Geometry and Angles: practice solutions, Grade 4

Sommaire

1 Line, ray or segment? ★★★

(a) A segment: two endpoints means a fixed length.

(b) A ray: one endpoint and one direction without end.

(c) A line: no endpoints.

2 Segments on a line ★★★

Start with P: \(\overline{PQ}\), \(\overline{PR}\), \(\overline{PS}\) (3 segments). Then Q: \(\overline{QR}\), \(\overline{QS}\) (2 more). Then R: \(\overline{RS}\) (1 more).

\[ 3 + 2 + 1 = 6 \]

There are 6 segments.

3 Name the angle ★★★

35° is acute (less than 90°). 90° is right. 112° is obtuse (between 90° and 180°). 89° is acute, just below a right angle. 180° is straight. 151° is obtuse.

4 True or false in everyday life ★★★

(a) True. Rails stay the same distance apart and never meet.

(b) True. At 3:00 the hands point at 12 and 3, a quarter turn of 90°.

(c) False. At 6:00 the hands point in opposite directions and form a straight angle of 180°.

5 Reading the protractor ★★★

(a) The orange ray crosses the outer scale at 70, so the angle measures 70°.

(b) 70° is less than 90°, so the angle is acute.

6 Right angles and parallel sides ★★★

A square has 4 right angles and 2 pairs of parallel sides. A rectangle also has 4 right angles and 2 pairs. A right triangle has 1 right angle and 0 pairs of parallel sides, because the three sides meet each other.

7 Counting lines of symmetry ★★★

(a) 4 (two through the middles of opposite sides and two diagonals). (b) 2. (c) 1, the line through the apex and the middle of the base. (d) 3, one from each vertex. (e) 0, no fold matches.

8 Missing angle in a corner ★★★

(a) \(90 - 24 = 66\), so 66°.

(b) \(180 - 128 = 52\), so 52°.

(c) \(90 - 66 = 24\), so 24°.

9 Opening a door ★★★

(a) \(25 + 40 = 65\). The door is open 65°.

(b) \(90 - 65 = 25\). It needs 25° more.

10 Triangles by sides ★★★

(a) All sides equal: equilateral. (b) Two equal sides: isosceles. (c) No equal sides: scalene. (d) Two equal sides: isosceles.

11 Triangles by angles ★★★

(a) All angles are less than 90°: acute (sum 180). (b) One right angle: right (sum 180). (c) 100° is obtuse: obtuse (sum 180). (d) All less than 90°: acute (sum 180).

12 Quadrilateral true or false ★★★

(a) True: a square has four right angles and opposite sides parallel.

(b) False: a 6 cm by 3 cm rectangle has unequal sides.

(c) True: that is the definition of a rhombus.

(d) False: a trapezoid has at least one pair of parallel sides; two pairs make a parallelogram.

13 Clock angles ★★★

The angle is 30° times the number of hours between the hands.

(a) \(2 \times 30 = 60\): 60°, acute.

(b) \(4 \times 30 = 120\): 120°, obtuse.

(c) At 9:00 the hands are 3 hours apart: \(3 \times 30 = 90\): 90°, right.

(d) \(5 \times 30 = 150\): 150°, obtuse.

14 A wide angle ★★★

(a) The ray crosses the outer scale at 128, so the angle is 128°.

(b) It is between 90° and 180°: obtuse.

(c) \(180 - 128 = 52\), so 52° more.

15 Letters and symmetry ★★★

(a) A, H, M and T can be folded left onto right. F and R cannot, and B and E cannot either.

(b) B, E and H can be folded top onto bottom. A, M, T, F and R cannot.

So H has both kinds.

16 Parallel or perpendicular streets ★★★

(a) Main Street and Oak Street are parallel.

(b) Elm Street is perpendicular to Main Street and to Oak Street (the square corners show it).

(c) No. Pine Street slants and crosses Main and Oak without making square corners.

17 Adjacent angles ★★★

\[ 125 - 47 = 78 \]

The unknown angle is 78°. Check: \(47 + 78 = 125\).

47° is acute, 78° is acute and 125° is obtuse.

18 Laptop screen ★★★

(a) \(105 - 38 = 67\), then \(67 + 21 = 88\). The final angle is 88°.

(b) 88° is less than 90°: acute.

19 Pie chart angles ★★★

(a) \(90 + 120 + 85 = 295\), then \(360 - 295 = 65\). The fourth sector is 65°.

(b) The 120° sector is obtuse. The 90° sector is right and the 85° and 65° sectors are acute.

20 Who am I? ★★★

(a) A square. (b) A rhombus. (c) A trapezoid. (d) A rectangle.

21 Robot turns ★★★

(a) \(40 + 75 + 65 = 180\). The total turn is 180°.

(b) \(\dfrac{180}{360} = \dfrac{1}{2}\): a half turn, which makes a straight angle.

22 Find the mistake ★★★

The angle is clearly smaller than a square corner, so it must be acute. Maya read the inner scale (which starts at 0 on the left), but the base ray points to 0 on the outer scale.

The correct measure is 50°. Indeed \(180 - 130 = 50\).

23 Folding a banner ★★★

(a) Folding along the line that cuts the 12 in sides in half gives 6 in by 8 in. Folding along the line that cuts the 8 in sides in half gives 12 in by 4 in.

(b) \(6 \times 8 = 48\) and \(12 \times 4 = 48\). Yes, the area is 48 square inches in both cases, half of \(12 \times 8 = 96\).