Angles with Parallel Lines and a Transversal

Learning Intentions

• To know the meaning of the terms transversal, corresponding, alternate, co-interior and parallel
• identify angles that are in a given relation to another angle (for example, identifying an angle co-interior to a given angle)
• find the size of angles when a transversal crosses parallel lines
• determine whether two lines are parallel using angles involving a transversal

Pre-requisite Summary

• Understand that lines can intersect and that some pairs of lines are parallel
• Know that parallel lines stay the same distance apart and never meet
• Be able to identify adjacent, supplementary and vertically opposite angles
• Know that angles on a straight line add to ${180}^{\circ }$
• Know that angles around a point add to ${360}^{\circ }$
• Be able to read labelled angle diagrams carefully
• Understand that a transversal is a line that crosses two other lines
• Be able to compare angle sizes and justify why two angles are equal or supplementary

Worked Examples

Worked Example 1

a) Define the terms transversal, corresponding, alternate, cointerior and parallel.
b) In a diagram of two parallel lines cut by a transversal, describe where corresponding angles are found.
c) Describe where alternate and cointerior angles are found.

Worked Example 2

In a labelled diagram of two parallel lines cut by a transversal, angle $a$ is given.
a) Identify one angle corresponding to angle $a$.
b) Identify one angle alternate to angle $a$.
c) Identify one angle cointerior to angle $a$.

Worked Example 3

Two parallel lines are crossed by a transversal. One angle is ${65}^{\circ }$.
a) Find an angle corresponding to it.
b) Find an angle alternate to it.
c) Find a cointerior angle related to it.

Worked Example 4

Two parallel lines are crossed by a transversal. One angle is ${118}^{\circ }$.
a) Find all angles equal to ${118}^{\circ }$.
b) Find all angles supplementary to ${118}^{\circ }$.
c) Explain the angle facts used.

Worked Example 5

A transversal crosses two lines. A pair of corresponding angles are both ${72}^{\circ }$.
a) State the relationship between the two lines.
b) Explain why this angle fact shows the lines are parallel.

Worked Example 6

A transversal crosses two lines. One pair of cointerior angles are ${110}^{\circ }$ and ${70}^{\circ }$.
a) Find their sum.
b) State the relationship between the two lines.
c) Explain why this angle fact shows the lines are parallel.

Problems

Problem 1

a) Define the terms transversal, corresponding, alternate, cointerior and parallel.
b) In a diagram of two parallel lines cut by a transversal, describe where corresponding angles are found.
c) Describe where alternate and cointerior angles are found.

Problem 2

In a labelled diagram of two parallel lines cut by a transversal, angle $x$ is given.
a) Identify one angle corresponding to angle $x$.
b) Identify one angle alternate to angle $x$.
c) Identify one angle cointerior to angle $x$.

Problem 3

Two parallel lines are crossed by a transversal. One angle is ${54}^{\circ }$.
a) Find an angle corresponding to it.
b) Find an angle alternate to it.
c) Find a cointerior angle related to it.

Problem 4

Two parallel lines are crossed by a transversal. One angle is ${127}^{\circ }$.
a) Find all angles equal to ${127}^{\circ }$.
b) Find all angles supplementary to ${127}^{\circ }$.
c) Explain the angle facts used.

Problem 5

A transversal crosses two lines. A pair of corresponding angles are both ${83}^{\circ }$.
a) State the relationship between the two lines.
b) Explain why this angle fact shows the lines are parallel.

Problem 6

A transversal crosses two lines. One pair of cointerior angles are ${104}^{\circ }$ and ${76}^{\circ }$.
a) Find their sum.
b) State the relationship between the two lines.
c) Explain why this angle fact shows the lines are parallel.

Exercises

Understanding and Fluency

1. State the meaning of each term:
   a) transversal
   b) parallel
   c) corresponding angles

2. State the meaning of each term:
   a) alternate angles
   b) cointerior angles
   c) parallel lines

3. In a diagram of two parallel lines cut by a transversal, state whether each pair is equal or supplementary:
   a) corresponding angles
   b) alternate angles
   c) cointerior angles

4. Identify the requested angle relationship in a labelled diagram:
   a) an angle corresponding to $\mathrm{\angle }a$
   b) an angle alternate to $\mathrm{\angle }a$
   c) an angle cointerior to $\mathrm{\angle }a$

5. Two parallel lines are cut by a transversal and one angle is ${47}^{\circ }$. Find:
   a) one corresponding angle
   b) one alternate angle
   c) one cointerior angle

6. Two parallel lines are cut by a transversal and one angle is ${132}^{\circ }$. Find:
   a) one corresponding angle
   b) one alternate angle
   c) one cointerior angle

7. Find the missing angle when a transversal crosses parallel lines:
   a) one angle is ${61}^{\circ }$, find its corresponding angle
   b) one angle is ${61}^{\circ }$, find its alternate angle
   c) one angle is ${61}^{\circ }$, find its cointerior angle

8. Find the missing angle when a transversal crosses parallel lines:
   a) one angle is ${109}^{\circ }$, find a corresponding angle
   b) one angle is ${109}^{\circ }$, find an alternate angle
   c) one angle is ${109}^{\circ }$, find a cointerior angle

9. Decide whether the lines are parallel:
   a) a pair of corresponding angles are equal
   b) a pair of alternate angles are equal
   c) a pair of cointerior angles add to ${180}^{\circ }$

10. Mixed practice:
    a) If alternate angles are ${73}^{\circ }$ and ${73}^{\circ }$, what can you conclude?
    b) If cointerior angles are ${95}^{\circ }$ and ${85}^{\circ }$, what can you conclude?
    c) If corresponding angles are ${64}^{\circ }$ and ${116}^{\circ }$, are the lines parallel?

Reasoning

11. Explain why corresponding angles are equal when a transversal crosses parallel lines.

12. A student says cointerior angles are always equal. Explain the mistake.

13. Explain why equal alternate angles can be used to prove that two lines are parallel.

14. A student says that if one angle is ${70}^{\circ }$, then every other angle formed by the transversal must also be ${70}^{\circ }$. Explain why this is incorrect.

15. Explain why cointerior angles on parallel lines add to ${180}^{\circ }$.

16. A student claims two lines are parallel because one corresponding angle is ${80}^{\circ }$ and another is ${100}^{\circ }$. Explain why this does not prove the lines are parallel.

Problem-solving

17. Two railway tracks are marked as parallel and a service road crosses them as a transversal. One interior angle is ${68}^{\circ }$. Find the matching alternate angle and the related cointerior angle.

18. In a street diagram, two roads are claimed to be parallel. A crossing road creates a pair of corresponding angles of ${75}^{\circ }$ and ${75}^{\circ }$. Explain whether the roads are parallel.

19. A carpenter draws two timber edges and a cross brace. The brace forms cointerior angles of ${112}^{\circ }$ and ${68}^{\circ }$. Determine whether the timber edges are parallel.

20. In a geometry diagram, one angle made by a transversal is ${121}^{\circ }$. Find all other possible angle sizes in the diagram.

21. An engineer checks two beams cut by a support bar. A pair of alternate angles are ${59}^{\circ }$ and ${59}^{\circ }$. What does this show about the beams?

22. Two lines are cut by a transversal. A pair of corresponding angles are ${84}^{\circ }$ and ${96}^{\circ }$. Determine whether the lines are parallel and justify your answer.

Potential Misunderstandings

• Students may confuse corresponding, alternate and cointerior angles
• Students may think cointerior angles are equal instead of supplementary
• Students may not recognise which line is the transversal in a diagram
• Students may assume any two equal angles imply lines are parallel, without checking the angle relationship
• Students may confuse vertically opposite angles with alternate angles
• Students may forget that corresponding and alternate angles are equal only when the lines are parallel
• Students may use the wrong angle sum when working with cointerior angles
• Students may identify the correct angle relationship but apply the wrong rule to find the angle size
• Students may think lines are parallel just because they look parallel in a sketch, rather than using angle facts

Next: 060. Finding Missing Angles in Diagrams