How to Solve: Properties of Triangles

How to Solve: Properties of Triangles

(For SSC, Bank, Railway Exams – Ace Your Exam with Confidence!)

Introduction

"Mastering triangle properties can get you 5-7 marks in SSC, Bank, or Railway exams—enough to push you from ‘just passing’ to ‘top ranker’! Whether it’s finding missing angles, checking if a triangle is possible, or using the Pythagorean theorem, these questions appear in every competitive exam. Let’s break it down so you never lose a mark again."

What You Need To Know First

Before diving in, make sure you understand: 1. Basic angle properties (sum of angles in a straight line = 180°, vertically opposite angles are equal). 2. Types of triangles (equilateral, isosceles, scalene, right-angled). 3. Algebra basics (solving simple equations like 2x + 30 = 180).

If any of these are shaky, pause and review them first—this guide assumes you’re solid on these.

Key Vocabulary

Formulas To Know

1. Angle Sum Property

Formula: ∠A + ∠B + ∠C = 180° - Variables: ∠A, ∠B, ∠C = angles of the triangle. - MEMORISE THIS – This is the most important property.

2. Exterior Angle Theorem

Formula: Exterior angle = Sum of two opposite interior angles -
Example: If ∠A = 50° and ∠B = 60°, then the exterior angle at C = 50° + 60° = 110°. - MEMORISE THIS – Saves time in angle problems.

3. Pythagorean Theorem (Right-Angled Triangles Only)

Formula: Hypotenuse² = Base² + Height² (or c² = a² + b²) - Variables: - c = hypotenuse (longest side, opposite 90°). - a, b = other two sides. - MEMORISE THIS – Used in every right-angled triangle problem.

4. Triangle Inequality Theorem

Formula: Sum of any two sides > Third side -
Example: If sides are 3, 4, 5 → 3+4 > 5 (7 > 5), 3+5 > 4 (8 > 4), 4+5 > 3 (9 > 3). Valid triangle. - MEMORISE THIS – Used to check if a triangle is possible.

5. Area of a Triangle

Formula 1 (Base & Height): Area = ½ × base × height Formula 2 (Heron’s Formula): Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 (semi-perimeter). - Given on exam sheet (but memorising saves time).

Step-by-Step Method

How to Solve Any Triangle Property Problem

Follow these steps in order for every question:

1. Read the question carefully. Underline given values (angles, sides).
2. Draw the triangle. Label all known sides/angles. If no diagram, sketch one.
3. Identify the type of triangle (equilateral, isosceles, right-angled, etc.).
4. Apply the relevant formula:
5. For angles → Use angle sum property or exterior angle theorem.
6. For sides → Use Pythagorean theorem or triangle inequality.
7. For area → Use ½ × base × height or Heron’s formula.
8. Solve for the unknown. Write equations clearly.
9. Check your answer. Does it make sense? (e.g., angles should add to 180°).

Worked Examples

Example 1 – Basic (Finding Missing Angle)

Question: In a triangle, two angles are 50° and 60°. Find the third angle.

Step-by-Step Solution: 1. Given: ∠A = 50°, ∠B = 60°, ∠C = ? 2. Draw the triangle. Label the angles. 3. Type of triangle: Scalene (all angles different). 4. Apply angle sum property: ∠A + ∠B + ∠C = 180°
→ 50° + 60° + ∠C = 180° 5. Solve: ∠C = 180° – 110° = 70° 6. Check: 50° + 60° + 70° = 180° ✔️

What we did and why: We used the angle sum property because the question gave two angles and asked for the third. This is the fastest way to solve such problems.

Example 2 – Medium (Exterior Angle Theorem)

Question: In a triangle, one interior angle is 40°, and the exterior angle adjacent to another angle is 110°. Find the remaining angles.

Step-by-Step Solution: 1. Given: ∠A = 40°, Exterior angle at B = 110°. 2. Draw the triangle. Label ∠A and the exterior angle at B. 3. Apply exterior angle theorem: Exterior angle = Sum of two opposite interior angles.
→ 110° = ∠A + ∠C
→ 110° = 40° + ∠C 4. Solve for ∠C: ∠C = 110° – 40° = 70° 5. Find ∠B using angle sum property: ∠A + ∠B + ∠C = 180°
→ 40° + ∠B + 70° = 180°
→ ∠B = 180° – 110° = 70° 6. Check: 40° + 70° + 70° = 180° ✔️

What we did and why: We used the exterior angle theorem first because it directly related the given exterior angle to the missing interior angles. Then, we used the angle sum property to find the last angle.

Example 3 – Exam-Style (Pythagorean Theorem + Triangle Inequality)

Question: A triangle has sides 5 cm, 12 cm, and x cm. If it is a right-angled triangle, find the possible values of x.

Step-by-Step Solution: 1. Given: Sides = 5 cm, 12 cm, x cm. Right-angled triangle. 2. Identify the hypotenuse: The longest side is the hypotenuse.
- If x is the hypotenuse → x² = 5² + 12² = 25 + 144 = 169 → x = 13 cm.
- If 12 cm is the hypotenuse → 12² = 5² + x² → 144 = 25 + x² → x² = 119 → x = √119 cm (≈10.9 cm). 3. Check triangle inequality for both cases:
- For x = 13 cm: 5 + 12 > 13 (17 > 13), 5 + 13 > 12 (18 > 12), 12 + 13 > 5 (25 > 5) ✔️
- For x = √119 cm: 5 + √119 > 12 (≈15.9 > 12), 5 + 12 > √119 (17 > 10.9), 12 + √119 > 5 (≈22.9 > 5) ✔️ 4. Possible values of x: 13 cm or √119 cm.

What we did and why: We considered both cases (x as hypotenuse or not) because the question didn’t specify. Then, we verified using the triangle inequality to ensure the sides form a valid triangle.

Common Mistakes

Exam Traps

1-Minute Recap (Night Before Exam)

"Listen up—this is all you need to remember for triangle properties: 1. Angles in a triangle always add to 180°. If you know two angles, subtract from 180° to find the third. 2. Exterior angle = Sum of two opposite interior angles. No need to overcomplicate it. 3. Right-angled triangle? Use Pythagoras (c² = a² + b²). Hypotenuse is the longest side. 4. Check if a triangle is possible: Sum of any two sides must be greater than the third side. 5. Area? Use ½ × base × height if you have height. Otherwise, Heron’s formula (but memorise it!). That’s it. Stay calm, draw diagrams, and apply these rules. You’ve got this!