Review: angles, triangles, polygons and the coordinate plane
Mathematics Prep 1 Term 1, Units 1–4, Review
GoalRevise Unit 4 with new mixed tasks.
New words
exterior angle · an angle formed by one side and the extension of the next side; it equals the sum of the two remote interior anglesparallelogram · a quadrilateral with both pairs of opposite sides parallelregular polygon · all sides equal and all angles equalmidpoint · the average of the coordinates of the two end points
Explanation
Angles: complementary add up to 90°, supplementary to 180°, round a point to 360°; vertically opposite angles are equal. With parallel lines, corresponding and alternate angles are equal and same-side interior angles add up to 180°; equal alternate angles also prove lines are parallel. A triangle has an angle sum of 180°, an exterior angle equals the sum of the two remote interior angles, and two sides together are longer than the third. A quadrilateral has 360°; a trapezium has exactly one pair of parallel sides; a parallelogram has diagonals that bisect each other; a rectangle has equal diagonals and a rhombus perpendicular ones. A polygon has (n − 2) × 180°, and each angle of a regular one is that sum ÷ n. In the plane, quadrants are decided by the signs, projections keep one coordinate, and the midpoint is the average.
Parallel lines. The lines are parallel and the alternate angles are (4x − 15)° and (2x + 25)°. They are equal: 4x − 15 = 2x + 25, so 2x = 40 and x = 20. Each angle is 4 × 20 − 15 = 65°.
A triangle. In triangle ABC, AB = AC and angle A = 36°. The base angles are equal: y = (180° − 36°) ÷ 2 = 72°. The exterior angle z at C is 36° + 72° = 108°. Check: 180° − 72° = 108°.
Polygons. A hexagon has an angle sum of (6 − 2) × 180° = 720°. Five angles are 100°, 120°, 130°, 115° and 125°. They add up to 590°, so the sixth angle is 720° − 590° = 130°. For a regular hexagon each angle is 720° ÷ 6 = 120°.
The triangle inequality. Two sides are 7 cm and 12 cm. The third side lies between 12 − 7 = 5 and 12 + 7 = 19. Can 5, 7 and 12 be sides? 5 + 7 = 12, which is not more than 12, so no.
The plane. A(−2, 3) is in quadrant II and B(6, −1) is in quadrant IV. The midpoint is M = ((−2 + 6) ÷ 2, (3 + (−1)) ÷ 2) = (2, 1). The projection of AB on the x-axis has length |6 − (−2)| = 8 and on the y-axis |−1 − 3| = 4.
Class activity
Geometry walk. With a grown-up, look at a floor tiled with squares or a honeycomb pattern. Name the polygons you see, say which are regular, and find one parallelogram, one trapezium and one pair of parallel lines.
Practice
1
Two angles on a straight line are 3x and (2x + 20)°. Find x and the angles.
5x + 20 = 180, so x = 32. The angles are 96° and 84°.
2
In rhombus ABCD, angle A = 70°. Find angle B and the angle between diagonal AC and side AB.
B = 180° − 70° = 110°. The diagonal bisects angle A, so it makes 35° with AB.
3
Each angle of a regular polygon is 144°. How many sides has it?
(n − 2) × 180 = 144n, so 36n = 360 and n = 10.
4
PQRS is a parallelogram with P(−3, 1), Q(2, −1) and R(5, 3). Find S.
The midpoint of PR is (1, 2). S = (2 × 1 − 2, 2 × 2 − (−1)) = (0, 5). Check: the midpoint of QS is ((2 + 0) ÷ 2, (−1 + 5) ÷ 2) = (1, 2).
5
Lina says 4, 5 and 10 make a triangle because each is shorter than 4 + 5 + 10. What is wrong?
We must compare the two shorter sides with the longest: 4 + 5 = 9, which is less than 10. No triangle exists.
