Grade 9 Angles of a Triangle: Short Notes

9 ශ්‍රේණිය · 2 වාරය · 16 පරිච්ඡේදය · ත්‍රිකෝණයක කෝණ

83%

of Grade 9 term tests include it (33 of 40)

300

marks in past papers (68 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Use the fact that the angles of a triangle add up to 180°
Explain why the angle sum is 180° using parallel lines
Use exterior angle = sum of the interior opposite angles
Use the sum of the exterior angles, 360°
Find angles by combining parallel-line facts with triangle facts

The roof's top angle

A carpenter in Kandy is cutting rafters for a new roof. Each side slopes up at 35° from the wall plate. What angle must he cut where the two rafters meet at the top? One fact about triangles gives it straight away: 110°.

The angle sum

The three interior angles of any triangle add up to 180°.Try it: cut out a paper triangle, tear off the three corners and put them side by side. They always make a straight line.Roof: 35° + 35° + top = 180°, so top = 110°.

Why it is always 180°

Draw PQ through A parallel to BC. Alternate angles are equal, so the two x's match and the two z's match. At A, x + y + z make a straight line: x + y + z = 180°.

✏️ Angles in algebra

The angles of a triangle are x, 2x and 60°. Find each angle.

1Angle sum: x + 2x + 60 = 180
23x = 120, so x = 40
3Angles: 40°, 80°, 60°. Check: 40 + 80 + 60 = 180 ✓

✅ 40°, 80°, 60°

The exterior angle

Produce (extend) one side of a triangle. The angle between the extended line and the next side is an exterior angle.The exterior angle = the sum of the two interior opposite angles (the two angles not next to it).Reason: exterior + its neighbour = 180° (straight line), and the three interior angles also make 180°.

Exterior = a + b

BC is produced to D. The exterior angle ACD equals a + b, the two interior opposite angles at A and B.

✏️ Using the exterior angle

In triangle ABC, BC is produced to D. ∠ACD = 115° and ∠ABC = 48°. Find ∠BAC and ∠ACB.

1Exterior angle = sum of interior opposite angles: 115 = ∠BAC + 48
2∠BAC = 115 − 48 = 67°
3Angles on a straight line: ∠ACB = 180 − 115 = 65°
4Check: 67 + 48 + 65 = 180 ✓

✅ ∠BAC = 67°, ∠ACB = 65°

⚠️ Opposite, not next door

In the 2012 and 2024 O/L papers many candidates did not use 'exterior angle = sum of the interior opposite angles'. Others added the angle next to the exterior angle. That neighbour is the one that makes 180° with it. The exterior angle equals the other two, the ones far away.

All three exterior angles

Take one exterior angle at each corner, going round the same way. Their sum is always 360°.Why? Each exterior + interior pair makes 180°, so the three pairs make 540°. Take away the interior 180°: 540 − 180 = 360°.Equilateral triangle: each exterior angle is 360 ÷ 3 = 120°.

⚠️ Give a reason for every angle

In one O/L paper only 23% could combine parallel-line facts with the triangle angle sum. Work in small steps: mark each angle you find on the diagram and write its reason, such as 'alternate angles', 'angles on a straight line', 'exterior angle of △PQR'. The next step usually appears once the angle is marked.

📌 Remember

• The interior angles of a triangle add up to 180°.• Exterior angle = sum of the two interior opposite angles.• An exterior angle and its neighbouring interior angle add to 180°.• The three exterior angles (one at each corner) add to 360°.• With parallel lines: find alternate or corresponding angles first, and write a reason for each step.

🧾 Quick revision sheet

Sum of the interior angles of a triangle?	180°
Exterior angle of a triangle equals…?	The sum of the two interior opposite angles.
Exterior angle + adjacent interior angle = ?	180° (angles on a straight line).
Sum of the three exterior angles?	360°
Two angles of a triangle are 50° and 60°. Third?	180 − 50 − 60 = 70°
Each angle of an equilateral triangle?	60° (exterior angle 120°).

Now test yourself: 5 quick checks and 1 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 10-minute lesson free ▶ Practise 5 past-paper questions