Parallel lines
🎯 In this topic you will
- Use geometric vocabulary for equal angles formed when lines intersect.
🧠 Key Words
- alternate angles
- corresponding angles
- geometric
- transversal
- vertically opposite angles
Show Definitions
- alternate angles: Angles that lie on opposite sides of a transversal and inside the two lines, and are equal if the lines are parallel.
- corresponding angles: Angles that are in the same position on two lines in relation to a transversal, and are equal if the lines are parallel.
- geometric: Related to the properties, measurement, and relationships of points, lines, angles, and shapes.
- transversal: A line that crosses two or more other lines at different points.
- vertically opposite angles: Angles that are opposite each other when two lines intersect, and are always equal.
This diagram shows two straight lines.
Angles $a$ and $c$ are equal. They are called vertically opposite angles.
Angles $b$ and $d$ are equal. They are also vertically opposite angles.
Vertically opposite angles are equal.
Angles $a$ and $b$ are not equal (unless they are both $90^\circ$).
They add up to $180^\circ$ because they are angles on a straight line.
The arrows on this diagram show that these two lines are parallel. The perpendicular distance between parallel lines is the same wherever you measure it.
Here, there is a third straight line crossing two parallel lines. It is called a transversal.
Where the transversal crosses the parallel lines, four angles are formed.
Angles $a$ and $e$ are called corresponding angles. Angles $d$ and $h$ are also corresponding angles. So are $b$ and $f$. So are $c$ and $g$.
Corresponding angles are equal.
Angles $d$ and $f$ are called alternate angles. Angles $c$ and $e$ are also alternate angles.
Alternate angles are equal.
These are important properties of parallel lines.
To help you remember:
- for vertically opposite angles, think of the letter X
- for corresponding angles, think of the letter F
- for alternate angles, think of the letter Z
❓ EXERCISES
1. Look at the diagram.
a. Write four pairs of corresponding angles.
b. Write two pairs of alternate angles.
👀 Show answer
b. $(q, w), (s, u)$
2.
a. One angle of $62^\circ$ is marked in the diagram.
Copy and complete these sentences.
i. Because corresponding angles are equal, angle $\_\_$ = $62^\circ$
ii. Because alternate angles are equal, angle $\_\_$ = $62^\circ$
b. Write the letters of a pair of vertically opposite angles.
👀 Show answer
a. ii. $d$ = $62^\circ$
b. $(a, c)$
❓ EXERCISES
3. The sizes of two angles are marked in the diagram.
a. Which other angles are $105^\circ$?
b. Which other angles are $75^\circ$?
👀 Show answer
b. $q$, $u$
4. Angle $APY$ is marked on the diagram.
Complete these sentences.
a. $APY$ and $CQY$ are .................... angles.
b. $APY$ and $XQD$ are .................... angles.
c. $APX$ and ........ are corresponding angles.
d. $CQX$ and ........ are alternate angles.
e. $CQP$ and ........ are vertically opposite angles.
👀 Show answer
b. Vertically opposite
c. $CQY$
d. $APX$
e. $XQD$
5. PQ and RS are parallel lines.
Find the sizes of angles $a$, $b$, $c$ and $d$.
Give a reason in each case.
👀 Show answer
$b = 136^\circ$ (vertically opposite to $c$)
$c = 136^\circ$ (corresponding to given $136^\circ$)
$d = 44^\circ$ (alternate to $a$)
6. Look at this diagram.
Explain why AB and CD cannot be parallel lines.
👀 Show answer
❓ EXERCISES
7. This diagram has three parallel lines and a transversal.
a. Write a set of three corresponding angles that includes angle $f$.
b. Write a pair of alternate angles that includes angle $c$.
c. Write another pair of alternate angles that includes angle $c$.
👀 Show answer
7b. Alternate angles including $c$: $(c,\, e)$.
7c. Another alternate pair including $c$: $(c,\, g)$.
8. Look at this diagram.
Write whether these are corresponding angles, alternate angles or neither.
a. $a$ and $d$
b. $b$ and $f$
c. $c$ and $g$
d. $d$ and $e$
e. $a$ and $h$
👀 Show answer
8b. Corresponding angles.
8c. Corresponding angles.
8d. Alternate angles.
8e. Alternate angles.
🧠 Think like a Mathematician
Scenario: Arun gives this explanation of why angles h and d are equal:
h = b because they are corresponding angles.
b = d because they are alternate angles.
Therefore h = d.
Tasks:
- Arun’s explanation is not correct. Write a correct version of the explanation.
- Write a different explanation of why h = d that does not use corresponding angles.
👀 Show answer
- 1: Correct explanation: h = f because they are alternate angles. f = d because they are vertically opposite angles. Therefore, h = d.
- 2: Alternative explanation without using corresponding angles: h and f are alternate angles, so they are equal. f and d are vertically opposite angles, so they are equal. Hence, h = d.
❓ EXERCISES
10. AB and CD are parallel.
a. Give a reason why $a$ and $d$ are equal.
b. Give a reason why $b$ and $e$ are equal.
c. Use your answers to a and b to show that the sum of the angles of triangle $ABC$ must be $180^\circ$.
👀 Show answer
b. $b$ and $e$ are equal because they are alternate angles.
c. $a b c = 180^\circ$ because they form the angles of triangle $ABC$.
11. Show that the sum of the angles of triangle $XYZ$ must be $180^\circ$.
💡 Tip
Use your answer to Question 10 as a guide.
👀 Show answer
❓ EXERCISES
💡 Tip
Extend the sides of the parallelogram.
12. $ABCD$ is a parallelogram.
a. Show that opposite angles of the parallelogram are equal.
b. Compare your answer to part $a$ with a partner’s answer. Can you improve his or her answer? Can you improve your own answer?
👀 Show answer
13b. A strong explanation should (i) state which sides are parallel, (ii) name the angle relationships used (alternate, corresponding, or interior–supplementary), and (iii) conclude explicitly that $\angle A = \angle C$ and $\angle B = \angle D$. Improve by adding a clear diagram marking the transversals and citing the angle facts used.
🧠 Think like a Mathematician
13. $ABCD$ is a trapezium. Two sides are extended to make the triangle $AXB$.
a. Show that the angles of triangles $ABX$ and $DCX$ are the same size.
b. Show that angles $A$ and $D$ of the trapezium add up to $180^\circ$.
c. What can you say about angles $B$ and $C$ of the trapezium? Give a reason for your answer.
👀 show answer
- a: Since $AB \parallel DC$, angle correspondences hold: $\angle ABX = \angle DCX$ and $\angle AXB = \angle DXC$ (alternate/corresponding angles with parallel lines). The third angles then match as well, so triangles $ABX$ and $DCX$ have the same angles (they are similar).
- b: Line $AD$ is a transversal of the parallels $AB$ and $DC$. Interior angles on the same side are supplementary, so $\angle A \angle D = 180^\circ$.
- c: Similarly, with transversal $BC$, interior angles give $\angle B \angle C = 180^\circ$ (co-interior angles on parallel lines).