103r. Review Angles in Parallel Lines

Learning Intentions

• To understand that parallel lines do not cross and that arrows are used to indicate this on a diagram
• To know the meaning of the terms transversal, corresponding, alternate and co-interior
• use properties of parallel lines to find unknown angles

Pre-requisite Summary

• Know that an angle measures the amount of turn between two lines
• Recall that angles are measured in degrees, written as ${}^{\circ }$
• Know that angles on a straight line add to ${180}^{\circ }$
• Know that angles at a point add to ${360}^{\circ }$
• Understand that vertically opposite angles are equal
• Know that perpendicular lines meet at ${90}^{\circ }$
• Be able to identify when two lines cross
• Be familiar with angle facts used to find unknown angles

Worked Examples

Worked Example 1

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

Worked Example 2

State the meaning of each term:
a) alternate angles
b) co-interior angles
c) arrows on a diagram showing parallel lines

Worked Example 3

Two parallel lines are cut by a transversal. One angle is ${68}^{\circ }$. Find:
a) the corresponding angle
b) an alternate angle
c) the co-interior angle on the same side of the transversal

Worked Example 4

Two parallel lines are cut by a transversal. One angle is ${117}^{\circ }$. Find:
a) the corresponding angle
b) an alternate angle
c) the co-interior angle

Worked Example 5

Find the unknown angle:
a) Corresponding angles are $x$ and ${52}^{\circ }$
b) Alternate angles are $y$ and ${109}^{\circ }$
c) Co-interior angles are $z$ and ${124}^{\circ }$

Worked Example 6

Use properties of parallel lines to find each unknown angle:
a) One angle is ${73}^{\circ }$ and its co-interior partner is $m$
b) One angle is ${96}^{\circ }$ and its corresponding angle is $n$
c) One angle is ${45}^{\circ }$ and its alternate angle is $p$

Problems

Problem 1

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

Problem 2

State the meaning of each term:
a) alternate angles
b) co-interior angles
c) arrows on a diagram showing parallel lines

Problem 3

Two parallel lines are cut by a transversal. One angle is ${74}^{\circ }$. Find:
a) the corresponding angle
b) an alternate angle
c) the co-interior angle on the same side of the transversal

Problem 4

Two parallel lines are cut by a transversal. One angle is ${128}^{\circ }$. Find:
a) the corresponding angle
b) an alternate angle
c) the co-interior angle

Problem 5

Find the unknown angle:
a) Corresponding angles are $x$ and ${67}^{\circ }$
b) Alternate angles are $y$ and ${121}^{\circ }$
c) Co-interior angles are $z$ and ${138}^{\circ }$

Problem 6

Use properties of parallel lines to find each unknown angle:
a) One angle is ${81}^{\circ }$ and its co-interior partner is $m$
b) One angle is ${102}^{\circ }$ and its corresponding angle is $n$
c) One angle is ${39}^{\circ }$ and its alternate angle is $p$

Exercises

Understanding and Fluency

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

2. State the meaning of each term:
   a) co-interior angles
   b) arrows on a diagram
   c) same side of a transversal

3. Decide whether each statement is true or false:
   a) Parallel lines cross if extended far enough
   b) A transversal cuts across two or more lines
   c) Corresponding angles in parallel lines are equal
   d) Co-interior angles in parallel lines add to ${180}^{\circ }$

4. One angle formed by a transversal across parallel lines is ${56}^{\circ }$. Find:
   a) a corresponding angle
   b) an alternate angle
   c) a co-interior angle

5. One angle formed by a transversal across parallel lines is ${133}^{\circ }$. Find:
   a) a corresponding angle
   b) an alternate angle
   c) a co-interior angle

6. Find each unknown angle:
   a) Corresponding angles are $a$ and ${48}^{\circ }$
   b) Alternate angles are $b$ and ${97}^{\circ }$
   c) Co-interior angles are $c$ and ${115}^{\circ }$

7. Find each unknown angle:
   a) Corresponding angles are $d$ and ${124}^{\circ }$
   b) Alternate angles are $e$ and ${63}^{\circ }$
   c) Co-interior angles are $f$ and ${71}^{\circ }$

8. Use angle properties to find each unknown:
   a) $g+{52}^{\circ }={180}^{\circ }$
   b) $h={88}^{\circ }$ because it is corresponding to an angle of ${88}^{\circ }$
   c) $j={37}^{\circ }$ because it is alternate to an angle of ${37}^{\circ }$

Reasoning

9. Explain why corresponding angles are equal when a transversal cuts parallel lines.

10. A student says that co-interior angles are always equal. Explain the mistake.

11. Noah says that alternate angles add to ${180}^{\circ }$. Is he correct? Explain.

12. Explain why arrows are useful on a diagram with parallel lines.

13. A student sees two equal angles and assumes they must be corresponding angles. Describe why this may not always be true.

Problem-solving

14. Two parallel roads are crossed by another road. One angle formed is ${65}^{\circ }$. Find the angle in the corresponding position at the second road crossing.

15. A railway line crosses two parallel streets. One interior angle is ${112}^{\circ }$. Find the co-interior angle on the same side of the transversal.

16. In a diagram, two parallel lines are cut by a transversal. An angle is marked $x$ and its alternate angle is ${84}^{\circ }$. Find $x$.

17. A builder marks two fence rails as parallel using arrows. A support beam crosses both rails. One angle formed is ${98}^{\circ }$. Find an alternate angle and a co-interior angle.

18. A student is told that two angles on the same side of a transversal inside parallel lines are co-interior. One angle is ${76}^{\circ }$. Find the other.

19. In a parallel line diagram, one angle is ${43}^{\circ }$. List the sizes of all other distinct angle values that can appear in the diagram.

Potential Misunderstandings

• Students may think parallel lines eventually cross if the diagram is extended
• Students may ignore the arrow markings that show lines are parallel
• Students may confuse a transversal with a parallel line
• Students may mix up corresponding and alternate angles
• Students may think co-interior angles are equal instead of supplementary
• Students may forget that co-interior angles add to ${180}^{\circ }$
• Students may use the wrong angle pair when finding an unknown angle
• Students may assume angle rules for parallel lines apply even when the lines are not marked parallel
• Students may confuse the position of angles inside and outside the parallel lines
• Students may rely on the appearance of a diagram instead of the stated angle properties
• Students may think any equal angles in the diagram must be corresponding
• Students may forget to use straight-line or vertically opposite angle facts together with parallel line properties