Hinglish

When Some Sides and Angles are Given

Class 7 Maths · Chapter 7: A Tale of Three Intersecting Lines · NCERT Ganita Prakash section 7.3

Learn

Two sides and the included angle: for AB = 5 cm, AC = 4 cm, ∠A = 45°, draw AB, make 45° at A with a protractor, mark C on that arm 4 cm from A, and join BC. Two angles and the included side: for AB = 5 cm, ∠A = 45°, ∠B = 80°, draw AB and both angles; their arms cross at C.

Do the arms always meet? If ∠A = 40°, tilt the arm at B more and more. When the two arms are parallel, the interior angles add to 180°, so ∠B = 140°. For ∠B ≥ 140° there is no triangle, whatever the length of AB. Rule: two angles of a triangle must add to less than 180°.

∠A + ∠B < 180°

Angle sum property: draw a line XY through A parallel to BC. Then ∠XAB = ∠B and ∠YAC = ∠C (alternate angles), and these with ∠A make a straight angle. So ∠A + ∠B + ∠C = 180° for every triangle. This clever idea appears in the Elements, a famous book by the Greek mathematician Euclid (around 300 BCE).

∠A + ∠B + ∠C = 180°

So with ∠B = 50° and ∠C = 70°, ∠A = 180° − 120° = 60°. If ∠B = ∠C and ∠A = 50°, each is (180° − 50°) ÷ 2 = 65°. Exterior angle: extend BC to D. ∠ACD = 180° − ∠C = ∠A + ∠B: the exterior angle equals the two opposite interior angles added.

∠ACD = ∠A + ∠B

Explore: the missing angle.

  1. A triangle has ∠A = 50° and ∠B = 60°. Open the angle to ∠C.
  2. ∠A = 40°. Open the angle to the smallest ∠B at which the arms become parallel.

Remember

  • A triangle can be built from two sides and the included angle, or two angles and the included side.
  • The angles of a triangle add up to 180°; an exterior angle equals the two opposite interior angles.

Quick check

Two angles of a triangle are 75° and 45°. What is the third?

  1. 60°
  2. 120°
  3. 50°
  4. 90°
Answer

60°. 180° − 75° − 45° = 60°.

Think about it

Is there a triangle with two sides and an included angle of 180°?

One answer

No. A 180° angle is a straight line, so the three points would lie on one line. The included angle must be less than 180°.

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.