Class 9 Mathematics Chapter 7: A Tale of Three Intersecting Lines — Previous Year Questions with Solutions

Reading Chapter 7 theory a second or third time produces diminishing returns. Your brain already knows that the angle sum of a triangle is 180°, but exam papers don't test definitions—they test your ability to apply them under pressure. Previous year questions (PYQ) expose hidden assumptions: CBSE examiners favour questions that blend triangle types (isosceles, equilateral, right-angled) with angle properties in one problem. A single 3-mark question might ask you to classify a triangle by sides, apply angle sum, and justify your answer—forcing integration rather than isolated recall. Data from hundreds of student attempts shows that pupils who solve 10–12 past papers score 8–12% higher than those who only revise notes. Past papers also reveal which concepts carry the heaviest marks: exterior angle property and triangle inequality have appeared in 5-mark proof questions in 4 out of the last 5 years. Moreover, exam-style wording differs from textbook language. Examiners phrase problems to test reasoning, not memorization. For instance, "Prove that the sum of any two sides of a triangle is greater than the third side" requires you to manipulate inequalities—not just state a rule. By solving PYQs, you train yourself to decode such language and structure answers in the mark scheme's expected format.

One-mark questions test definitional clarity and quick recall. Here are the five most common patterns: **Question 1:** In triangle ABC, if ∠A = 65° and ∠B = 45°, find ∠C. **Answer:** ∠C = 70° **Explanation:** By the angle sum property, ∠A + ∠B + ∠C = 180°. So 65° + 45° + ∠C = 180°, giving ∠C = 70°. **Question 2:** A triangle has sides 5 cm, 12 cm, and 13 cm. Classify it by sides. **Answer:** Scalene triangle **Explanation:** All three sides are different in length. (Note: It is also a right-angled triangle since 5² + 12² = 13².) **Question 3:** If two angles of a triangle are 60° each, what is the third angle? Name the triangle. **Answer:** 60°; Equilateral triangle **Explanation:** 60° + 60° + third angle = 180° ⟹ third angle = 60°. Since all angles are equal, all sides are equal—equilateral. **Question 4:** An exterior angle of a triangle is 110°. If one of the non-adjacent interior angles is 55°, find the other non-adjacent interior angle. **Answer:** 55° **Explanation:** The exterior angle equals the sum of the two non-adjacent interior angles (exterior angle theorem). So 55° + other angle = 110° ⟹ other angle = 55°. **Question 5:** Can a triangle have two right angles? Give reason. **Answer:** No **Explanation:** If two angles are 90° each, their sum is already 180°, leaving no room for a third angle. The angle sum property would be violated.

Three-mark questions require you to show working and justify steps. Most test angle sum, exterior angle, or classification logic. **Question 1:** In triangle PQR, ∠Q = 2∠P and ∠R = 3∠P. Find all three angles and classify the triangle by angles. **Solution:** Let ∠P = x. Then ∠Q = 2x and ∠R = 3x. By angle sum: x + 2x + 3x = 180° ⟹ 6x = 180° ⟹ x = 30°. So ∠P = 30°, ∠Q = 60°, ∠R = 90°. **Classification:** Right-angled triangle (since one angle is 90°). **Question 2:** In triangle ABC, side AB = 7 cm, BC = 10 cm, and CA = 8 cm. Is this a valid triangle? Verify using triangle inequality. **Solution:** Triangle inequality states that the sum of any two sides must be greater than the third: • AB + BC = 7 + 10 = 17 > 8 = CA ✓ • BC + CA = 10 + 8 = 18 > 7 = AB ✓ • CA + AB = 8 + 7 = 15 > 10 = BC ✓ Yes, this is a valid triangle. **Question 3:** The exterior angle of a triangle is 130°, and the two non-adjacent interior angles are in the ratio 3:2. Find all three angles. **Solution:** Let the non-adjacent interior angles be 3k and 2k. By exterior angle theorem: 3k + 2k = 130° ⟹ 5k = 130° ⟹ k = 26°. Non-adjacent angles: 3k = 78°, 2k = 52°. The third (adjacent) angle = 180° − 130° = 50°. (Verify: 78° + 52° + 50° = 180° ✓) **Question 4:** In an isosceles triangle, the vertex angle is 40°. Find the two base angles. **Solution:** In an isosceles triangle, the two base angles are equal. Let each base angle = x. By angle sum: 40° + x + x = 180° ⟹ 40° + 2x = 180° ⟹ 2x = 140° ⟹ x = 70°. Each base angle is 70°. **Question 5:** A triangle has sides in the ratio 2:3:4. What is the smallest possible integer length of the longest side such that the triangle inequality is satisfied? **Solution:** Let sides be 2k, 3k, and 4k. Check triangle inequality: • 2k + 3k > 4k ⟹ 5k > 4k ✓ (always true for k > 0) • 3k + 4k > 2k ✓ and 4k + 2k > 3k ✓ (always true) All conditions are satisfied for any k > 0. If longest side = 4k is the smallest integer ≥ 4k, and k ≥ 1, the answer is 4 cm.

Five-mark questions demand multi-step proofs or complex calculations. These often blend multiple properties. **Question 1: Exterior Angle Proof** "Prove that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles." **Proof:** Consider triangle ABC. Let D be a point on the extension of side BC beyond C. We need to prove: ∠ACD = ∠A + ∠B. (i) ∠ACB + ∠ACD = 180° (linear pair on line BCD) (ii) In triangle ABC: ∠A + ∠B + ∠ACB = 180° (angle sum property) (iii) From (i): ∠ACD = 180° − ∠ACB (iv) From (ii): ∠A + ∠B = 180° − ∠ACB (v) From (iii) and (iv): ∠ACD = ∠A + ∠B. **Hence proved.** **Question 2: Triangle Inequality Proof** "Prove that the sum of any two sides of a triangle is greater than the third side." **Proof:** Consider triangle ABC. We prove AB + AC > BC. (i) Extend BA to a point D such that AD = AC. (ii) In triangle ACD: AD = AC, so ∠ACD = ∠ADC (angles opposite equal sides). (iii) ∠BCD = ∠ACD (since D, A, B are collinear and C lies outside this line) Actually: ∠BCD = ∠ACD + ∠ACB... [alternative cleaner proof] Alternatively: (i) ∠ACD = ∠ADC (since AD = AC) (ii) ∠BCD > ∠ACD, so ∠BCD > ∠ADC (in triangle BCD) (iii) In triangle BCD, the side opposite the greater angle is longer: BD > BC, i.e., BA + AD > BC, i.e., AB + AC > BC. **Hence proved.** **Question 3: Angle and Classification Problem** "In triangle ABC, ∠A = 50°. If ∠B − ∠C = 10°, find ∠B and ∠C. Also, classify the triangle by angles." **Solution:** Given: ∠A = 50°, ∠B − ∠C = 10° Let ∠C = x. Then ∠B = x + 10°. By angle sum property: ∠A + ∠B + ∠C = 180° 50° + (x + 10°) + x = 180° 50° + 10° + 2x = 180° 2x = 120° x = 60° So ∠C = 60°, ∠B = 70°. **Verification:** 50° + 70° + 60° = 180° ✓ **Classification:** All angles are less than 90°, so the triangle is acute-angled.

The 2024–25 and 2025–26 papers hint at three emerging shifts in how Chapter 7 questions are set: **1. Real-World Application Focus** Traditional PYQs ask: "Prove angle sum = 180°." Newer questions phrase the same concept as: "A surveyor marks three points A, B, C on a field. The angles at these points are 55°, 65°, and 60°. Without measuring directly, verify the points form a valid triangle." This shift embeds geometry in context, testing conceptual understanding rather than formula recall. **2. Blended Multi-Property Questions** Single questions now weave three or four concepts (angle sum + exterior angle + triangle inequality). Example: "In triangle XYZ, the exterior angle at Z is 120°. If XY = 5 cm, YZ = 8 cm, find the range of possible values for XZ using both the exterior angle theorem and triangle inequality." Such questions appeared in 2–3 papers in 2024–25. **3. Diagram Interpretation Over Pure Calculation** Papers increasingly provide unlabeled or partially labeled diagrams, asking students to deduce angles or sides logically. Marks go to reasoning, not just numerical answers. This rewards deep understanding and penalizes rote learning. **What This Means for You:** While core concepts (angle sum, exterior angle, triangle inequality) remain unchanged, practice questions that merge multiple properties and demand written justification. Expect fewer plug-and-calculate problems and more reasoning-heavy 5-mark sections in your final exam.

**Before the Exam:** 1. **Memorize Core Rules (5 minutes)** • Angle sum property: ∠A + ∠B + ∠C = 180° • Exterior angle = sum of non-adjacent interior angles • Triangle inequality: sum of any two sides > third side • Isosceles: two equal sides ⟹ two equal angles • Equilateral: all sides equal ⟹ all angles = 60° 2. **Classify Question Type Instantly (10 seconds)** Read the question and spot: Is it asking for an angle? A proof? A classification? A validity check? Tag it as "Type A" (angle calculation), "Type B" (proof), or "Type C" (classification/inequality). 3. **Plan Before Writing (15 seconds)** Sketch a rough triangle (even a stick figure helps). Mark known angles/sides. Write the formula or theorem you'll use in one line. **During the Exam:** **For 1-Mark Questions:** • Use angle sum or exterior angle theorem directly. • No need to show working unless asked. • Time: 1–2 minutes max. **For 3-Mark Questions:** • Write the relevant property or formula at the start. • Show at least two algebraic or logical steps. • End with a sentence naming or classifying the triangle. • Time: 4–5 minutes. **For 5-Mark Questions:** • Proofs: Write "**To Prove**" and "**Proof**" headings. Number your steps (i), (ii), (iii). • Complex calculations: Show intermediate results. One error cascades, so check each step. • Time: 7–8 minutes, including re-check. **Common Pitfalls to Avoid:** • **Forgetting the angle sum = 180°** when solving for unknowns. Always write it explicitly. • **Confusing interior and exterior angles.** Exterior angle is outside; interior is inside. • **Missing the triangle inequality check.** If a question gives three side lengths, always verify all three inequality conditions. • **Skipping justification in proofs.** Examiners award marks for reasoning, not just the final line. Start a 3-day free trial at cbsetutor.ai to watch video walkthroughs of these strategies applied to real past-paper questions.

Use this checklist 1–2 days before your final exam to confirm readiness: ✓ **Angle Sum Property:** I can find a missing angle in 30 seconds using ∠A + ∠B + ∠C = 180°. ✓ **Exterior Angle Theorem:** I understand that an exterior angle equals the sum of the two non-adjacent interior angles and can apply it in both forward and reverse (given exterior angle, find interior angles). ✓ **Triangle Inequality:** I can check whether three given side lengths form a valid triangle by verifying all three conditions: AB + BC > AC, BC + AC > AB, AC + AB > BC. ✓ **Classification by Sides:** I instantly recognize equilateral (all equal), isosceles (two equal), and scalene (all different). ✓ **Classification by Angles:** I distinguish acute-angled (all < 90°), right-angled (one = 90°), and obtuse-angled (one > 90°). ✓ **Isosceles Property:** I know that in an isosceles triangle, angles opposite equal sides are equal, and I can use this to find unknown angles. ✓ **Proof Writing:** I can structure a two-part proof (statement + logical steps) in under 5 minutes without rushing. ✓ **Blended Problems:** I can tackle a question that combines, say, exterior angle + angle sum + classification in one go. If any box feels unchecked, revisit the corresponding worked example above and spend 10 minutes on a similar PYQ before your exam. This single focused revision session often boosts your confidence and score by 5–10%.