Solved Exercises on Understanding Congruent Figures in Grade 8

Master congruent figures: SSS, SAS, ASA, AAS, HL criteria, properties, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 SSS Congruence Criterion
Exercise 1
Triangle ABC has sides AB = 5 cm, BC = 7 cm, AC = 8 cm. Triangle DEF has sides DE = 5 cm, EF = 7 cm, DF = 8 cm. Are the triangles congruent?
Definition:

SSS Congruence: Two triangles are congruent if all three pairs of corresponding sides are equal. This is the Side-Side-Side congruence criterion.

SSS Congruence Method:
  1. Compare all three pairs of corresponding sides
  2. Verify that each pair of sides is equal
  3. Conclude that triangles are congruent
Given Sides
AB = DE = 5, BC = EF = 7, AC = DF = 8
Congruence
△ABC ≅ △DEF
Step 1: Compare all three pairs of sides

AB = DE = 5 cm

BC = EF = 7 cm

AC = DF = 8 cm

Step 2: Apply SSS criterion

Since all three pairs of corresponding sides are equal, triangles are congruent by SSS criterion

Step 3: Write the congruence statement

△ABC ≅ △DEF

△ABC ≅ △DEF by SSS criterion
Final answer:

Yes, the triangles are congruent by SSS criterion: △ABC ≅ △DEF

Applied rules:

• SSS Congruence: All three pairs of corresponding sides must be equal

• Congruent Figures: Same shape and same size

• Corresponding Parts: Equal sides correspond to equal sides

2 SAS Congruence Criterion
Exercise 2
Triangle PQR has PQ = 6 cm, QR = 8 cm, and ∠Q = 60°. Triangle XYZ has XY = 6 cm, YZ = 8 cm, and ∠Y = 60°. Are the triangles congruent?
Definition:

SAS Congruence: Two triangles are congruent if two pairs of corresponding sides are equal and the included angles are equal. This is the Side-Angle-Side congruence criterion.

Given Sides
PQ = XY = 6, QR = YZ = 8
Included Angle
∠Q = ∠Y = 60°
Congruence
△PQR ≅ △XYZ
Step 1: Identify equal corresponding sides

PQ = XY = 6 cm and QR = YZ = 8 cm

Step 2: Verify the included angle

∠Q = ∠Y = 60°

Step 3: Apply SAS criterion

Since two pairs of corresponding sides are equal and the included angle is equal, triangles are congruent by SAS criterion

△PQR ≅ △XYZ by SAS criterion
Final answer:

Yes, the triangles are congruent by SAS criterion: △PQR ≅ △XYZ

Applied rules:

• SAS Congruence: Two equal sides with equal included angle

• Included Angle: The angle between the two equal sides

• Corresponding Parts: Equal sides and angles correspond to equal parts

3 ASA Congruence Criterion
Exercise 3
Triangle LMN has ∠L = 45°, ∠M = 65°, and LM = 10 cm. Triangle UVW has ∠U = 45°, ∠V = 65°, and UV = 10 cm. Are the triangles congruent?
Definition:

ASA Congruence: Two triangles are congruent if two pairs of corresponding angles are equal and the included side is equal. This is the Angle-Side-Angle congruence criterion.

Given Angles
∠L = ∠U = 45°, ∠M = ∠V = 65°
Included Side
LM = UV = 10 cm
Congruence
△LMN ≅ △UVW
Step 1: Identify equal corresponding angles

∠L = ∠U = 45° and ∠M = ∠V = 65°

Step 2: Verify the included side

LM = UV = 10 cm

Step 3: Apply ASA criterion

Since two pairs of corresponding angles are equal and the included side is equal, triangles are congruent by ASA criterion

△LMN ≅ △UVW by ASA criterion
Final answer:

Yes, the triangles are congruent by ASA criterion: △LMN ≅ △UVW

Applied rules:

• ASA Congruence: Two equal angles with equal included side

• Included Side: The side between the two equal angles

• Third Angle: Automatically equal by angle sum property

Rules and methods, laws,...
\(SSS: AB = DE, BC = EF, AC = DF \Rightarrow \triangle ABC \cong \triangle DEF\)
SSS Congruence
SSS Criterion
AB = DE, BC = EF, AC = DF
Side-Side-Side Congruence
SAS Criterion
AB = DE, ∠B = ∠E, BC = EF
Side-Angle-Side Congruence
ASA Criterion
∠A = ∠D, AB = DE, ∠B = ∠E
Angle-Side-Angle Congruence
AAS Criterion
∠A = ∠D, ∠B = ∠E, BC = EF
Angle-Angle-Side Congruence
HL Criterion
∠C = ∠F = 90°, AB = DE, BC = EF
Hypotenuse-Leg Congruence
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Equal Measures: If △ABC ≅ △DEF, then AB = DE, BC = EF, AC = DF, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Key definitions:

Congruent Figures: Figures that have the same shape and the same size. They can be perfectly superimposed on each other.

Congruence Symbol: ≅ means "is congruent to"

Corresponding Parts: Parts that match when one figure is placed on top of another

Complete methodology:
  1. Identify given information: Check for equal sides and angles
  2. Choose the criterion: Select SSS, SAS, ASA, AAS, or HL based on available information
  3. Verify the conditions: Ensure all requirements of the chosen criterion are met
  4. State the conclusion: Write the congruence statement with correct vertex correspondence
Tip 1: Remember that congruent figures have both equal sides AND equal angles.
Tip 2: The order of vertices in congruence statements is crucial.
Tip 3: Use CPCTC to find unknown measures once congruence is established.
Solution: Exercises 4 to 5
4 AAS Congruence Criterion
Exercise 4
Triangle GHI has ∠G = 50°, ∠H = 70°, and GI = 12 cm. Triangle JKL has ∠J = 50°, ∠K = 70°, and JL = 12 cm. Are the triangles congruent?
Definition:

AAS Congruence: Two triangles are congruent if two pairs of corresponding angles are equal and a pair of corresponding non-included sides are equal. This is the Angle-Angle-Side congruence criterion.

Given Angles
∠G = ∠J = 50°, ∠H = ∠K = 70°
Non-included Side
GI = JL = 12 cm
Congruence
△GHI ≅ △JKL
Step 1: Identify equal corresponding angles

∠G = ∠J = 50° and ∠H = ∠K = 70°

Step 2: Identify equal non-included side

GI = JL = 12 cm (side not between the two known angles)

Step 3: Apply AAS criterion

Since two pairs of corresponding angles are equal and a pair of corresponding non-included sides are equal, triangles are congruent by AAS criterion

△GHI ≅ △JKL by AAS criterion
Final answer:

Yes, the triangles are congruent by AAS criterion: △GHI ≅ △JKL

Applied rules:

• AAS Congruence: Two equal angles and a pair of equal non-included sides

• Non-included Side: Side that is not between the two known angles

• Third Angle: Automatically equal by angle sum property

5 HL Congruence for Right Triangles
Exercise 5
Triangle ABC is a right triangle with ∠C = 90°, hypotenuse AB = 13 cm, and leg BC = 5 cm. Triangle DEF is a right triangle with ∠F = 90°, hypotenuse DE = 13 cm, and leg EF = 5 cm. Are the triangles congruent?
Definition:

HL Congruence: Two right triangles are congruent if their hypotenuses are equal and one pair of legs are equal. This is the Hypotenuse-Leg congruence criterion, exclusive to right triangles.

Given Information
∠C = ∠F = 90°, AB = DE = 13, BC = EF = 5
Hypotenuse-Leg
Hypotenuse = 13, Leg = 5
Congruence
△ABC ≅ △DEF
Step 1: Identify right angles

∠C = ∠F = 90° (both triangles are right triangles)

Step 2: Identify equal hypotenuses

AB = DE = 13 cm (hypotenuses of right triangles)

Step 3: Identify equal legs

BC = EF = 5 cm (one pair of legs in right triangles)

Step 4: Apply HL criterion

Since both triangles are right triangles with equal hypotenuses and one equal leg, they are congruent by HL criterion

△ABC ≅ △DEF by HL criterion
Final answer:

Yes, the triangles are congruent by HL criterion: △ABC ≅ △DEF

Applied rules:

• HL Congruence: Only for right triangles with equal hypotenuse and one leg

• Right Triangle Property: One angle is 90°

• Exclusive to Right Triangles: Cannot be applied to acute or obtuse triangles

Congruent Figures Laws, Methods, and Properties
\(SSS: AB = DE, BC = EF, AC = DF \Rightarrow \triangle ABC \cong \triangle DEF\)
SSS Congruence
Key definitions:

Congruent Figures: Geometric figures that have the same shape and the same size. They can be perfectly superimposed on each other through rigid transformations (translations, rotations, reflections).

Congruent Triangles: Triangles with equal corresponding sides and equal corresponding angles. If △ABC ≅ △DEF, then AB = DE, BC = EF, AC = DF, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

CPCTC: Corresponding Parts of Congruent Triangles are Congruent. Once triangles are proven congruent, all corresponding parts are equal.

Complete methodology:
  1. Analyze given information: Identify known equal sides and angles
  2. Select congruence criterion: Choose SSS, SAS, ASA, AAS, or HL based on available information
  3. Verify conditions: Confirm that all requirements of the chosen criterion are satisfied
  4. State congruence: Write the congruence statement with correct vertex correspondence
  5. Apply CPCTC: Use the fact that corresponding parts are equal to solve for unknowns
Tip 1: Always match vertices in the same order when writing congruence statements.
Tip 2: Remember that SSA (Side-Side-Angle) is NOT a valid congruence criterion!
Tip 3: Use the congruence criteria to prove triangles are identical in all aspects.
Tip 4: If triangles are congruent, all corresponding measurements are equal.
Common errors: Confusing congruence with similarity, using SSA as a criterion, misidentifying corresponding parts, forgetting that congruence requires BOTH equal sides AND equal angles.
Exam preparation: Master all five congruence criteria, practice identifying corresponding parts, solve proof problems, understand when SSA is insufficient.
Congruence criteria to know:

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

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

• ASA (Angle-Side-Angle): Two pairs of equal corresponding angles with equal included side

• AAS (Angle-Angle-Side): Two pairs of equal corresponding angles with equal non-included side

• HL (Hypotenuse-Leg): Equal hypotenuses and one pair of equal legs in right triangles

• Properties: If △ABC ≅ △DEF, then all corresponding sides and angles are equal.

Exercise with Visualization: Congruent Triangle Criteria
Exercise 6: Congruent Triangle Comparisons
Consider triangles with different congruence criteria:
SSS: All sides equal (3, 4, 5)
SAS: Two sides and included angle equal
ASA: Two angles and included side equal

Analysis: The chart shows how different congruence criteria uniquely determine triangle shapes and sizes.

  • SSS: Triangle completely determined by side lengths
  • SAS: Triangle determined by two sides and included angle
  • ASA: Triangle determined by two angles and included side

Questions & Answers

Question: Why isn't SSA (Side-Side-Angle) a valid congruence criterion? It seems like having two sides and an angle should be enough to determine a triangle.

Answer: Great question! SSA is not sufficient because it can lead to two different triangles. Here's why:

  • When you have two sides and a non-included angle, the third side might not be uniquely determined
  • You could potentially draw two different triangles that satisfy the SSA condition
  • This creates ambiguity, so SSA doesn't guarantee uniqueness like the valid criteria do

Example: If you have sides of length 5 and 7 with a non-included angle of 30°, you might be able to construct two different triangles depending on how you arrange the sides.

However, AAS works because the second angle is determined by the angle sum property, giving you the third angle and converting it to ASA.

Question: How do I remember which sides and angles are "included" versus "non-included" in the criteria?

Answer: Think of it this way:

  • Included side: The side BETWEEN two known angles (ASA)
  • Included angle: The angle BETWEEN two known sides (SAS)
  • Non-included side: A side that is NOT between the two known angles (AAS)

Memory trick: "Included" means "between" - the side or angle is sandwiched between the other two elements mentioned in the criterion.

Example: In SAS, the angle is "sandwiched" between the two sides. In ASA, the side is "sandwiched" between the two angles.

This distinction is crucial because it determines which congruence criterion applies!

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

Answer: The key differences are:

  • Congruent Triangles: Same shape AND same size, equal sides, equal angles. They can be perfectly superimposed.
  • Similar Triangles: Same shape but different sizes, proportional sides, equal angles.

Think of congruent triangles as identical twins (same everything) while similar triangles are like a photo and its enlargement (same shape, different sizes).

Criteria comparison:

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

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

Question: What does CPCTC mean and how do I use it?

Answer: CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent."

Once you've proven two triangles are congruent using one of the criteria (SSS, SAS, etc.), you can use CPCTC to conclude that:

  • All corresponding sides are equal
  • All corresponding angles are equal
  • Any corresponding parts (medians, altitudes, etc.) are equal

Example: If you prove △ABC ≅ △DEF by SAS, then you can immediately conclude: AB = DE, BC = EF, AC = DF, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F

This is very useful for finding unknown measurements after establishing congruence!

Question: When do I use HL instead of other criteria for right triangles?

Answer: HL (Hypotenuse-Leg) is specifically for right triangles and is very efficient:

  • Use HL when you know you have a right triangle
  • You only need the hypotenuse and one leg to be equal
  • It's simpler than proving SSS (which would require finding the third side)

You can still use other criteria (SAS, ASA, etc.) for right triangles if you have the necessary information, but HL is unique to right triangles and often the most direct path.

Example: If you know two right triangles have equal hypotenuses and one equal leg, you can immediately conclude they're congruent by HL without needing to find the other leg.

This makes HL particularly useful in word problems involving right triangles!