Moderate Level Worksheet Questions
Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.
Congruent triangles are fundamental in geometry. Understanding the criteria for congruency (SSS, SAS, ASA, RHS) and their applications is essential for geometric proofs and real-world problem solving.
First, let's explore the basic definitions and criteria for triangle congruency.
1. Define congruent triangles.
Perfect! Congruent triangles have exactly the same shape and size.
2. Write any one condition for congruency of triangles.
Excellent! SSS (Side-Side-Side) is one of the congruency criteria.
3. State the meaning of the symbol "≅".
Great! The symbol ≅ means "congruent to".
4. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, which criterion is used?
Correct! SAS (Side-Angle-Side) criterion applies when the angle is between the two equal sides.
5. Name the criterion for congruency in which two angles and one side are equal.
Well done! ASA (Angle-Side-Angle) uses two angles and the included side.
6. Which side of a triangle is opposite to the largest angle?
Perfect! In any triangle, the longest side is always opposite to the largest angle.
7. If two triangles are congruent, what can you say about their corresponding sides?
Excellent! Corresponding sides of congruent triangles are always equal.
8. Write the full form of RHS criterion.
Correct! RHS stands for Right angle-Hypotenuse-Side criterion.
9. Name the property used when we say ΔABC ≅ ΔPQR ⇒ ∠A = ∠P.
Perfect! CPCT means Corresponding Parts of Congruent Triangles are equal.
10. If AB = PQ, BC = QR, and AC = PR, which criterion is satisfied?
Outstanding! When all three sides are equal, we use SSS criterion.
Drag each piece of information to its correct congruency criterion:
Part B: Short Answer Questions (2 Marks Each)
1. In ΔABC and ΔDEF, AB = DE, BC = EF, AC = DF. State the congruency criterion and write the correspondence of vertices.
Step 1: Identify the criterion
Given: AB = DE, BC = EF, AC = DF
Congruency criterion:
Step 2: Find correspondence
A corresponds to:
B corresponds to:
C corresponds to:
Excellent! Using SSS criterion: A ↔ D, B ↔ E, C ↔ F.
2. Two sides of a triangle are 6 cm and 8 cm, and the included angle is 60°. Another triangle has two sides 6 cm and 8 cm, and the included angle 60°. Are the triangles congruent?
Step 1: Analyze given information
Both triangles have: Two sides and included angle equal
Are they congruent?
Step 2: Identify criterion
Criterion used:
Perfect! Yes, they are congruent by SAS criterion.
3. ΔXYZ has XY = 5 cm, YZ = 7 cm, and ∠Y = 50°. ΔPQR has PQ = 5 cm, QR = 7 cm, and ∠Q = 50°. Are the triangles congruent?
Step 1: Compare given information
Triangle XYZ: XY = 5 cm, YZ = 7 cm, ∠Y = 50°
Triangle PQR: PQ = 5 cm, QR = 7 cm, ∠Q = 50°
Are they congruent?
Step 2: Identify criterion
The angle is between the two equal sides, so criterion:
Great work! Yes, congruent by SAS criterion.
4. In ΔABC and ΔPQR, AB = PQ, ∠B = ∠Q, and BC = QR. Write the congruency criterion.
Analysis: Two sides and included angle
AB = PQ (first side)
∠B = ∠Q (included angle)
BC = QR (second side)
Criterion:
Correct! SAS criterion applies.
5. Write any two real-life examples where congruent triangles are seen.
Example 1:
Example 2:
Excellent examples! Congruent triangles provide structural stability.
Part C: Long Answer Questions (4 Marks Each)
1. In ΔABC, AB = AC and ∠B = 50°. In ΔDEF, DE = DF and ∠E = 50°. Prove the triangles are congruent.
Step 1: Identify triangle types
ΔABC: AB = AC (isosceles triangle)
ΔDEF: DE = DF (isosceles triangle)
Step 2: Use isosceles triangle property
In ΔABC: AB = AC, so ∠B = ∠C =
In ΔDEF: DE = DF, so ∠E = ∠F =
Step 3: Find remaining angles
In ΔABC: ∠A = 180° - 50° - 50° =
In ΔDEF: ∠D = 180° - 50° - 50° =
Step 4: Apply congruency criterion
We have: AB = DE (given equal sides), ∠B = ∠E = 50°, AC = DF (given equal sides)
Criterion used:
Therefore: ΔABC ≅
Perfect proof! ΔABC ≅ ΔDEF by SAS criterion.
2. ΔLMN is right-angled at M and ΔXYZ is right-angled at Y. If LM = XY and LN = XZ, prove that the triangles are congruent.
Step 1: Identify given information
∠M = ∠Y =
LM = XY (one side equal)
LN = XZ (this is the
Step 2: Identify criterion
We have: Right angle, hypotenuse, and one side
Criterion:
Step 3: Conclusion
Therefore: ΔLMN ≅
Excellent! ΔLMN ≅ ΔXYZ by RHS criterion.
3. The sides of a triangle are 7 cm, 8 cm, and 9 cm, and the sides of another triangle are 9 cm, 8 cm, and 7 cm. Are they congruent?
Step 1: Compare sides
Triangle 1: 7 cm, 8 cm, 9 cm
Triangle 2: 9 cm, 8 cm, 7 cm
Step 2: Check equality
All three sides are equal (just in different order):
Step 3: Apply criterion
Since all three sides are equal: Criterion =
Step 4: Conclusion
The triangles are:
Note: Correspondence depends on how vertices are
Perfect! The triangles are congruent by SSS criterion.
Test your understanding with these multiple choice questions:
For each question, click on the correct answer:
1. If two triangles are congruent, their corresponding angles are:
(a) Equal (b) Unequal (c) Supplementary (d) Complementary
Correct! By CPCT property, corresponding angles of congruent triangles are equal.
2. In SSS criterion, how many sides are equal?
(a) 1 (b) 2 (c) 3 (d) None
Correct! SSS means all three sides are equal.
3. If ΔABC ≅ ΔPQR, then side BC is equal to:
(a) PQ (b) QR (c) PR (d) None
Correct! In the correspondence ABC ↔ PQR, BC corresponds to QR.
4. In SAS, the angle given must be:
(a) Opposite to equal sides (b) Between the equal sides (c) Largest angle (d) Any angle
Correct! SAS requires the angle to be included (between) the two equal sides.
5. Which congruency rule cannot be used in triangles?
(a) SSA (b) SSS (c) SAS (d) ASA
Correct! SSA (Side-Side-Angle) is not a valid congruency criterion as it can lead to ambiguous cases.