Triangles: Angle Properties, Sides and Pythagoras
Triangles: Angle Properties, Sides and Pythagoras
Every triangle question rests on four facts: the angles add to 180°, an exterior angle equals the two opposite interior angles, any two sides together exceed the third, and in a right triangle the square of the hypotenuse equals the sum of the squares of the other sides. NTPC asks isosceles angle problems directly; the rest appears inside mensuration and height questions.
1. Properties Box
| Property | Statement |
|---|---|
| Angle sum | ∠A + ∠B + ∠C = 180° |
| Exterior angle | Exterior angle = sum of the two opposite interior angles |
| Isosceles | Equal sides face equal angles |
| Side-angle order | The largest side faces the largest angle |
| Triangle inequality | Each side < sum of the other two and > their difference |
| Pythagoras (right angle at C) | c² = a² + b² |
| Acute or obtuse test (c longest) | c² < a² + b² acute; c² > a² + b² obtuse |
| Bisectors of ∠B and the exterior angle at C | Meet at angle ∠A/2 |
| Internal bisectors of ∠B and ∠C | Meet at angle 90° + ∠A/2 |
2. Pythagorean Triplets
| Triplet | Multiples seen in papers |
|---|---|
| 3, 4, 5 | 6-8-10, 9-12-15, 15-20-25 |
| 5, 12, 13 | 10-24-26 |
| 7, 24, 25 | 14-48-50 |
| 8, 15, 17 | 16-30-34 |
| 9, 40, 41 | 18-80-82 |
| 20, 21, 29 | 40-42-58 |
Isosceles right triangle: sides a, a, a√2 30-60-90 triangle: sides 1 : √3 : 2
✗ Sides 6 and 10, so the third side can be 4 to 16 | ✓ Strictly between: 4 < x < 16, so 5 to 15 for integers
✗ Exterior angle at C = ∠A + ∠C | ✓ It equals the two opposite interior angles: ∠A + ∠B
हिंदी नोट: त्रिभुज के कोणों का योग 180° और बाह्य कोण दोनों अंतः सम्मुख कोणों के योग के बराबर होता है। किन्हीं दो भुजाओं का योग तीसरी से बड़ा होना चाहिए। समकोण त्रिभुज में कर्ण² = आधार² + लंब²।
Exam Pointer: Verified NTPC pattern: base angles of an isosceles triangle from its vertex angle (June 2025 Graduate CBT-1). Exterior angle, triangle inequality and Pythagoras questions had no verified NTPC shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Angle sum and isosceles triangles
[PYQ: NTPC Graduate CBT-1 21-Jun-2025 Shift-3]
EXAM LEVEL
Q. The vertex angle of an isosceles triangle is 50°. Find each base angle.
The base angles are equal and share 180° − 50° = 130°, so each is 65°.
Answer: 65°
EXAMATLAS LEVEL
Q. In an isosceles triangle each base angle is 15° more than twice the vertex angle. Find all the angles.
Let the vertex angle be v. Then v + 2(2v + 15) = 180 gives 5v = 150, so v = 30° and each base angle is 75°.
Answer: 30°, 75°, 75°
Pattern 2: Exterior angle theorem and bisector angles
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. An exterior angle of a triangle is 110° and one of the opposite interior angles is 45°. Find the other two angles of the triangle.
The other opposite angle = 110° − 45° = 65°, and the angle next to the exterior angle = 180° − 110° = 70°.
Answer: 65° and 70°
EXAMATLAS LEVEL
Q. In triangle ABC, BC is extended to D so that ∠ACD = 120°, and ∠A : ∠B = 3 : 1. Find the angles of the triangle, and the angle at which the bisector of ∠B meets the bisector of ∠ACD.
∠A + ∠B = 120° in the ratio 3 : 1, so ∠A = 90°, ∠B = 30° and ∠C = 60°. The two bisectors meet at ∠A/2 = 45°.
Answer: 90°, 30°, 60°; 45°
Pattern 3: Triangle inequality and the acute or obtuse test
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Can sides 4 cm, 7 cm and 12 cm form a triangle?
4 + 7 = 11, which is less than 12, so no.
Answer: No
EXAMATLAS LEVEL
Q. Two sides of a triangle are 6 cm and 10 cm, and the third side is a whole number. How many triangles are possible, and how many of them are obtuse?
4 < x < 16 gives x = 5 to 15, eleven triangles. If x is the longest side, it is obtuse when x² > 36 + 100 = 136: x = 12, 13, 14, 15. If 10 is the longest, it is obtuse when 100 > 36 + x², that is x² < 64: x = 5, 6, 7. Obtuse = 7.
Answer: 11 triangles; 7 obtuse
Pattern 4: Pythagoras: ladders, poles and diagonals
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A 13 m ladder reaches a window 12 m high. How far is its foot from the wall?
√(169 − 144) = 5 m (5-12-13 triplet).
Answer: 5 m
EXAMATLAS LEVEL
Q. A 25 m ladder rests against a wall with its top 24 m up. The foot slips 8 m further from the wall. How far does the top slide down?
The foot is first √(625 − 576) = 7 m away, then 15 m. The new height is √(625 − 225) = 20 m, so the top slides 4 m.
Answer: 4 m
CBT-2 LEVEL
Pattern 5: Angle bisector theorem and the median length
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. In triangle ABC, AB = 6 cm, AC = 9 cm and BC = 10 cm. The bisector of ∠A meets BC at D. Find BD and DC.
BD : DC = AB : AC = 2 : 3, so BD = 4 cm and DC = 6 cm.
Answer: 4 cm and 6 cm
EXAMATLAS LEVEL
Q. In triangle ABC, AB = 8 cm, AC = 6 cm and BC = 10 cm, and AD is the median to BC. Find AD by Apollonius' theorem and check it another way.
AB² + AC² = 2(AD² + BD²) gives 100 = 2(AD² + 25), so AD = 5 cm. Check: 6-8-10 is right-angled at A, and the median to the hypotenuse is half of it, 5 cm.
Answer: 5 cm
60-Second Revision
- Angles add to 180°; the exterior angle is the sum of the two opposite interior angles.
- The third side lies strictly between the difference and the sum.
- Compare c² with a² + b² to decide acute, right or obtuse.
- Learn the triplets 3-4-5, 5-12-13, 7-24-25, 8-15-17.