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Lesson 12 ~25 min Unit 3 · Geometry +85 XP

Using Parallel Line Properties

When a transversal cuts two parallel lines, three powerful angle relationships appear: corresponding angles equal, alternate angles equal, and co-interior angles supplementary. We'll use these to find unknowns AND to prove lines are parallel.

Today's hook: One transversal. Eight angles. But only TWO different sizes, and they're related by three simple rules.
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Think First
warm-up

Two parallel railway tracks. A road crosses both at the same angle. Where the road meets the first track, an angle of $65^{\circ}$ is formed on the upper side. What angle is formed on the upper side where the road meets the SECOND track? Why must it be the same?

Record your answer in your workbook.
1
Parallel Lines and a Transversal
+5 XP

A transversal is a straight line that cuts across two (or more) other lines. When the two lines are parallel, the transversal creates 8 angles, but only TWO different sizes. Three rules connect them: corresponding angles equal, alternate angles equal, and co-interior angles supplementary (sum to $180^{\circ}$).

The transversal $t$ crosses parallel lines $\ell_1$ and $\ell_2$. Corresponding angles sit in matching positions (same "corner" of each intersection), they are equal. Alternate angles sit on OPPOSITE sides of the transversal, BETWEEN the two parallel lines, they are equal. Co-interior angles sit on the SAME side of the transversal, BETWEEN the parallel lines, they add to $180^{\circ}$.

Parallel lines cut by a transversal l1 l2 t a b c d e f g h Corresponding: a=e, b=f, c=g, d=h | Alternate: c=f, d=e | Co-interior: c+e=180
If $\ell_1 \parallel \ell_2$: corresponding $=$, alternate $=$, co-interior sum $= 180^{\circ}$
F-shape for corresponding
Trace an "F" (or backwards F) along the angles, matching corners are equal.
Z-shape for alternate
Trace a "Z" through the middle, the two "elbows" mark equal alternate angles.
C-shape for co-interior
Trace a "C", the two angles inside the C sum to $180^{\circ}$.
2
What You'll Master
objectives

Know

  • The three parallel-line rules (corresponding, alternate, co-interior)
  • The correct reason phrase for each rule
  • The converses: if a pair of angles obeys the rule, the lines are parallel

Understand

  • Why only two sizes appear when a transversal cuts parallel lines
  • How to spot F, Z and C shapes in a diagram
  • How to chain rules together to find an angle several steps away

Can Do

  • Find an unknown angle using one parallel-line rule
  • Find an unknown angle using a chain of two or three rules
  • Prove two lines are parallel given an angle pair
3
Words You Need
vocabulary
TransversalA line that cuts across two (or more) other lines.
Parallel linesTwo lines in the same plane that never meet; shown with matching arrowheads.
Corresponding anglesSame position at each intersection; equal when lines are parallel (F-shape).
Alternate anglesOpposite sides of the transversal, between the parallel lines; equal (Z-shape).
Co-interior anglesSame side of the transversal, between the parallel lines; supplementary (C-shape).
SupplementaryTwo angles that add to $180^{\circ}$.
Converse"If, then" reversed: if angles satisfy the rule, the lines must be parallel.
Reason phraseShort justification: "corresponding $\angle$s, $\ell_1 \parallel \ell_2$".
4
Reason Phrases You Must Write
+5 XP

In NSW geometry, EVERY step that uses a parallel-line rule must be followed by a short reason. Markers look for these exact phrases:

• "corresponding $\angle$s, $\ell_1 \parallel \ell_2$"use when two angles are in the same position at each intersection.
• "alternate $\angle$s, $\ell_1 \parallel \ell_2$"use when two angles form a Z.
• "co-interior $\angle$s, $\ell_1 \parallel \ell_2$"use when two angles form a C (then write $= 180^{\circ}$).
Always name the parallel lines after the rule. Without a parallel-line reason, the step earns no mark.

Sample working with reasons x = 70 (corresponding angles, AB || CD) y = 55 (alternate angles, AB || CD) z + 110 = 180 (co-interior, AB || CD) z = 70 Tick: rule + parallel lines named + result. Without "AB || CD" the step loses the mark.
Result $+$ Rule $+$ Parallel lines named.
Name the lines
"$AB \parallel CD$" tells the marker which pair you're using.
Use $\angle$s, not "angles"
Abbreviate with the angle symbol, quick and clear.
Co-interior $\neq$ equal
Co-interior angles SUM to $180^{\circ}$, don't write them equal.
Quick book-notes · Reason phrases
  • Corresponding $\angle$s, $\ell_1 \parallel \ell_2$ → angles equal.
  • Alternate $\angle$s, $\ell_1 \parallel \ell_2$ → angles equal.
  • Co-interior $\angle$s, $\ell_1 \parallel \ell_2$ → angles sum to $180^{\circ}$.
Micro-check: Which reason fits two angles that form a "Z" between parallel lines?
5
Finding Unknown Angles
+5 XP

To find an unknown angle, look for the F, Z or C shape that connects it to a known angle. Sometimes one rule is enough. Sometimes you need to chain together TWO rules, for example, an alternate-angle equality followed by an angle-on-a-straight-line subtraction.

Step 1: Identify the parallel lines and the transversal. Step 2: Find a known angle that pairs with the unknown via one of the three rules (or via co-angles like vertically opposite, angle on a straight line). Step 3: Write the equation, solve, then add the reason. If the path is more than one step, write each step on its own line with its own reason.

Find x when AB || CD A B C D 110 x x = 110 (corresponding angles, AB || CD)
Spot the shape → write the rule → solve.
Highlight the transversal
Trace it in colour to see the F, Z or C clearly.
One step per line
If you chain rules, each new line gets its own reason.
Sanity check
Acute should look acute; obtuse should look obtuse on the diagram.
Quick book-notes · Finding unknowns
  • Identify which pair of lines is parallel.
  • Spot F (corresponding), Z (alternate) or C (co-interior).
  • One step per line, reason after each step.
True or false: Co-interior angles between parallel lines are always equal.

Co-interior angles SUM to $180^{\circ}$, they are equal only in the special case of two $90^{\circ}$ angles.

6
Proving Lines Are Parallel
+5 XP

Each rule has a converse that runs backwards: if a transversal makes a pair of corresponding angles equal, the lines MUST be parallel. Same for alternate angles. For co-interior angles, the converse is: if they sum to $180^{\circ}$, the lines are parallel.

To prove $\ell_1 \parallel \ell_2$, find a transversal and a pair of angles that satisfies ONE of these:
• A pair of corresponding angles equal $\Rightarrow$ lines parallel (converse).
• A pair of alternate angles equal $\Rightarrow$ lines parallel.
• A pair of co-interior angles summing to $180^{\circ}$ $\Rightarrow$ lines parallel.
The reason phrase swaps: now it's "$\ell_1 \parallel \ell_2$ (corresponding $\angle$s equal)".

If alternate angles are equal -> lines parallel 72 72 l1 || l2 because alternate angles are equal.
Equal alternate (or corresponding) $\angle$s $\Rightarrow$ parallel
Flip the logic
Forward: parallel → angles equal. Converse: angles equal → parallel.
State the conclusion
End your proof with "Therefore $AB \parallel CD$".
Any one rule works
You only need ONE pair to satisfy a converse, pick the easiest.
Quick book-notes · Proving lines parallel
  • Converses turn the rule around: angles equal $\Rightarrow$ parallel.
  • Co-interior converse: sum to $180^{\circ}$ $\Rightarrow$ parallel.
  • End with: "Therefore $AB \parallel CD$".
Micro-check: A transversal cuts two lines so that a pair of co-interior angles are $105^{\circ}$ and $75^{\circ}$. Are the lines parallel?

Watch Me Solve It · 3 examples

Watch Me Solve It · Corresponding angles
+15 XP per step
Q1
PROBLEM
$AB \parallel CD$. The transversal $PQ$ makes a $68^{\circ}$ angle with $AB$ above-right. Find the corresponding angle with $CD$.
  1. 1
    Identify the shape
    Two angles in the same "above-right" position $\rightarrow$ an F (corresponding).
  2. 2
    Apply the rule
    $x = 68^{\circ}$ (corresponding $\angle$s, $AB \parallel CD$)
  3. 3
    Conclude
    The required angle is $68^{\circ}$.
    No subtraction needed, corresponding angles are simply equal.
Answer$x = 68^{\circ}$.
Watch Me Solve It · Co-interior angles
+15 XP per step
Q2
PROBLEM
$EF \parallel GH$. A transversal makes co-interior angles of $115^{\circ}$ and $y^{\circ}$. Find $y$.
  1. 1
    Identify the shape
    Same side of transversal, between parallel lines $\rightarrow$ a C (co-interior).
  2. 2
    Apply the rule
    $y + 115 = 180$ (co-interior $\angle$s, $EF \parallel GH$)
  3. 3
    Solve
    $y = 180 - 115 = 65^{\circ}$
    Co-interior $\neq$ equal, remember they sum to $180^{\circ}$.
Answer$y = 65^{\circ}$.
Watch Me Solve It · Proving lines parallel
+15 XP per step
Q3
PROBLEM
A transversal cuts lines $\ell_1$ and $\ell_2$. The alternate angles measured are $42^{\circ}$ and $42^{\circ}$. Prove $\ell_1 \parallel \ell_2$.
  1. 1
    State the given facts
    The pair of alternate angles between $\ell_1$ and $\ell_2$ are both $42^{\circ}$.
  2. 2
    Apply the converse
    If alternate $\angle$s are equal, the lines are parallel (converse of alternate-angle rule).
  3. 3
    Conclude
    $\therefore \ell_1 \parallel \ell_2$.
    The converse is the key, it turns "angles equal" into "lines parallel".
Answer$\therefore \ell_1 \parallel \ell_2$ (alternate $\angle$s equal).

Common Pitfalls

8
Common Pitfalls
heads-up
Treating co-interior as equal
Students often write $x = 110^{\circ}$ when the angles are co-interior, forgetting the rule is SUM to $180^{\circ}$.
Fix: Spot the C-shape. If you see a C, write $a + b = 180^{\circ}$, never $a = b$.
Missing the reason phrase
A correct numeric answer with no reason loses the working mark in NSW geometry.
Fix: Always end each step with "(rule, $\ell_1 \parallel \ell_2$)".
Assuming lines are parallel from the picture
Lines might LOOK parallel but unless marked with arrowheads OR proved, you cannot use parallel-line rules.
Fix: Check for arrowheads or read the question; if neither, you must prove parallel first.
Copy Into Your Books

The three rules

  • Corresponding $\angle$s equal (F)
  • Alternate $\angle$s equal (Z)
  • Co-interior $\angle$s sum to $180^{\circ}$ (C)

Reason phrases

  • "corresponding $\angle$s, $\ell_1 \parallel \ell_2$"
  • "alternate $\angle$s, $\ell_1 \parallel \ell_2$"
  • "co-interior $\angle$s, $\ell_1 \parallel \ell_2$"

Converses (prove parallel)

  • Corresponding equal $\Rightarrow$ parallel
  • Alternate equal $\Rightarrow$ parallel
  • Co-interior sum $180^{\circ}$ $\Rightarrow$ parallel

Method

  • Find F, Z or C
  • Equation + reason
  • One step per line

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Parallel Line Drills
4 problems

Four quick drills using the parallel-line rules. Solve, then reveal.

  1. 1 $AB \parallel CD$. A pair of corresponding angles is marked $x$ and $124^{\circ}$. Find $x$.

    Corresponding $\angle$s, $AB \parallel CD$.$x = 124^{\circ}$

  2. 2 $PQ \parallel RS$. Co-interior angles are $y$ and $58^{\circ}$. Find $y$.

    $y + 58 = 180$ (co-interior $\angle$s, $PQ \parallel RS$).$y = 122^{\circ}$

  3. 3 Alternate angles on a transversal are $3x$ and $48^{\circ}$. Find $x$.

    $3x = 48$ (alternate $\angle$s).$x = 16$

  4. 4 Two lines are cut by a transversal making corresponding angles of $63^{\circ}$ and $67^{\circ}$. Are the lines parallel?

    Corresponding $\angle$s NOT equal.No, the lines are NOT parallel.
Complete in your workbook.

Quick Check · 5 questions

1
When a transversal crosses two parallel lines, co-interior angles:
+10 XP
2
$AB \parallel CD$. A pair of corresponding angles measure $x$ and $72^{\circ}$. Find $x$.
+10 XP
3
A pair of angles forms a "Z" between two parallel lines. They are called:
+10 XP
4
$EF \parallel GH$. Co-interior angles are $117^{\circ}$ and $y$. Find $y$.
+10 XP
5
Which condition is enough to prove two lines are parallel?
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Easy 3 MARKS

Q6. $AB \parallel CD$. A transversal makes an angle of $134^{\circ}$ above the upper line on the left of the transversal. Find, with reasons, the size of:
(a) the corresponding angle below the lower line on the same side,
(b) the co-interior angle on the right of the transversal between the lines,
(c) the alternate angle on the right of the transversal between the lines.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. $\ell_1 \parallel \ell_2$. A transversal makes co-interior angles of $(2x + 10)^{\circ}$ and $(3x - 5)^{\circ}$.
(a) Set up an equation.
(b) Solve for $x$.
(c) State both angles.

Answer in your workbook.
Reason Hard 3 MARKS

Q8. A transversal cuts lines $AB$ and $CD$. The corresponding angles are $(5x - 20)^{\circ}$ and $(3x + 40)^{\circ}$.
(a) Find the value of $x$ that would make $AB \parallel CD$.
(b) Compute each angle for that $x$.
(c) Briefly justify why these values force the lines to be parallel.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. C Sum to $180^{\circ}$.

2. A$72^{\circ}$ (corresponding angles equal).

3. D Alternate angles.

4. B$63^{\circ}$ ($180 - 117$).

5. C Alternate angles equal.

Show Your Working Model Answers

Q6 (3 marks): (a) $134^{\circ}$ (corresponding $\angle$s, $AB \parallel CD$) [1]. (b) $180 - 134 = 46^{\circ}$ (co-interior $\angle$s, $AB \parallel CD$) [1]. (c) $134^{\circ}$ (alternate $\angle$s, $AB \parallel CD$) [1].

Q7 (3 marks): (a) $(2x + 10) + (3x - 5) = 180$ (co-interior $\angle$s, $\ell_1 \parallel \ell_2$) [1]. (b) $5x + 5 = 180 \Rightarrow x = 35$ [1]. (c) Angles: $2(35) + 10 = 80^{\circ}$ and $3(35) - 5 = 100^{\circ}$. Check: $80 + 100 = 180^{\circ}$ ✓ [1].

Q8 (3 marks): (a) $5x - 20 = 3x + 40 \Rightarrow 2x = 60 \Rightarrow x = 30$ [1]. (b) $5(30) - 20 = 130^{\circ}$ and $3(30) + 40 = 130^{\circ}$ [1]. (c) The corresponding angles are equal, so by the converse of the corresponding-angles rule, $AB \parallel CD$ [1].

Stretch Challenge · +25 XP, +10 coins

The Zig-Zag Path

A delivery robot travels from $A$ to $B$ along a zig-zag track between two parallel walls $\ell_1$ and $\ell_2$. At point $P$ on the lower wall, the angle between the robot's path going up and the wall is $52^{\circ}$. At the next bend $Q$ on the upper wall, the robot turns through an angle and continues to the next bend $R$ on the lower wall. (a) If $\ell_1 \parallel \ell_2$, what angle does the robot's path make with the upper wall at $Q$ (on the same side as $52^{\circ}$, going DOWN the next leg)? (b) The robot's TURN angle at $Q$ is measured between the incoming and outgoing legs. If the path is symmetric (the angle going up to $Q$ matches the angle going down from $Q$), find the size of that turn angle. (c) Explain how the turn angle changes if the angle at $P$ were $40^{\circ}$ instead.

Reveal solution

(a) The path from $P$ to $Q$ acts as a transversal. The angle at $Q$ between the incoming path and $\ell_2$ on the corresponding side is $52^{\circ}$ (alternate angles, $\ell_1 \parallel \ell_2$). The outgoing leg of the path makes the same $52^{\circ}$ with $\ell_2$ on the opposite side (symmetry). (b) The straight line $\ell_2$ at $Q$ measures $180^{\circ}$. The path bends inward by $180 - 52 - 52 = 76^{\circ}$, so the turn angle is $76^{\circ}$. (c) If the angle at $P$ were $40^{\circ}$, the turn angle becomes $180 - 40 - 40 = 100^{\circ}$, a shallower approach means a sharper turn.

R
Quick Review

Transversal

A line that cuts across two (or more) lines.

Corresponding (F)

Equal when lines are parallel.

Alternate (Z)

Equal when lines are parallel.

Co-interior (C)

Sum to $180^{\circ}$ when lines are parallel.

Converse

Angles equal $\Rightarrow$ lines parallel.

Reason phrase

State rule + name parallel lines.

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