057. Angle Relationships and Angle Sums

Learning Intentions

• To know the meaning of the terms adjacent, complementary, supplementary, vertically opposite and perpendicular
• work with vertically opposite angles and perpendicular lines
• find angles at a point using angle sums of ${90}^{\circ },{180}^{\circ }$ and ${360}^{\circ }$

Pre-requisite Summary

• Understand that an angle is formed by two rays meeting at a vertex
• Be able to measure and classify angles such as acute, right, obtuse and straight
• Know that a right angle is ${90}^{\circ }$ and a straight angle is ${180}^{\circ }$
• Understand that angles can be named from diagrams with labelled points
• Be able to identify when lines intersect
• Know that angles around a point make a full turn of ${360}^{\circ }$

Worked Examples

Worked Example 1

a) Define the terms adjacent, complementary, supplementary, vertically opposite and perpendicular.
b) State whether two angles of ${35}^{\circ }$ and ${55}^{\circ }$ are complementary.
c) State whether two angles of ${110}^{\circ }$ and ${70}^{\circ }$ are supplementary.

Worked Example 2

Two lines intersect and one angle is ${48}^{\circ }$.
a) State the size of the vertically opposite angle.
b) State the size of each adjacent angle.
c) Explain your reasoning.

Worked Example 3

A diagram shows two perpendicular lines. One angle is labelled $x$ and another adjacent angle is labelled ${37}^{\circ }$.
a) Write an equation using the angle sum of ${90}^{\circ }$.
b) Find $x$.

Worked Example 4

A straight line is split into two adjacent angles, one of which is ${126}^{\circ }$.
a) Write an equation using the angle sum of ${180}^{\circ }$.
b) Find the other angle.

Worked Example 5

Angles around a point are ${95}^{\circ }$, ${140}^{\circ }$, and $x$.
a) Write an equation using the angle sum of ${360}^{\circ }$.
b) Find $x$.

Worked Example 6

Two intersecting lines create angles labelled $x^{\circ }$, ${72}^{\circ }$, $y^{\circ }$, and ${72}^{\circ }$.
a) State the value of $x$.
b) Find $y$.
c) State which angles are vertically opposite and which are supplementary.

Problems

Problem 1

a) Define the terms adjacent, complementary, supplementary, vertically opposite and perpendicular.
b) State whether two angles of ${41}^{\circ }$ and ${49}^{\circ }$ are complementary.
c) State whether two angles of ${125}^{\circ }$ and ${55}^{\circ }$ are supplementary.

Problem 2

Two lines intersect and one angle is ${63}^{\circ }$.
a) State the size of the vertically opposite angle.
b) State the size of each adjacent angle.
c) Explain your reasoning.

Problem 3

A diagram shows two perpendicular lines. One angle is labelled $x$ and another adjacent angle is labelled ${58}^{\circ }$.
a) Write an equation using the angle sum of ${90}^{\circ }$.
b) Find $x$.

Problem 4

A straight line is split into two adjacent angles, one of which is ${134}^{\circ }$.
a) Write an equation using the angle sum of ${180}^{\circ }$.
b) Find the other angle.

Problem 5

Angles around a point are ${85}^{\circ }$, ${115}^{\circ }$, and $x$.
a) Write an equation using the angle sum of ${360}^{\circ }$.
b) Find $x$.

Problem 6

Two intersecting lines create angles labelled $x^{\circ }$, ${68}^{\circ }$, $y^{\circ }$, and ${68}^{\circ }$.
a) State the value of $x$.
b) Find $y$.
c) State which angles are vertically opposite and which are supplementary.

Exercises

Understanding and Fluency

1. State the meaning of each term:
   a) adjacent
   b) complementary
   c) supplementary

2. State the meaning of each term:
   a) vertically opposite
   b) perpendicular
   c) adjacent angles

3. Decide whether each pair of angles is complementary, supplementary, or neither:
   a) ${25}^{\circ }$ and ${65}^{\circ }$
   b) ${102}^{\circ }$ and ${78}^{\circ }$
   c) ${40}^{\circ }$ and ${30}^{\circ }$

4. Decide whether each pair of angles is complementary, supplementary, or neither:
   a) ${15}^{\circ }$ and ${75}^{\circ }$
   b) ${91}^{\circ }$ and ${89}^{\circ }$
   c) ${120}^{\circ }$ and ${60}^{\circ }$

5. Two lines intersect. One angle is ${52}^{\circ }$. Find:
   a) the vertically opposite angle
   b) one adjacent angle
   c) the other adjacent angle

6. Two lines intersect. One angle is ${109}^{\circ }$. Find:
   a) the vertically opposite angle
   b) one adjacent angle
   c) the other adjacent angle

7. Find the missing angle on a right angle:
   a) $x+{27}^{\circ }={90}^{\circ }$
   b) $x+{64}^{\circ }={90}^{\circ }$
   c) $x+{13}^{\circ }={90}^{\circ }$

8. Find the missing angle on a straight line:
   a) $x+{45}^{\circ }={180}^{\circ }$
   b) $x+{118}^{\circ }={180}^{\circ }$
   c) $x+{97}^{\circ }={180}^{\circ }$

9. Find the missing angle around a point:
   a) ${120}^{\circ }+{85}^{\circ }+x={360}^{\circ }$
   b) ${140}^{\circ }+{90}^{\circ }+x={360}^{\circ }$
   c) ${75}^{\circ }+{145}^{\circ }+x={360}^{\circ }$

10. Mixed practice:
    a) If two lines are perpendicular, what is the size of each angle formed?
    b) If one of two vertically opposite angles is ${44}^{\circ }$, what is the other?
    c) If two angles are supplementary and one is ${73}^{\circ }$, what is the other?

Reasoning

11. Explain why vertically opposite angles are equal.

12. A student says that two adjacent angles must always be supplementary. Explain the mistake.

13. Explain why perpendicular lines create four right angles.

14. A student says that angles of ${35}^{\circ }$ and ${55}^{\circ }$ are supplementary. Explain why this is incorrect.

15. Explain why angles around a point add to ${360}^{\circ }$.

16. A student finds the angle next to ${128}^{\circ }$ on a straight line by adding $128+180$. Explain the error.

Problem-solving

17. At an intersection, one angle is ${74}^{\circ }$. Find all four angles formed by the intersecting lines.

18. A carpenter checks two pieces of timber that meet at a right angle. One marked angle is ${32}^{\circ }$ inside the corner. Find the other angle needed to complete the right angle.

19. A road junction forms four angles. One angle is ${116}^{\circ }$. Find the other three angles.

20. Three angles around a point are ${122}^{\circ }$, ${88}^{\circ }$, and $x$. Find $x$.

21. A straight line is split into three adjacent angles of ${35}^{\circ }$, $x$, and ${68}^{\circ }$. Find $x$.

22. Two perpendicular lines are crossed by another line. One acute angle formed is ${24}^{\circ }$. Find the angle needed to complete the right angle.

Potential Misunderstandings

• Students may confuse adjacent angles with vertically opposite angles
• Students may think complementary angles must be next to each other, when they only need to add to ${90}^{\circ }$
• Students may think supplementary angles must be on a straight line, rather than understanding that they are any two angles summing to ${180}^{\circ }$
• Students may forget that vertically opposite angles are equal
• Students may think perpendicular lines mean parallel lines
• Students may use ${180}^{\circ }$ instead of ${90}^{\circ }$ when working with perpendicular lines
• Students may use the wrong angle sum, such as using ${360}^{\circ }$ for angles on a straight line
• Students may add angles when they should subtract from ${90}^{\circ }$, ${180}^{\circ }$, or ${360}^{\circ }$ to find a missing angle

Next: 058.