Geometry Angles and Shapes

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Formula and Equations Reference

Use these rules and formulas to help solve the questions.

Straight-line angles: a + b = 180°

Angles around a point: a + b + c + d = 360°

Triangle angle sum: a + b + c = 180°

Quadrilateral angle sum: the interior angles add to 360°

Polygon interior-angle sum: (n − 2) × 180°, where n is the number of sides

Rectangle area: A = l × w

Rectangle perimeter: P = 2(l + w)

Triangle area: A = ½ × b × h

Similarity scale factor: k = corresponding length ÷ original length

Part 1: Core Geometry Skills

Curriculum links: AC9M7M04, AC9M7M05, AC9M8M01 and AC9M8SP01.

Worked example: When two parallel lines are crossed by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°. In a triangle, the interior angles add to 180°. In a quadrilateral, they add to 360°.

1. Parallel lines: Diagram description: lines AB and CD are horizontal and parallel. A sloping transversal crosses AB at P and CD at Q. The angle above AB and to the right of the transversal at P is 68°. Find the corresponding angle above CD and to the right of the transversal at Q. Remember that corresponding angles are equal.

Answer: __________°

2. In the same diagram, find the co-interior angle below AB and to the right of the transversal at P if the angle above CD and to the right of the transversal at Q is 68°. Use the straight-line equation a + b = 180°.

Answer: __________°

3. A triangle has angles of 47° and 86°. Calculate the third angle. Use the triangle angle-sum equation a + b + c = 180°.

Answer: __________°

4. A regular pentagon has five equal interior angles. Calculate the size of each interior angle. Use the polygon sum rule: (n − 2) × 180°, then divide by the number of equal angles.

Answer: __________°

5. Select all statements that are always true.

Vertically opposite angles are equal.

Angles on a straight line add to 180°.

All quadrilaterals have four equal sides.

The interior angles of a triangle add to 180°.

Part 2: Shapes, Measurement and Challenge

Method reminder: For a composite shape, split it into familiar rectangles or triangles. Add the areas. For perimeter, add the lengths of the outside edges only.

6. A composite garden is made from a 10 m by 6 m rectangle with a 4 m by 2 m rectangle removed from one corner. Calculate the area of the remaining garden. Use the rectangle area formula A = l × w for each rectangle.

Answer: __________ m²

7. The same garden has outside edge lengths of 10 m, 4 m, 2 m, 6 m, 4 m and 8 m. Calculate its perimeter by adding the outside edges. For a complete rectangle, the perimeter formula is P = 2(l + w).

Answer: __________ m

8. A trapezium has parallel sides of 8 cm and 14 cm and a perpendicular height of 5 cm. Use area = ½ × (sum of parallel sides) × height. If a shape is split into a triangle, remember that triangle area is A = ½ × b × h. Find the area of the trapezium.

Answer: __________ cm²

9. Challenge — congruence: Two triangles each have side lengths 5 cm, 7 cm and 9 cm. Are they congruent? Explain which congruence rule supports your answer.

10. Challenge — similarity: Triangle A has side lengths 3 cm, 4 cm and 5 cm. Triangle B has side lengths 6 cm, 8 cm and 10 cm. Are the triangles similar? Use the scale-factor equation k = corresponding length ÷ original length. State the scale factor from Triangle A to Triangle B and explain.

11. Draw two similar triangles with a scale factor of 2. Label one pair of corresponding sides and one pair of equal corresponding angles. You may use k = corresponding length ÷ original length to check the scale factor.

Teacher Answer Key

1. 68°. Corresponding angles in parallel lines are equal.

2. 112°. Co-interior angles add to 180°, so 180° − 68° = 112°.

3. Using a + b + c = 180°: 47° + 86° = 133°; 180° − 133° = 47°.

4. Polygon sum = (5 − 2) × 180° = 540°. Each angle = 540° ÷ 5 = 108°.

5. Vertically opposite angles are equal; angles on a straight line add to 180°; triangle angles add to 180°.

6. Whole rectangle area = l × w = 10 × 6 = 60 m². Removed area = 4 × 2 = 8 m². Remaining area = 52 m².

7. Add the outside edges: 10 + 4 + 2 + 6 + 4 + 8 = 34 m.

8. ½ × (8 + 14) × 5 = ½ × 22 × 5 = 55 cm².

9. Yes. The triangles are congruent by SSS because all three corresponding side lengths are equal.

10. Yes. Using k = corresponding length ÷ original length, 6 ÷ 3 = 2. Each side in Triangle B is twice the corresponding side in Triangle A, so the scale factor is 2. The corresponding angles are equal.

11. Responses should show the same angle sizes, with every corresponding side in the larger triangle twice the length of the smaller triangle.