Geometry of 2D Shapes – Part 2

About this video

Part 2 of a Grade 9 worksheet on 2D shapes: the properties of triangles and quadrilaterals — an isosceles triangle, a rectangle, a square with a kite, and a trapezium — used to find unknown angles and sides, with a reason for every step.

What it covers

Angles in a triangle add up to 180°, with angles such as 5x + 20°
Isosceles triangle: equal angles lie opposite equal sides
Rectangle, square and kite properties as reasons
Angles in a quadrilateral add up to 360°
Trapezium: one pair of parallel sides, so alternate angles

Extra worked examples

✍️ Extra example: in △ABC, Â = 4x + 10°, B̂ = x + 25° and Ĉ = 3x − 15°. Find x and each angle.

1
(4x + 10°) + (x + 25°) + (3x − 15°) = 180°
∠s of a △ add up to 180°.
2
8x + 20° = 180° → 8x = 160° → x = 20°
Group the like terms, then use the inverses.
3
Â = 4(20°) + 10° = 90°, B̂ = 20° + 25° = 45°, Ĉ = 3(20°) − 15° = 45°
Substitute in brackets. Check: 90° + 45° + 45° = 180° ✓
4
B̂ = Ĉ, so AB = AC.
Equal angles lie opposite equal sides (isos △) — a right-angled isosceles triangle.

✍️ Extra example: ABCD is a kite with AB = AD and CB = CD. Â = 70° and B̂ = 115°. Find D̂ and Ĉ.

1
D̂ = B̂ = 115°
Kite: the angles between the unequal sides are equal.
2
Ĉ = 360° − 70° − 115° − 115° = 60°
∠s of a quad add up to 360°.

Exercises to try

Work each one out, then click to reveal the answer.

1
In △PQR, PQ = PR and Q̂ = 52°. Find R̂ and P̂.
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R̂ = 52° (isos △); P̂ = 180° − 52° − 52° = 76° (∠s of a △)
2
PQRS is a rectangle. The diagonal PR splits the corner at P into QP̂R = 2x + 10° and RP̂S = x + 20°. Find x.
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(2x + 10°) + (x + 20°) = 90° (corner of a rectangle) → 3x = 60° → x = 20°, so QP̂R = 50° and RP̂S = 40°
3
ABCD is a trapezium with AD ∥ BC. The diagonal BD makes AD̂B = 38°. Find DB̂C, with a reason.
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DB̂C = 38° (alt ∠s; AD ∥ BC)
4
△XYZ has X̂ = 2x − 10°, Ŷ = x + 20° and Ẑ = x + 30°. Find x, then classify the triangle.
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4x + 40° = 180° (∠s of a △) → x = 35° → 60°, 55°, 65°: no equal angles, so scalene
5
A quadrilateral has angles 3x, 2x + 10°, x + 50° and 2x − 20°. Find x and the largest angle.
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8x + 40° = 360° (∠s of a quad) → x = 40° → 120°, 90°, 90° and 60°; the largest is 120°

Read the full lesson

🔺Grade 8–9 · Geometry of 2D ShapesAngles in Triangles and Quadrilaterals: Full Guide

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