102r. Classifying and Relating Angles

Learning Intentions

• To know the meaning of the terms complementary, supplementary, vertically opposite and perpendicular
• classify angles as acute, right, obtuse, straight, reflex or a revolution
• relate compass bearings to angles
• determine the angles at a point using angle properties

Pre-requisite Summary

• Know that an angle measures the amount of turn between two lines or rays
• Understand that angles are measured in degrees, written as ${}^{\circ }$
• Recall benchmark angles such as ${90}^{\circ }$, ${180}^{\circ }$ and ${360}^{\circ }$
• Be able to compare the size of two angles
• Know the four main compass directions: north, east, south and west
• Understand that a full turn is ${360}^{\circ }$ and a half-turn is ${180}^{\circ }$
• Be able to identify intersecting lines and points where lines meet
• Know that some angle relationships can be found without measuring

Worked Examples

Worked Example 1

State the meaning of each term:
a) complementary angles
b) supplementary angles
c) vertically opposite angles
d) perpendicular lines

Worked Example 2

Classify each angle:
a) ${35}^{\circ }$
b) ${90}^{\circ }$
c) ${128}^{\circ }$

Worked Example 3

Classify each angle:
a) ${180}^{\circ }$
b) ${250}^{\circ }$
c) ${360}^{\circ }$

Worked Example 4

Relate each compass turn to an angle:
a) a quarter-turn clockwise from north
b) a half-turn from east
c) a three-quarter turn clockwise from north

Worked Example 5

Find the unknown angle at a point:
a) Angles around a point are ${110}^{\circ }$, ${95}^{\circ }$ and $x$
b) Angles around a point are ${75}^{\circ }$, ${140}^{\circ }$, ${65}^{\circ }$ and $y$

Worked Example 6

Use angle properties to find unknown angles:
a) Two complementary angles are $x$ and ${28}^{\circ }$
b) Two supplementary angles are $m$ and ${133}^{\circ }$
c) One of a pair of vertically opposite angles is ${47}^{\circ }$. Find the other angle.

Problems

Problem 1

State the meaning of each term:
a) complementary angles
b) supplementary angles
c) vertically opposite angles
d) perpendicular lines

Problem 2

Classify each angle:
a) ${42}^{\circ }$
b) ${90}^{\circ }$
c) ${154}^{\circ }$

Problem 3

Classify each angle:
a) ${180}^{\circ }$
b) ${300}^{\circ }$
c) ${360}^{\circ }$

Problem 4

Relate each compass turn to an angle:
a) a quarter-turn anticlockwise from east
b) a half-turn from north
c) a three-quarter turn clockwise from west

Problem 5

Find the unknown angle at a point:
a) Angles around a point are ${125}^{\circ }$, ${85}^{\circ }$ and $x$
b) Angles around a point are ${90}^{\circ }$, ${110}^{\circ }$, ${70}^{\circ }$ and $y$

Problem 6

Use angle properties to find unknown angles:
a) Two complementary angles are $x$ and ${36}^{\circ }$
b) Two supplementary angles are $m$ and ${121}^{\circ }$
c) One of a pair of vertically opposite angles is ${62}^{\circ }$. Find the other angle.

Exercises

Understanding and Fluency

1. Match each term to its meaning:
   a) complementary
   b) supplementary
   c) vertically opposite
   d) perpendicular

2. Classify each angle:
   a) ${18}^{\circ }$
   b) ${90}^{\circ }$
   c) ${146}^{\circ }$
   d) ${180}^{\circ }$

3. Classify each angle:
   a) ${275}^{\circ }$
   b) ${360}^{\circ }$
   c) ${89}^{\circ }$
   d) ${91}^{\circ }$

4. Find the missing angle:
   a) An angle complementary to ${24}^{\circ }$
   b) An angle supplementary to ${58}^{\circ }$
   c) An angle vertically opposite to ${73}^{\circ }$

5. Determine the angle of each turn:
   a) a quarter-turn
   b) a half-turn
   c) a full turn
   d) a three-quarter turn

6. Relate compass directions to angles:
   a) clockwise from north to east
   b) clockwise from north to south
   c) clockwise from north to west
   d) clockwise from east to north

7. Find the unknown angle at a point:
   a) ${140}^{\circ }$, ${120}^{\circ }$ and $x$
   b) ${85}^{\circ }$, ${135}^{\circ }$, ${40}^{\circ }$ and $y$
   c) ${90}^{\circ }$, ${90}^{\circ }$, ${110}^{\circ }$ and $z$

8. Use angle properties to find each unknown:
   a) $a+{41}^{\circ }={90}^{\circ }$
   b) $b+{107}^{\circ }={180}^{\circ }$
   c) Vertically opposite to ${129}^{\circ }$

Reasoning

9. Explain why two complementary angles cannot include an obtuse angle.

10. A student says that supplementary angles add to ${360}^{\circ }$. Explain the mistake.

11. Noah says that all angles bigger than ${180}^{\circ }$ are revolutions. Is he correct? Explain.

12. Explain why perpendicular lines always form right angles.

13. A student says that if one angle at a point is ${200}^{\circ }$, the remaining angles at that point must add to ${200}^{\circ }$. Describe the error.

Problem-solving

14. A compass needle turns clockwise from north to east, then from east to south. What total angle has it turned?

15. Two roads meet at right angles. Explain what this tells you about the roads.

16. At a point, three angles are ${150}^{\circ }$, ${95}^{\circ }$ and $x$. Find $x$.

17. A ship changes direction by making a three-quarter clockwise turn from north. What direction is it now facing, and what angle has it turned through?

18. Two intersecting lines form one angle of ${68}^{\circ }$. Find the vertically opposite angle.

19. A mechanic turns a wheel through one full revolution and then a quarter-turn more. What total angle has the wheel turned through?

Potential Misunderstandings

• Students may think complementary angles add to ${180}^{\circ }$ instead of ${90}^{\circ }$
• Students may think supplementary angles add to ${360}^{\circ }$ instead of ${180}^{\circ }$
• Students may confuse vertically opposite angles with adjacent angles
• Students may think perpendicular lines can meet at any angle
• Students may confuse obtuse angles with reflex angles
• Students may think a straight angle is the same as a right angle
• Students may classify any large angle as a revolution, instead of recognising that a revolution is exactly ${360}^{\circ }$
• Students may confuse clockwise and anticlockwise turns on a compass
• Students may forget that bearings and compass turns are measured from a starting direction
• Students may think angles at a point add to ${180}^{\circ }$ instead of ${360}^{\circ }$
• Students may use the wrong angle property when finding unknown angles
• Students may assume vertically opposite angles are supplementary, rather than equal