Unit 12 Section 1 : Recap: Angles and Scale Drawing

The concepts in this unit rely heavily on knowledge acquired previously, particularly for angles and scale drawings, so in this first section we revise these two topics.

Example 1

In the diagram opposite, determine the size of each of the unknown angles.

Since
c + 180° = 180° (BCD is a straight line)
c = 180° – 100°
c = 80°
Also b = c, since the triangle is isosceles, so b = 80°. Finally, since
a + b + c = 180° (angles in a triangle add up to 180°)
then
a = 180° – (80° + 80°)
so a = 20°

Example 2

In the diagram opposite, given that a = 65°, determine the size of each of the unknown angles.

b = 180° – a (angles on a straight line are
supplementary, i.e. they add up to 180°)
b = 180° – 65°
b = 115°
c = a = 65° (vertically opposite angles)
d = b = 115° (corresponding angles, as the lines are parallel)
e = a = 65° (corresponding angles)
f = a = 65° (alternate angles)

Example 3

Draw an accurate plan of the car park which is sketched here. Use the scale

1 cm ≡ 10 m.

Estimate the distance AB.

The equivalent lengths are:

100 m ≡ 10 cm, 80 m ≡ 8 cm, 60 m ≡ 6 cm,

giving the following scale drawing:

In the scale drawing, AB = 11.7 cm, which gives an actual distance AB = 117 m in the car park.

Exercises

Question 1

Determine the size of each of the angles marked with a letter in the following diagrams.

(a)
a = °
b = °
c = °
c = 180° – 63°   (supplementary angles)
= 117°
a = 63°   (opposite, then corresponding angles)
b = 180° – a   (supplementary angles)
= 180° – 63°
= 117°
(b)
a = °
b = °
c = °
a = 48°   (opposite, then corresponding angles)
b = a   (alternate angles)
= 48°
c = 180° – 48°   (angle corresponding to c
is supplementary to 48°)
= 132°
Question 2

Determine the size of each of the angles marked with a letter in the following diagram:

a = °
b = °
c = °
d = °
e = °
Question 3

BCDE is a trapezium. Determine the size of each of the angles marked with a letter in the diagram.

p = °
q = °
r = °
s = °
x = °
y = °
z = °
Question 4

(a)

The time on this clock is 3 o'clock.

What is the size of the angle between the hands?

°
(b)

Write down the whole number missing from this sentence:

At o'clock the size of the angle between the hands is 180°.

(c)

What is the size of the angle between the hands at 1 o'clock?

°
(d)

What is the size of the angle between the hands at 5 o'clock?

°
(e)

How long does it take for the minute hand to move 360° ?

minutes
Question 5

(a)

Which two of these angles are the same size?

and
(b)

Kelly is facing North. She turns clockwise through 2 right angles. Which direction is she facing now?

(c)

Aled is facing West. He turns clockwise through 3 right angles. Which direction is he facing now?

Question 6

The shape below has 3 identical white tiles and 3 identical grey tiles. The sides of each tile are all the same length. Opposite sides of each tile are parallel. One of the angles is 70° .

(a)

Calculate the size of angle k.

k = °
(b)

Calculate the size of angle m.

m = °
3m + 3 × 70° = 360°   ⇒   m = 50°
Question 7

Kay is drawing shapes on her computer.

(a)

She wants to draw this triangle. She needs to know angles a, b and c. Calculate angles a, b and c.

a = °
b = °
c = °
(b)

Kay draws a rhombus: Calculate angles d and e.

d = °
e = °
(c)

Kay types the instructions to draw a regular pentagon:

repeat 5 [forward 10, left turn 72°]

Complete the following instructions to draw a regular hexagon:

repeat 6 [forward 10, left turn °]

Question 8

Look at the diagram:

Side AB is the same length as side AC.
Side BD is the same length as side BC.
Calculate the value of x.

x = °
ABC = 3x°   (AB = AC)
⇒ DBC = 2x°
⇒ BDC = BCD = 3x°   (BD = BC)
2x + 3x + 3x = 180°   (angles of triangle)
⇒ x = 22.5°