Geometry of 2D Shapes – Part 1

About this video

Part 1 of a Grade 9 worksheet on 2D shapes: six questions on parallelograms, a rhombus and triangles — alternate and co-interior angles, the exterior angle of a triangle, vertically opposite angles and isosceles triangles — each step with its reason.

What it covers

Parallelogram: opposite sides parallel, so alternate angles; opposite angles equal
Exterior angle of a triangle = the sum of the two interior opposite angles
Rhombus: the diagonals bisect the angles
Co-interior angles add up to 180°
Isosceles triangles and angles on a straight line

Extra worked examples

✍️ Extra example: ABCD is a parallelogram with diagonal BD. AD̂B = 35° and Â = 115°. Find DB̂C, AB̂D and Ĉ.

1
DB̂C = AD̂B = 35°
Alt ∠s; AD ∥ BC (opposite sides of a parallelogram are parallel).
2
AB̂D = 180° − 115° − 35° = 30°
∠s of △ABD add up to 180°.
3
Ĉ = Â = 115°
Opp ∠s of a parallelogram are equal.
4
Check: Â + AB̂C = 115° + 65° = 180° ✓
Co-interior angles between AD ∥ BC add up to 180°.

✍️ Extra example: PQRS is a rhombus with diagonal PR. QP̂R = x and Q̂ = 70°. Find x.

1
RP̂S = QP̂R = x
Rhombus: the diagonals bisect the angles.
2
x + x + 70° = 180°
Co-int ∠s; PS ∥ QR.
3
2x = 110° → x = 55°
So the angle at P is 110°, and the opposite angles are 110° and 70° — they add up to 360° ✓

Exercises to try

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

1
ABCD is a parallelogram with Â = 3x + 10° and Ĉ = 100°. Find x.
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3x + 10° = 100° (opp ∠s of a parallelogram) → x = 30°
2
ABCD is a parallelogram with Â = 2y + 20° and B̂ = y + 40°. Find y.
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(2y + 20°) + (y + 40°) = 180° (co-int ∠s; AD ∥ BC) → 3y = 120° → y = 40°
3
Two lines cross. One angle is 5x − 20° and the vertically opposite angle is 3x + 30°. Find x and the angle.
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5x − 20° = 3x + 30° (vert opp ∠s) → x = 25°; each angle is 105°
4
In △ABC, BC is extended to D. AĈD = 5x + 10°, Â = 2x + 15° and B̂ = 2x + 5°. Find x.
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5x + 10° = 4x + 20° (ext ∠ of △) → x = 10°: 60° = 35° + 25° ✓
5
In △PQR, PQ = PR, P̂ = 4x and Q̂ = 3x. Find x and every angle.
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R̂ = 3x (isos △); 10x = 180° → x = 18° → P̂ = 72°, Q̂ = R̂ = 54°

Read the full lesson

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

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