior angles of a triangle sum to 180°.

• Parallel Lines: Lines in a plane that never intersect.

• Alternate Interior Angles: Angles on opposite sides of a transversal between parallel lines.

• Corresponding Angles: Angles occupying matching positions when a transversal crosses two lines.

• Cross-Multiplication: A method for solving proportions.

• Perimeter: The total distance around a figure.

• Area: The amount of two-dimensional space enclosed by a figure.

• Area Scale Factor: The square of the linear scale factor.

• Pythagorean Theorem: The relationship a² + b² = c² in a right triangle.

• Indirect Measurement: Finding an unknown distance using mathematical relationships instead of direct measurement.

• Scale Drawing: A drawing representing an object using proportional dimensions.

• Scale Model: A physical model with dimensions proportional to the original object.

• Geometric Proof: A logical argument establishing a mathematical relationship.

• Proportional Reasoning: Using ratios and proportions to solve problems.

• Mathematical Verification: Checking whether a calculated result satisfies the original conditions.

Key Takeaways

• Similar triangles have the same shape but are not necessarily the same size.

• Corresponding angles in similar triangles are equal.

• Corresponding sides in similar triangles are proportional.

• The order of vertices in similarity notation identifies the correct correspondence.

• Congruent triangles are always similar, but similar triangles are not always congruent.

• AA similarity requires two pairs of equal corresponding angles.

• SSS similarity requires three pairs of proportional corresponding sides.

• SAS similarity requires two proportional side pairs and an equal included angle.

• Scale factors describe how corresponding lengths change between similar figures.

• A scale factor greater than 1 represents enlargement.

• A scale factor between 0 and 1 represents reduction.

• Corresponding side lengths change by the scale factor k.

• Perimeters of similar figures also change by k.

• Areas of similar figures change by k².

• Similarity allows missing side lengths to be calculated using proportions.

• Cross-multiplication is a useful method for solving proportional equations.

• Parallel lines can create similar triangles through equal corresponding or alternate interior angles.

• Similarity can be combined with the Pythagorean theorem, angle-sum relationships, and algebra.

• Similar triangles are useful for measuring heights, distances, and dimensions indirectly.

• Maps, photographs, scale drawings, and architectural models rely on proportional relationships.

• Mathematical proofs should establish similarity before using corresponding side ratios.

• Real-world solutions should include appropriate units and reasonableness checks.

• The central skill in similarity problems is identifying corresponding parts, establishing a valid similarity relationship, and using consistent proportions to calculate unknown quantities.

Check Your Understanding

Recall and Understanding

1. Define similar triangles and explain how they differ from congruent triangles.

2. State the three main triangle similarity criteria.

3. Explain why two equal pairs of corresponding angles are sufficient to prove triangle similarity.

4. Define scale factor and explain the difference between enlargement and reduction.

5. Explain why the areas of similar triangles change by the square of the scale factor.

Application

6. Two triangles have corresponding sides of 3 cm, 5 cm, and 7 cm, and 6 cm, 10 cm, and 14 cm. Determine whether they are similar and justify your answer.

7. A triangle has sides of 4 cm, 6 cm, and 9 cm. A similar triangle has a corresponding shortest side of 12 cm. Find its other two side lengths.

8. Two similar triangles have corresponding sides of 5 cm and 15 cm. If another side of the smaller triangle is 8 cm, find the corresponding side of the larger triangle.

9. Two triangles have corresponding angles of 45° and 70°. Explain why they are similar and calculate the third angle.

10. A triangle has an area of 18 cm². A similar triangle has a scale factor of 3. Find the larger triangle's area.

Analysis and Challenge

11. In triangle ABC, DE is parallel to BC. Given AD = 6 cm, AB = 15 cm, and AE = 8 cm, calculate AC.

12. Two similar triangles have corresponding sides measuring 2x + 3 cm and 18 cm, and another corresponding pair measuring 5 cm and 10 cm. Find x.

13. A student who is 1.5 m tall casts a shadow of 1.8 m. At the same time, a building casts a shadow of 21.6 m. Calculate the building's height and explain why similarity applies.

14. A scale model of a tower is constructed at a scale of 1 : 250. The model is 32 cm tall. Find the actual tower height in metres.

15. Two similar triangular supports have corresponding bases of 6 m and 10 m. The smaller support has a height of 4.5 m and an area of 13.5 m². Determine the larger support's height and area, showing all calculations.

Final Summary

Similar triangles have the same shape, equal corresponding angles, and proportional corresponding side lengths.

Similarity can be established using AA, SSS, or SAS criteria.

Once similarity has been proven, scale factors and proportions can be used to calculate unknown side lengths and solve geometric problems.

Similar triangles also provide important relationships between corresponding perimeters and areas.

Parallel lines, angle relationships, and the Pythagorean theorem can be combined with similarity to solve more advanced problems.

In practical situations, similarity allows heights, distances, and dimensions to be calculated using indirect measurement and scale models.

The central principle is that similar triangles preserve their shape through equal corresponding angles and proportional side lengths, allowing unknown measurements to be determined through mathematical reasoning.