Hard Level Worksheet Questions
Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.
Triangle congruency is a fundamental concept in geometry involving advanced applications of SSS, SAS, ASA, and RHS criteria. Understanding these principles is essential for geometric proofs, construction problems, and real-world engineering applications where precise measurements and structural integrity are crucial.
Let's explore advanced concepts of triangle congruency and their applications.
1. Write the definition of congruent triangles in your own words.
Perfect! Congruent triangles have exactly the same shape and size, meaning all corresponding sides and angles are equal.
2. Which congruency criterion requires a right angle as a condition?
Correct! RHS (Right angle-Hypotenuse-Side) criterion specifically applies to right-angled triangles.
3. ΔABC ≅ ΔPQR. If AB = 5 cm, BC = 4 cm, and AC = 6 cm, what is the length of PQ?
Step 1: In congruent triangles, corresponding sides are equal
Step 2: AB corresponds to
Step 3: Therefore, PQ = AB =
Excellent! By CPCT property, corresponding sides of congruent triangles are equal.
4. Name the congruency rule when two sides and the included angle are equal.
Perfect! SAS (Side-Angle-Side) criterion uses two sides and the included angle.
5. In ΔPQR, PQ = QR and PR = 6 cm. Name the type of triangle and state if it can be congruent to another triangle with sides 6 cm, 6 cm, and 8 cm.
Step 1: Since PQ = QR, triangle type is
Step 2: For congruency, all three sides must be
Step 3: Can it be congruent?
Great! Isosceles triangle with base 6 cm cannot be congruent to triangle with sides 6, 6, 8.
6. In ΔLMN and ΔXYZ, LM = XY, MN = YZ, and LN = XZ. Which criterion is satisfied?
Correct! When all three sides are equal, SSS criterion applies.
7. Which part of the congruency statement ΔABC ≅ ΔDEF corresponds to vertex B?
Step 1: In congruency statements, order
Step 2: Vertex B corresponds to
Perfect! In the correspondence ABC ↔ DEF, B corresponds to E.
8. Write the meaning of CPCT in congruency.
Excellent! CPCT stands for Corresponding Parts of Congruent Triangles (are equal).
9. True or False: SSA is a valid congruency criterion for triangles.
Correct! SSA (Side-Side-Angle) is NOT a valid criterion as it can lead to ambiguous cases.
Drag each scenario to its correct congruency criterion:
Part B: Short Answer Questions (2 Marks Each)
1. Two triangles have all three sides equal. State the congruency rule and give an example.
Step 1: Identify the rule
When all three sides are equal, the congruency rule is
Step 2: Provide example
Triangle 1 has sides 3 cm, 4 cm, 5 cm
Triangle 2 has sides
Perfect! SSS criterion requires all three corresponding sides to be equal.
2. ΔPQR has PQ = PR and ∠Q = ∠R. Show it's isosceles and state if it can be congruent to ΔXYZ with XY = XZ and ∠Y = ∠Z.
Step 1: Analyze triangle type
Since PQ = PR, triangle is
Step 2: Check congruency possibility
Both triangles are isosceles with equal base angles
They
Excellent! Both are isosceles triangles with similar properties.
3. In ΔLMN, LM = MN, and ∠L = 40°. Find ∠N. If ΔXYZ has XY = YZ and ∠X = 40°, are they congruent?
Step 1: Use isosceles triangle property
Since LM = MN, base angles are equal: ∠L = ∠N = 40°
Step 2: Find ∠M
∠M = 180° - 40° - 40° =
Step 3: Analyze ΔXYZ
Similarly, ΔXYZ has ∠Z = 40°, ∠Y = 100°
Step 4: Check congruency
Triangles are
Outstanding! Both triangles have the same angles and proportional sides.
Part C: Long Answer Questions (4 Marks Each)
1. In ΔABC, AB = AC and ∠B = 50°. In ΔPQR, PQ = PR and ∠Q = 50°. Prove congruency.
Step 1: Identify triangle types
Both triangles are
Step 2: Use isosceles triangle properties
In ΔABC: AB =
In ΔPQR: PQ =
Step 3: Calculate vertex angles
∠A = 180° - 50° - 50° =
∠P = 180° - 50° - 50° =
Step 4: Apply congruency criterion
We have: AB
Congruency criterion:
Therefore: ΔABC ≅
Excellent! Both isosceles triangles are congruent by SAS criterion.
2. ΔPQR and ΔXYZ are right-angled at Q and Y respectively. PQ = XY and PR = XZ. Prove congruency using RHS.
Step 1: Identify given conditions
∠Q = ∠Y =
Hypotenuse: PR =
One side: PQ =
Step 3: Apply RHS criterion
By RHS criterion: ΔPQR ≅
Perfect! RHS criterion applies when right angle, hypotenuse, and one side are equal.
3. ΔLMN has LM = MN and ∠L = 70°. ΔXYZ has XY = YZ and ∠X = 70°. Analyze congruency.
Step 1: Identify triangle types
Both triangles are
Step 2: Calculate angles in ΔLMN
Since LM = MN: ∠L = ∠N =
∠M = 180° - 70° - 70° =
Step 3: Calculate angles in ΔXYZ
Since XY =
∠Y = 180° - 70° - 70° =
Step 4: Determine congruency
All corresponding angles are equal, so congruency by
Corresponding parts: LM = XY, MN = YZ, LN =
Outstanding! Both triangles are congruent with all corresponding parts equal.
4. Triangle 1 has sides 7 cm, 9 cm, 10 cm. Triangle 2 has sides 10 cm, 7 cm, 9 cm. Are they congruent?
Step 1: Compare side lengths
Triangle 1: 7 cm, 9 cm, 10 cm
Triangle 2: 10 cm, 7 cm, 9 cm
Step 2: Check if all sides are equal
All three sides are equal
Step 3: Apply congruency rule
Congruency rule:
Step 4: State conclusion
The triangles are
Note: Correspondence depends on how vertices are
Perfect! Order of sides doesn't matter - all three sides are equal, so SSS applies.
Test your understanding with these multiple choice questions:
For each question, click on the correct answer:
1. If ΔABC ≅ ΔPQR, then ∠B equals:
(a) ∠P (b) ∠Q (c) ∠R (d) None
Correct! In ΔABC ≅ ΔPQR, the correspondence is A↔P, B↔Q, C↔R.
2. Which of these is not a valid congruency rule?
(a) SSS (b) SAS (c) SSA (d) ASA
Correct! SSA (Side-Side-Angle) is not a valid congruency criterion as it can create ambiguous cases.
3. In RHS congruency, the side given along with hypotenuse must be:
(a) Base (b) Opposite side (c) Any side adjacent to right angle (d) Median
Correct! RHS criterion requires the right angle, hypotenuse, and one side adjacent to the right angle.
4. If two triangles have two angles and the included side equal, the rule is:
(a) SSS (b) SAS (c) ASA (d) RHS
Correct! ASA (Angle-Side-Angle) criterion uses two angles and the included side.
5. The property "corresponding parts of congruent triangles are equal" is abbreviated as:
(a) CPC (b) CPCT (c) CPCTE (d) CP
Correct! CPCT stands for Corresponding Parts of Congruent Triangles (are equal).
🎉 Outstanding! You've Mastered Hard Level Triangle Congruency! Here's what you accomplished:
✓ Advanced Congruency Criteria: Deep understanding of SSS, SAS, ASA, and RHS applications
✓ Complex Triangle Analysis: Working with isosceles triangles and special cases
✓ Correspondence Mastery: Understanding vertex and side correspondence in congruent triangles
✓ CPCT Applications: Using properties of congruent triangles for problem solving
✓ Invalid Criteria Recognition: Understanding why SSA is not a valid congruency rule
✓ Real-World Applications: Connecting congruency to construction, engineering, and design
✓ Proof Construction: Building logical arguments for triangle congruency
✓ Problem-Solving Strategies: Analyzing complex scenarios and choosing appropriate criteria
Your expertise in triangle congruency prepares you for advanced geometric proofs, engineering applications, and sophisticated mathematical reasoning!