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Properties of Triangles (Angles & Sides)

Introduction

Geometry में triangles सबसे basic shapes होते हैं। इनके angle और side relationships को समझना कई aptitude और geometry problems को आसानी से solve करने में मदद करता है।

यह pattern खास तौर पर sum of angles, Pythagoras theorem, और side relationships पर ध्यान देता है - खासकर equilateral, isosceles, scalene और right-angled triangles में।

Pattern: Properties of Triangles (Angles & Sides)

Pattern

Main idea: किसी भी triangle में angles का sum हमेशा 180° होता है, और right-angled triangles में Pythagoras theorem लागू होती है: a² + b² = c².

Common Properties:
• Interior angles का sum = 180°
• Equilateral triangle में सभी angles = 60° और सभी sides equal होती हैं।
• Isosceles triangle में दो sides और दो angles equal होते हैं।
• Scalene triangle में सभी sides और सभी angles अलग होते हैं।
• Right-angled triangle में (Hypotenuse)² = (Base)² + (Perpendicular)².

Step-by-Step Example

Question

Triangle ABC में angle A = 50° और angle B = 60° है। Angle C find करें।

Solution

  1. Step 1: Triangle angle sum property याद करें।

    Triangle में सभी तीन angles का sum = 180°।

  2. Step 2: दिए गए values substitute करें।

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

  3. Step 3: Simplify करके unknown angle निकालें।

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

  4. Final Answer:

    Angle C = 70°.

  5. Quick Check:

    50° + 60° + 70° = 180° ✅

Quick Variations

1. दो angles दिए हों तो तीसरा angle find करना।

2. Right triangle में missing side find करने के लिए Pythagoras theorem का उपयोग।

3. दिए गए sides या angles से triangle का type पहचानना (equilateral, isosceles, scalene)।

4. 30°-60°-90° और 45°-45°-90° triangles में special ratios apply करना।

Trick to Always Use

  • Step 1: हमेशा angle sum rule (A + B + C = 180°) से शुरुआत करें।
  • Step 2: Right triangles में Pythagoras theorem apply करें।
  • Step 3: Isosceles या equilateral triangles में sides/angles की equality का उपयोग करें।

Summary

Summary

Properties of Triangles (Angles & Sides) pattern में:

  • Triangle के interior angles का sum हमेशा 180° होता है।
  • Pythagoras theorem सिर्फ right-angled triangles पर लागू होती है।
  • Equilateral → सभी sides और angles equal; Isosceles → दो sides equal; Scalene → सभी sides अलग।
  • Quick check: हमेशा total angles = 180° verify करें।

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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