Grade 10 Triangles: Short Notes

10 ශ්‍රේණිය · 1 වාරය · 8 පරිච්ඡේදය · ත්‍රිකෝණ

100%

of Grade 10 term tests include it (54 of 54)

908

marks in past papers (221 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Prove and use: the angles of a triangle add up to 180°
Use the exterior angle theorem
Find angles in isosceles triangles using equal base angles
Use the converse to show sides are equal or find a length
Combine these results in term-test problems

A roof that won't lean

A carpenter in Kurunegala is building a roof frame. He cuts the two sloping rafters exactly the same length. Without a protractor he already knows both rafters meet the wall beam at the same angle, and that the three angles of the frame add up to 180°. Today you learn why.

Three angles make 180°

In every triangle, the three interior angles add up to 180°: long or short, thin or fat.So if you know two angles, the third is 180° minus their sum.This is a theorem: we can prove it using parallel lines, as the next card shows.

Why it is 180°

Draw PQ through A parallel to BC. ∠PAB = x and ∠QAC = y (alternate angles). The angles on the straight line PQ give x + z + y = 180°, and these are exactly the triangle's three angles.

✏️ Finding the third angle

Two angles of a triangle are 48° and 75°. Find the third angle.

1Sum of the known angles: 48° + 75° = 123°.
2Third angle = 180° − 123° = 57°.
3Check: 48 + 75 + 57 = 180 ✓

✅ 57°

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 two angles inside the triangle that are not next to it are its interior opposite angles.Theorem: the exterior angle equals the sum of the two interior opposite angles.

Exterior angle = a + b

BC is produced to D. Proof: a + b + c = 180° (triangle) and ∠ACD + c = 180° (straight line). So ∠ACD = a + b.

🧮 Exterior angle theorem

Use it to jump straight to an angle without first finding the third interior angle. The adjacent interior angle and the exterior angle add up to 180°.

∠ACD = ∠BAC + ∠ABC

✏️ Using the exterior angle

In △ABC, BC is produced to D. ∠ACD = 125° and ∠BAC = 50°. Find (i) ∠ABC (ii) ∠ACB.

1Exterior angle = sum of interior opposite angles: 125° = 50° + ∠ABC.
2(i) එබැවින් ∠ABC = 125° − 50° = 75°.
3(ii) Angles on a straight line: ∠ACB = 180° − 125° = 55°.

✅ (i) 75° (ii) 55°

⚠️ Opposite, not adjacent

A common slip is adding the exterior angle to the wrong angles, or setting it equal to the interior angle next to it. The neighbour and the exterior angle make a straight line (sum 180°). The exterior angle equals the sum of the other two, the ones far from it.

Isosceles triangles

A triangle with two equal sides is isosceles. Equal sides are marked with the same tick.Theorem: if two sides of a triangle are equal, the angles opposite those sides are equal.These equal angles sit at the two ends of the third side (the base angles).

Proving it with congruence

AB = AC. Draw AD bisecting ∠BAC. In △ABD and △ACD: AB = AC, ∠BAD = ∠CAD, AD is common. So △ABD ≡ △ACD (SAS), giving ∠ABC = ∠ACB.

⚠️ Which angles are equal?

Examiners report students who know the theorem but use the wrong pair. The equal angles are opposite the equal sides, not between them. Mark the equal sides first, then look across the triangle.Also spot hidden isosceles triangles: two radii of a circle are equal, so a triangle with two radii is isosceles.

The converse

It works backwards too: if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.Use it to prove two lengths are equal, or to find a missing length.If all three angles are 60°, all three sides are equal: an equilateral triangle.

📌 Remember

• The three angles of a triangle add up to 180°.• Exterior angle = sum of the two interior opposite angles.• Two equal sides → the angles opposite them are equal.• Two equal angles → the sides opposite them are equal.• Look for hidden isosceles triangles, such as two radii.

🧾 Quick revision sheet

Sum of the interior angles of a triangle	180°
Exterior angle theorem	The exterior angle equals the sum of the two interior opposite angles.
Exterior angle + adjacent interior angle	180° (angles on a straight line)
Isosceles triangle theorem	If two sides are equal, the angles opposite them are equal.
Converse of the isosceles theorem	If two angles are equal, the sides opposite them are equal.
Apex angle 40° in an isosceles triangle: base angles?	70° each: (180 − 40) ÷ 2
Each angle of an equilateral triangle	60°