Solved Exercises on Triangle Similarity in Grade 8

Master triangle similarity: AA, SAS, SSS criteria, proportional sides, angle relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 AA Similarity Criterion
Exercise 1
In triangles ABC and DEF, ∠A = 50°, ∠B = 60°, ∠D = 50°, ∠E = 60°. Are the triangles similar? If yes, write the similarity statement and find ∠C and ∠F.
Definition:

AA Similarity: Two triangles are similar if two pairs of corresponding angles are equal. Since the sum of angles in a triangle is 180°, the third angles will automatically be equal.

AA Similarity Method:
  1. Identify two pairs of equal corresponding angles
  2. Conclude that triangles are similar
  3. Find remaining angles using angle sum property
Given Angles
∠A = ∠D = 50°, ∠B = ∠E = 60°
Similarity
△ABC ~ △DEF
Step 1: Identify equal angles

∠A = ∠D = 50° and ∠B = ∠E = 60°

Step 2: Apply AA criterion

Since two pairs of corresponding angles are equal, triangles are similar: △ABC ~ △DEF

Step 3: Find remaining angles

∠C = 180° - 50° - 60° = 70°

∠F = 180° - 50° - 60° = 70°

△ABC ~ △DEF with ∠C = ∠F = 70°
Final answer:

Yes, the triangles are similar by AA criterion. △ABC ~ △DEF, and ∠C = ∠F = 70°

Applied rules:

• AA Similarity: Two pairs of equal corresponding angles guarantee similarity

• Angle Sum Property: Sum of angles in a triangle is 180°

• Corresponding Parts: Equal angles correspond to equal angles

2 SAS Similarity Criterion
Exercise 2
Triangle ABC has sides AB = 6 cm, AC = 8 cm, and ∠A = 45°. Triangle DEF has sides DE = 9 cm, DF = 12 cm, and ∠D = 45°. Are the triangles similar?
Definition:

SAS Similarity: Two triangles are similar if two pairs of corresponding sides are proportional and the included angles are equal.

Given Sides
AB = 6, AC = 8, DE = 9, DF = 12
Ratio Check
AB/DE = 6/9 = 2/3, AC/DF = 8/12 = 2/3
Similarity
△ABC ~ △DEF
Step 1: Check the ratios of corresponding sides

AB/DE = 6/9 = 2/3 and AC/DF = 8/12 = 2/3

Step 2: Verify the included angles

∠A = ∠D = 45°

Step 3: Apply SAS similarity

Since AB/DE = AC/DF = 2/3 and ∠A = ∠D, the triangles are similar by SAS criterion

△ABC ~ △DEF by SAS similarity with ratio 2:3
Final answer:

Yes, the triangles are similar by SAS criterion with a ratio of 2:3

Applied rules:

• SAS Similarity: Proportional sides and equal included angle

• Proportionality: Ratios of corresponding sides must be equal

• Equal Included Angle: The angle between the proportional sides

3 SSS Similarity Criterion
Exercise 3
Triangle PQR has sides PQ = 4 cm, QR = 6 cm, PR = 8 cm. Triangle STU has sides ST = 6 cm, TU = 9 cm, SU = 12 cm. Are the triangles similar?
Definition:

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

Given Sides
PQ = 4, QR = 6, PR = 8, ST = 6, TU = 9, SU = 12
Ratios
PQ/ST = 4/6 = 2/3, QR/TU = 6/9 = 2/3, PR/SU = 8/12 = 2/3
Similarity
△PQR ~ △STU
Step 1: Calculate ratios of all corresponding sides

PQ/ST = 4/6 = 2/3

QR/TU = 6/9 = 2/3

PR/SU = 8/12 = 2/3

Step 2: Check if all ratios are equal

All ratios equal 2/3, so the sides are proportional

Step 3: Apply SSS similarity

Since all corresponding sides are proportional, triangles are similar by SSS criterion

△PQR ~ △STU by SSS similarity with ratio 2:3
Final answer:

Yes, the triangles are similar by SSS criterion with a ratio of 2:3

Applied rules:

• SSS Similarity: All three pairs of corresponding sides are proportional

• Proportionality: All side ratios must be equal

• Corresponding Sides: Match sides opposite to equal angles

Rules and methods, laws,...
\(AA: \angle A = \angle D, \angle B = \angle E \Rightarrow \triangle ABC \sim \triangle DEF\)
AA Similarity
AA Criterion
∠A = ∠D, ∠B = ∠E
Angle-Angle Similarity
SAS Criterion
AB/DE = AC/DF, ∠A = ∠D
Side-Angle-Side Similarity
SSS Criterion
AB/DE = BC/EF = AC/DF
Side-Side-Side Similarity
Proportional Sides: If △ABC ~ △DEF, then AB/DE = BC/EF = AC/DF
Equal Angles: If △ABC ~ △DEF, then ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Key definitions:

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

Similarity Ratio: The constant of proportionality between corresponding sides

Corresponding Parts: Parts that match when triangles are superimposed

Complete methodology:
  1. Identify given information: Check for equal angles or proportional sides
  2. Choose the criterion: AA, SAS, or SSS based on available information
  3. Verify the conditions: Ensure all requirements of the chosen criterion are met
  4. State the conclusion: Write the similarity statement with correct vertex correspondence
Tip 1: Remember that similarity implies proportional sides and equal angles.
Tip 2: The order of vertices in similarity statements matters.
Tip 3: Use the similarity ratio to find missing measurements.
Solution: Exercises 4 to 5
4 Using Similarity to Find Unknown Sides
Exercise 4
Triangle ABC is similar to triangle DEF. If AB = 5 cm, BC = 7 cm, AC = 8 cm, and DE = 10 cm, find EF and DF.
Definition:

Proportional Sides Property: If two triangles are similar, their corresponding sides are proportional, meaning the ratios of corresponding sides are equal.

Given
△ABC ~ △DEF, AB = 5, BC = 7, AC = 8, DE = 10
Ratio
AB/DE = 5/10 = 1/2
Unknown Sides
BC/EF = 1/2, AC/DF = 1/2
Step 1: Find the similarity ratio

Since △ABC ~ △DEF, the ratio of corresponding sides is AB/DE = 5/10 = 1/2

Step 2: Set up proportions for unknown sides

BC/EF = 1/2, so 7/EF = 1/2

AC/DF = 1/2, so 8/DF = 1/2

Step 3: Solve for unknown sides

From 7/EF = 1/2: EF = 7 × 2 = 14 cm

From 8/DF = 1/2: DF = 8 × 2 = 16 cm

EF = 14 cm, DF = 16 cm
Final answer:

EF = 14 cm and DF = 16 cm

Applied rules:

• Proportional Sides: Corresponding sides of similar triangles are proportional

• Cross Multiplication: Solve proportions by cross-multiplying

• Vertex Correspondence: Match sides correctly based on vertex order

5 Real-World Application
Exercise 5
A tree casts a shadow of 12 meters. At the same time, a 2-meter tall pole casts a shadow of 3 meters. How tall is the tree?
Definition:

Shadow Problems: Objects and their shadows form similar triangles due to parallel sun rays, allowing us to set up proportions between heights and shadow lengths.

Given
Pole height = 2m, Pole shadow = 3m, Tree shadow = 12m
Proportion
Height₁/Shadow₁ = Height₂/Shadow₂
Tree Height
2/3 = h/12
Step 1: Recognize similar triangles

The sun rays create similar triangles between the pole and its shadow, and the tree and its shadow

Step 2: Set up the proportion

Pole height/Pole shadow = Tree height/Tree shadow

2/3 = h/12 where h is the tree height

Step 3: Solve for the unknown height

2/3 = h/12

h = (2 × 12)/3 = 24/3 = 8 meters

Step 4: Verify the answer

Check: 2/3 = 8/12 → 2/3 = 2/3 ✓

Tree height = 8 meters
Final answer:

The tree is 8 meters tall

Applied rules:

• Similar Triangles in Shadows: Parallel light rays create similar triangles

• Proportional Relationships: Heights are proportional to shadow lengths

• Real-World Applications: Use similar triangles to measure inaccessible heights

Triangle Similarity Laws, Methods, and Properties
\(AA: \angle A = \angle D, \angle B = \angle E \Rightarrow \triangle ABC \sim \triangle DEF\)
AA Similarity
Key definitions:

Similarity: Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional

Similarity Ratio: The constant of proportionality between corresponding sides of similar triangles

Corresponding Parts: Angles and sides that match when one triangle is transformed to match the other

Complete methodology:
  1. Analyze given information: Identify known angles and sides
  2. Select similarity criterion: Choose AA, SAS, or SSS based on available information
  3. Verify conditions: Confirm that all requirements of the chosen criterion are satisfied
  4. Write similarity statement: Use correct vertex correspondence
  5. Apply properties: Use proportional sides and equal angles to solve for unknowns
Tip 1: Always match vertices in the same order when writing similarity statements.
Tip 2: The similarity ratio helps find unknown sides in proportional triangles.
Tip 3: Real-world problems often involve shadows or mirrors creating similar triangles.
Tip 4: If triangles are similar, all corresponding parts are in the same ratio.
Common errors: Confusing similarity with congruence, misidentifying corresponding parts, forgetting that similarity implies proportional sides and equal angles.
Exam preparation: Master all three similarity criteria, practice identifying corresponding parts, solve real-world applications.
Similarity criteria to know:

• AA (Angle-Angle): Two pairs of equal corresponding angles

• SAS (Side-Angle-Side): Two pairs of proportional sides with equal included angle

• SSS (Side-Side-Side): Three pairs of proportional corresponding sides

• Properties: If △ABC ~ △DEF, then ∠A=∠D, ∠B=∠E, ∠C=∠F and AB/DE=BC/EF=AC/DF

Exercise with Visualization: Similar Triangle Properties
Exercise 6: Similar Triangle Ratios
Consider similar triangles with different similarity ratios:
Small triangle sides: 3, 4, 5
Medium triangle sides: 6, 8, 10 (ratio 2:1)
Large triangle sides: 9, 12, 15 (ratio 3:1)

Analysis: The chart shows how corresponding sides maintain proportional relationships across similar triangles.

  • Small: 3, 4, 5 (right triangle)
  • Medium: 6, 8, 10 (ratio 2:1 with small)
  • Large: 9, 12, 15 (ratio 3:1 with small)

Questions & Answers

Question: I'm confused about how to identify which sides are corresponding when triangles are rotated differently. How do I match them correctly?

Answer: Great question! To identify corresponding sides:

  • Look for equal angles - sides opposite to equal angles are corresponding sides
  • Pay attention to the order of vertices in similarity statements like △ABC ~ △DEF
  • Corresponding parts appear in the same position in the names: A corresponds to D, B to E, C to F
  • If you know △ABC ~ △DEF, then AB corresponds to DE, BC to EF, and AC to DF

Example: If △PQR ~ △XYZ, then P corresponds to X, Q to Y, R to Z. So side PQ corresponds to XY, QR to YZ, and PR to XZ.

Visual tip: Draw the triangles separately and label the vertices consistently to make matching easier!

Question: When solving for missing sides using similarity, how do I set up the proportion correctly? Sometimes I get confused about which ratios to use.

Answer: Here's a systematic approach to setting up proportions:

  • Start with the similarity statement (e.g., △ABC ~ △DEF)
  • Write the proportion as: AB/DE = BC/EF = AC/DF
  • This means small triangle side / large triangle side = ratio for all corresponding sides
  • If you know AB = 5, DE = 10, and BC = 7, and need EF, use AB/DE = BC/EF: 5/10 = 7/EF

Example: If △PQR ~ △STU with PQ = 4, QR = 6, and ST = 8, to find TU: PQ/ST = QR/TU → 4/8 = 6/TU → TU = 12

Key: Always match the positions in the similarity statement!

Question: What's the difference between similar triangles and congruent triangles? They seem very similar!

Answer: The key differences are:

  • Similar Triangles: Same shape, proportional sides, equal angles. The triangles are the same shape but different sizes.
  • Congruent Triangles: Same shape AND same size, equal sides, equal angles. The triangles are identical copies.

Think of similar triangles as scaled versions of each other (like a photo enlargement), while congruent triangles are exact copies that can be perfectly overlaid.

Criteria comparison:

  • Similar: AA, SAS, SSS (proportional sides)
  • Congruent: SSS, SAS, ASA, AAS, HL (equal sides)

Every congruent triangle is also similar (with ratio 1:1), but not every similar triangle is congruent!

Question: How do I know which similarity criterion to use when I'm given a problem?

Answer: Look for specific information in the problem:

  • AA Criterion: If you're given two pairs of equal angles (or can determine them), use AA. This is often the easiest!
  • SAS Criterion: If you have two pairs of proportional sides AND the included angles are equal, use SAS.
  • SSS Criterion: If all three pairs of sides are proportional, use SSS.

Decision tree:

  1. Do you see two pairs of equal angles? → Use AA
  2. Do you see two pairs of proportional sides with equal angle between them? → Use SAS
  3. Do you see three pairs of proportional sides? → Use SSS

If you're proving similarity, choose the criterion that matches your given information. If you're checking if triangles are similar, verify if any of these criteria apply.

Question: Can you use similarity to find areas of triangles? How does that work?

Answer: Yes! There's a special relationship between areas of similar triangles:

If two triangles are similar with a similarity ratio of k (meaning corresponding sides are in the ratio k:1), then their areas are in the ratio k²:1.

For example:

  • If △ABC ~ △DEF with sides in ratio 2:1, then their areas are in ratio 4:1
  • If one triangle has area 10 and the similarity ratio is 3:1, the other triangle has area 10×9 = 90 or 10÷9 depending on which is larger

This happens because area involves two dimensions (length × width), so when linear dimensions scale by k, areas scale by k².

This is very useful in advanced problems involving similar figures!