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Properties of Triangles (Angles & Sides)

Introduction

Triangles are the most fundamental shapes in geometry. Understanding their angle and side relationships helps in solving many aptitude and geometry problems efficiently.

This pattern focuses on properties such as sum of angles, Pythagoras theorem, and side relationships in different types of triangles - equilateral, isosceles, scalene, and right-angled.

Pattern: Properties of Triangles (Angles & Sides)

Pattern

The key idea is: The sum of angles in any triangle is 180°, and in right-angled triangles, the Pythagoras theorem applies: a² + b² = c².

Common Properties:
• Sum of interior angles = 180°
• In an equilateral triangle, all angles = 60° and all sides are equal.
• In an isosceles triangle, two sides and two angles are equal.
• In a scalene triangle, all sides and all angles are different.
• In a right-angled triangle, (Hypotenuse)² = (Base)² + (Perpendicular)².

Step-by-Step Example

Question

In a triangle ABC, angle A = 50° and angle B = 60°. Find angle C.

Solution

  1. Step 1: Recall the triangle angle sum property.

    The sum of all three angles in a triangle = 180°.

  2. Step 2: Substitute the given values.

    A + B + C = 180° → 50° + 60° + C = 180°.

  3. Step 3: Simplify and find the unknown angle.

    C = 180° - (50° + 60°) = 180° - 110° = 70°.

  4. Final Answer:

    Angle C = 70°.

  5. Quick Check:

    50° + 60° + 70° = 180° ✅

Quick Variations

1. Find the missing angle when two angles are known.

2. Use Pythagoras theorem to find the missing side in a right triangle.

3. Identify the triangle type (equilateral, isosceles, scalene) from given sides or angles.

4. Apply special ratios in 30°-60°-90° and 45°-45°-90° triangles.

Trick to Always Use

  • Step 1: Always start by applying the angle sum rule (A + B + C = 180°).
  • Step 2: In right triangles, use Pythagoras theorem for side relations.
  • Step 3: For isosceles or equilateral triangles, apply equality of sides or angles.

Summary

Summary

In the Properties of Triangles (Angles & Sides) pattern:

  • The sum of interior angles of a triangle is always 180°.
  • Pythagoras theorem applies only to right-angled triangles.
  • Equilateral → all equal sides and angles; Isosceles → two equal sides; Scalene → all sides different.
  • Quick check: Always verify that total angles = 180°.

Practice

(1/5)
1. In a triangle ABC, angle A = 40° and angle B = 90°. Find angle C.
easy
A. 50°
B. 40°
C. 60°
D. 70°

Solution

  1. Step 1: Apply the angle sum property.

    The sum of angles in a triangle is 180° (A + B + C = 180°).

  2. Step 2: Substitute the given values.

    40° + 90° + C = 180°.

  3. Step 3: Simplify to find C.

    C = 180° - 130° = 50°.

  4. Final Answer:

    Angle C = 50° → Option A.

  5. Quick Check:

    40° + 90° + 50° = 180° ✅
Hint: Use 180° - (sum of known angles).
Common Mistakes: Forgetting to subtract the sum of known angles from 180°.
2. In a triangle, two angles measure 55° and 65°. Find the third angle.
easy
A. 70°
B. 60°
C. 65°
D. 55°

Solution

  1. Step 1: Use the sum of angles formula.

    A + B + C = 180°.

  2. Step 2: Substitute given values.

    55° + 65° + C = 180°.

  3. Step 3: Simplify.

    C = 180° - 120° = 60°.

  4. Final Answer:

    Angle C = 60° → Option B.

  5. Quick Check:

    55° + 65° + 60° = 180° ✅
Hint: Subtract the sum of the two angles from 180°.
Common Mistakes: Adding all three angles to get 180° instead of subtracting the known sum.
3. In a right-angled triangle, if one side is 3 cm and the other side is 4 cm, find the hypotenuse.
easy
A. 6 cm
B. 7 cm
C. 5 cm
D. 8 cm

Solution

  1. Step 1: Recall the Pythagoras theorem.

    For a right triangle, (hypotenuse)² = (side1)² + (side2)².

  2. Step 2: Substitute values.

    h² = 3² + 4² = 9 + 16 = 25.

  3. Step 3: Take square root.

    h = √25 = 5 cm.

  4. Final Answer:

    Hypotenuse = 5 cm → Option C.

  5. Quick Check:

    3² + 4² = 9 + 16 = 25 = 5² ✅
Hint: Remember the 3-4-5 right-triangle pattern.
Common Mistakes: Adding side lengths directly instead of squaring them.
4. If two sides of an isosceles triangle are each 10 cm and the base is 12 cm, find its perimeter.
medium
A. 30 cm
B. 28 cm
C. 34 cm
D. 32 cm

Solution

  1. Step 1: Recall perimeter formula.

    Perimeter = sum of all sides.

  2. Step 2: Substitute values for the two equal sides and base.

    Perimeter = 10 + 10 + 12.

  3. Step 3: Compute.

    Perimeter = 32 cm.

  4. Final Answer:

    Perimeter = 32 cm → Option D.

  5. Quick Check:

    2 × 10 + 12 = 20 + 12 = 32 ✅
Hint: For isosceles, perimeter = 2 × equal side + base.
Common Mistakes: Forgetting to add both equal sides before adding the base.
5. In a right-angled triangle, if hypotenuse = 13 cm and one side = 5 cm, find the other side.
medium
A. 12 cm
B. 11 cm
C. 10 cm
D. 9 cm

Solution

  1. Step 1: Apply Pythagoras theorem.

    (Hypotenuse)² = (side1)² + (side2)².

  2. Step 2: Substitute known values.

    13² = 5² + x² ⇒ 169 = 25 + x².

  3. Step 3: Solve for x.

    x² = 169 - 25 = 144 ⇒ x = √144 = 12 cm.

  4. Final Answer:

    Other side = 12 cm → Option A.

  5. Quick Check:

    5² + 12² = 25 + 144 = 169 = 13² ✅
Hint: Use common Pythagorean triples (5,12,13) to spot answers quickly.
Common Mistakes: Adding squares instead of subtracting to isolate the unknown.

Mock Test

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