Lesson 11 ~25 min Unit 3 · Geometry +100 XP

Parallel Lines and Transversals

When a transversal cuts a pair of parallel lines, eight angles are formed, and they line up in three special families: alternate (equal, Z-shape), co-interior (supplementary, C-shape) and corresponding (equal, F-shape).

Today's hook: Why do railway tracks stay the same distance apart forever, and what does this tell us about angles?

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Think First

warm-up

Draw two horizontal parallel lines, about 5 cm apart. Now draw a slanted line cutting through BOTH of them, this is called a "transversal". Count how many angles are formed at the crossings. Which ones look equal? Which ones look like they'd add to $180^{\circ}$?

Record your answer in your workbook.

What's a Transversal?

+5 XP

A transversal is a line that crosses two (or more) other lines. When the lines being crossed are parallel, the transversal creates EIGHT angles, four at each crossing, and they fall into three special pairs: alternate, co-interior, corresponding.

Two parallel lines (mark with arrows or use $\parallel$ symbol) are cut by a transversal. At each crossing four angles form. Compared across the two crossings: alternate angles are equal (form a Z), co-interior angles sum to $180^{\circ}$ (form a C), and corresponding angles are equal (form an F).

Parallel lines + transversal ⇒ alternate (=), co-int. ($180^{\circ}$), corresponding (=).

Alternate = Z

Trace a Z across the diagram, the corners are equal.

Co-int. = C

Trace a C, the two inside-same-side angles sum to $180^{\circ}$.

Corresponding = F

Trace an F, same-position angles at each crossing are equal.

What You'll Master

objectives

Know

What a transversal is
The three pairs: alternate, co-interior, corresponding
The Z, C and F shape memory aids

Understand

Why parallel lines force these angle relationships
How the three families connect (e.g. alternate + co-int. $= 180^{\circ}$ on same line)
When a diagram does NOT imply parallel lines

Can Do

Identify pairs of alt., co-int. and corr. angles
Find unknown angles using parallel-line rules
Cite the reason "(alt. angles, $AB \parallel CD$)" in working

Words You Need

vocabulary

Parallel ($\parallel$)Two lines that never meet, same direction.

TransversalA line crossing two (or more) other lines.

Alternate anglesOn OPPOSITE sides of the transversal, between the two lines. Z-shape. EQUAL.

Co-interior anglesOn the SAME side of the transversal, between the two lines. C-shape. Sum $180^{\circ}$.

Corresponding anglesSame position at each crossing. F-shape. EQUAL.

Vertically opposite"X" angles at a single crossing, always equal.

SupplementaryTwo angles that sum to $180^{\circ}$.

Reason namee.g. "(alt. angles, $AB \parallel CD$)" written in brackets.

Alternate Angles (Z-shape), EQUAL

+5 XP

Alternate angles sit on OPPOSITE sides of the transversal and BETWEEN the two parallel lines. They trace out a Z (or backward Z). They are EQUAL.

To spot them: look for the "inside corners" on different sides of the transversal, they make a Z. $\angle x = \angle y$ (alt. angles, $AB \parallel CD$). If one is $70^{\circ}$, the other is also $70^{\circ}$.

$\angle x = \angle y$ (alt. angles, $AB \parallel CD$).

Z-test

If you can draw a Z linking the two angles, they're alternate ⇒ equal.

Between the lines

BOTH angles must be inside, not above or below the parallel lines.

Opposite sides

If they're on the same side of the transversal, they're co-interior, not alternate.

Book notes · Card 4

Alternate angles sit on opposite sides of the transversal, between the parallel lines (Z-shape).
Alternate angles are EQUAL when the lines are parallel.
Reason name: "(alt. angles, $AB \parallel CD$)".

Quick check+1 coin

True or False: Alternate angles formed by a transversal cutting two parallel lines are equal in size.

Co-interior Angles (C-shape), SUM $180^{\circ}$

+5 XP

Co-interior angles sit on the SAME side of the transversal, BETWEEN the two parallel lines. They trace out a C. They are supplementary they sum to $180^{\circ}$.

Co-interior angles share the SAME side of the transversal. Both are between the parallel lines. $\angle x + \angle y = 180^{\circ}$ (co-int. angles, $AB \parallel CD$). If one is $110^{\circ}$, the other is $70^{\circ}$. (You met these in the parallelogram lesson, same idea.)

$\angle x + \angle y = 180^{\circ}$ (co-int. angles, $AB \parallel CD$).

C-test

Draw a C linking the two inside-same-side angles.

Sum to $180^{\circ}$

Not equal, supplementary. Don't confuse with alternate (Z).

Also called "allied"

Some older textbooks call them "allied angles", same thing.

Book notes · Card 5

Co-interior angles: same side of transversal, between the parallel lines (C-shape).
They are SUPPLEMENTARY: sum to $180^{\circ}$ (NOT equal).
Reason name: "(co-int. angles, $AB \parallel CD$)".

Quick check+1 coin

Two parallel lines are cut by a transversal. One co-interior angle is $115^{\circ}$. The other is:

Corresponding Angles (F-shape), EQUAL

+5 XP

Corresponding angles are in the SAME POSITION at each crossing, e.g. both above the parallel line and both to the right of the transversal. They trace out an F. They are EQUAL.

To check if two angles correspond: are they on the SAME side of the transversal AND in the same position (both above, or both below) at their respective crossings? If yes, $\angle x = \angle y$ (corr. angles, $AB \parallel CD$). The classic visual: an "F" formed by following the transversal and the parallel line.

$\angle x = \angle y$ (corr. angles, $AB \parallel CD$).

F-test

Trace an F using the transversal and one parallel line.

Same position

Both above-and-right? Both below-and-left? ⇒ corresponding.

4 pairs

There are 4 pairs of corresponding angles at any pair of crossings.

Book notes · Card 6

Corresponding angles: same position at each crossing (F-shape).
Corresponding angles are EQUAL when lines are parallel.
Reason name: "(corr. angles, $AB \parallel CD$)".

Fill the blank+1 coin

Two parallel lines are cut by a transversal. Corresponding angles are in size (one word).

Three Parallel Lines at Once

+5 XP

So far every diagram had ONE transversal cutting a single PAIR of parallel lines. The same three families still work when a transversal cuts three (or more) parallel lines at once, you just have to be careful which crossings you compare. Corresponding angles stay equal all the way down the transversal, through every line. Co-interior angles only apply between ONE adjacent pair of lines at a time, not straight across all three.

Label the parallel lines $\ell_1, \ell_2, \ell_3$ (top to bottom), all cut by the same transversal $t$, so $\ell_1 \parallel \ell_2 \parallel \ell_3$. Because every line is parallel to every other line, the corresponding angle in the same position (say, above the line and to the right of $t$) is equal at ALL THREE crossings: $\angle x = \angle y = \angle z$. But co-interior angles are worked out ONE PAIR at a time, the co-interior pair between $\ell_1$ and $\ell_2$ is separate from the co-interior pair between $\ell_2$ and $\ell_3$, each pair still sums to $180^{\circ}$ on its own.

$\angle x = \angle y = \angle z$ (corr. angles, $\ell_1 \parallel \ell_2 \parallel \ell_3$).

Corresponding chains through

Same position at every crossing on the same transversal ⇒ all equal, even across 3+ lines.

Co-interior is pairwise

Only compare ONE adjacent pair of lines at a time, don't jump from line 1 straight to line 3.

Same transversal, same rules

Every extra parallel line just adds another crossing, the Z/C/F tests still apply at each one.

Book notes · Card 7

A transversal can cut three (or more) parallel lines, not just two.
Corresponding angles stay EQUAL at every crossing on the transversal, however many parallel lines there are.
Co-interior angles only relate ONE adjacent pair of lines at a time, each pair sums to $180^{\circ}$ on its own.

Quick check+1 coin

A transversal cuts three parallel lines $\ell_1 \parallel \ell_2 \parallel \ell_3$. The angle above $\ell_1$, on the right of the transversal, is $47^{\circ}$. What is the corresponding angle above $\ell_3$, also on the right of the transversal?

Watch Me Solve It · 3 examples

Watch Me Solve It · Find x (alternate)

+15 XP per step

PROBLEM

Two parallel lines $AB \parallel CD$ are cut by a transversal. An angle of $72^{\circ}$ is marked on the top line, between the lines, on the right of the transversal. The angle $x$ is on the bottom line, between the lines, on the LEFT of the transversal. Find $x$.

1
Spot the relationship
Both angles are BETWEEN the lines, on OPPOSITE sides ⇒ they form a Z ⇒ alternate.
2
Apply the rule
$x = 72^{\circ}$ (alt. angles, $AB \parallel CD$).
3
State
$x = 72^{\circ}$.
Equal because alternate angles are equal whenever the lines are parallel.

Answer$x = 72^{\circ}$.

Watch Me Solve It · Find x (co-interior)

+15 XP per step

PROBLEM

Lines $\ell \parallel m$ are cut by a transversal. Two angles are formed, BOTH between the lines on the SAME side of the transversal. One is $108^{\circ}$, the other is $x^{\circ}$. Find $x$.

1
Spot the relationship
Same side, between the lines ⇒ C-shape ⇒ co-interior.
2
Apply the rule
$x + 108 = 180$ (co-int. angles, $\ell \parallel m$).
3
Solve
$x = 180 - 108 = 72^{\circ}$.
Co-interior angles sum to $180^{\circ}$, not equal, check the C.

Answer$x = 72^{\circ}$.

Watch Me Solve It · Find x (corresponding)

+15 XP per step

PROBLEM

Lines $AB \parallel CD$ are cut by transversal $EF$. An $85^{\circ}$ angle sits ABOVE the top line, on the right of the transversal. Angle $x$ sits ABOVE the bottom line, on the right of the transversal. Find $x$.

1
Spot the relationship
Both above and on the right ⇒ same position ⇒ F-shape ⇒ corresponding.
2
Apply the rule
$x = 85^{\circ}$ (corr. angles, $AB \parallel CD$).
3
State
$x = 85^{\circ}$.
Corresponding angles are equal when the lines are parallel.

Answer$x = 85^{\circ}$.

Common Pitfalls

heads-up

Confusing Z with C

Z (alternate) = equal. C (co-interior) = sum $180^{\circ}$. Mixing them gives the wrong answer.

Fix: If both angles are on the SAME side of the transversal, it's a C (co-int.).

Using these rules WITHOUT parallel lines

These three rules only work when the two lines being cut are parallel. The diagram MUST show arrow marks or state $\parallel$.

Fix: Check for the parallel-arrow symbol or $\parallel$ in the problem before applying any rule.

Forgetting to state which lines are parallel

"(alt. angles)" alone isn't enough. Write "(alt. angles, $AB \parallel CD$)" to be complete.

Fix: Always include the parallel-line names in the reason.

Copy Into Your Books

Alternate (Z)

Opposite sides of transversal
Between the parallel lines
EQUAL
(alt. angles, $AB \parallel CD$)

Co-interior (C)

Same side of transversal
Between the parallel lines
SUM $180^{\circ}$
(co-int. angles, $AB \parallel CD$)

Corresponding (F)

Same position at each crossing
EQUAL
(corr. angles, $AB \parallel CD$)

Reminders

Rules need parallel lines!
Mark parallels with arrows or $\parallel$
Cite parallel-line names in reason

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Z, C, F

4 problems

Four quick drills. Solve, then reveal.

1 Alt. angles, one is $54^{\circ}$. The other is?

Alt. angles equal.$54^{\circ}$
2 Co-int. angles, one is $123^{\circ}$. The other is?

$180 - 123$.$57^{\circ}$
3 Corr. angles, one is $98^{\circ}$. The other is?

Corr. angles equal.$98^{\circ}$
4 An angle of $63^{\circ}$ is above the top parallel line. Find the angle below the BOTTOM parallel line directly under it (i.e. vertically opp. to the corresponding angle).

Corr. = $63^{\circ}$, vert. opp. = $63^{\circ}$.$63^{\circ}$

Complete in your workbook.

Quick Check · 6 questions

A transversal is:

+10 XP

Alternate angles between two parallel lines are:

+10 XP

Two parallel lines are cut by a transversal. One co-interior angle is $65^{\circ}$. The other is:

+10 XP

Corresponding angles trace out the letter:

+10 XP

Which of these is a REQUIRED condition for the alt./co-int./corr. rules to work?

+10 XP

A transversal cuts three parallel lines $\ell_1 \parallel \ell_2 \parallel \ell_3$. Between $\ell_1$ and $\ell_2$, on the same side of the transversal, one angle is $75^{\circ}$. Its co-interior partner (at the crossing with $\ell_2$) is:

+10 XP

Show Your Working · 4 questions

Show Your Working

11 marks total

Apply Easy 3 MARKS

Q6. Lines $AB \parallel CD$ are cut by transversal $EF$. At the top crossing the angle above-right is $78^{\circ}$. Find, with a reason in each case:
(a) The angle below the top line, same side, on the right.
(b) The corresponding angle on the bottom line.
(c) The co-interior angle to the original $78^{\circ}$.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. Two parallel lines are cut by a transversal. Two co-interior angles are $(2x + 10)^{\circ}$ and $(3x - 20)^{\circ}$.
(a) Set up an equation.
(b) Solve for $x$.
(c) State the actual angle values.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. For each statement, decide TRUE or FALSE and explain why in one sentence.
(a) Vertically opposite angles are always equal, even without parallel lines.
(b) If alternate angles are equal, then the two lines must be parallel.
(c) Two co-interior angles must each be obtuse.

Answer in your workbook.

Reason Medium 2 MARKS

Q9. In the Think First activity at the start of this lesson, you sketched a transversal crossing two parallel lines and noticed some angle pairs looked equal, while others looked like they added to $180^{\circ}$. Explain, in your own words, WHY co-interior angles between two parallel lines MUST sum to $180^{\circ}$, using what you know about corresponding angles and angles on a straight line. Do not just state the rule, justify it.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B A line that crosses two other lines.

2. A Equal.

3. C$115^{\circ}$.

4. A F (equal).

5. D The two lines being cut must be parallel.

6. B $105^{\circ}$, co-interior angles between $\ell_1$ and $\ell_2$ are supplementary, pair by pair.

Show Your Working Model Answers

Q6 (3 marks): (a) $180 - 78 = 102^{\circ}$ (angles on a straight line at the crossing) [1]. (b) Corresponding angle below-right on bottom line $= 78^{\circ}$ (corr. angles, $AB \parallel CD$) [1]. (c) Co-interior angle $= 180 - 78 = 102^{\circ}$ (co-int. angles, $AB \parallel CD$) [1].

Q7 (3 marks): (a) $(2x + 10) + (3x - 20) = 180$ (co-int. angles) [1]. (b) $5x - 10 = 180 \Rightarrow 5x = 190 \Rightarrow x = 38$ [1]. (c) Angles are $86^{\circ}$ and $94^{\circ}$ (check $86 + 94 = 180$) [1].

Q8 (3 marks): (a) TRUE, vertically opposite angles are always equal at any crossing, regardless of parallelism [1]. (b) TRUE, equal alternate angles is the converse: if they're equal, the lines must be parallel [1]. (c) FALSE, co-interior pairs sum to $180^{\circ}$, so they could be $90^{\circ}$ each, or one acute and one obtuse, only the SUM is fixed [1].

Q9 (2 marks): A co-interior angle and the corresponding angle sitting at the SAME position on the SAME crossing lie next to each other on a straight line, so they sum to $180^{\circ}$ (angles on a straight line) [1]. But the corresponding angle is EQUAL to the co-interior angle's partner on the other parallel line (corr. angles, lines are parallel), so substituting that equal value shows the two co-interior angles themselves must also sum to $180^{\circ}$ [1].

Stretch Challenge · +25 XP, +10 coins

Three-Lines Puzzle

(a) Three lines $\ell_1, \ell_2, \ell_3$ all run in the same direction (all parallel to each other). A transversal cuts all three. At $\ell_1$ it makes an angle of $40^{\circ}$ above-right. What angle does it make above-right at $\ell_2$ and $\ell_3$? Justify. (b) Two parallel lines $AB \parallel CD$ are cut by transversal $XY$ at points $P$ (on $AB$) and $Q$ (on $CD$). Triangle $PQR$ is formed where $R$ lies on $CD$ to the right of $Q$. If $\angle APQ = 65^{\circ}$ (above $AB$, left of transversal) and $\angle PQR = 50^{\circ}$, find $\angle QPR$. (c) Show how the angle-sum of a triangle ($180^{\circ}$) can be PROVED using alternate angles between parallel lines.

Reveal solution

(a) All three above-right angles equal $40^{\circ}$ (corresponding angles transfer through every pair of parallel lines). (b) $\angle APQ$ and $\angle PQR'$ (where $R'$ is to the LEFT of $Q$) are alternate, so $\angle PQR' = 65^{\circ}$. Then $\angle PQR = 180 - 65 = 115^{\circ}$... but actually if $\angle PQR = 50^{\circ}$ is GIVEN as something between $Q$ and another arrangement, $\angle QPR = 180 - 65 - 50 = 65^{\circ}$ via ∠ sum of $\triangle$. (c) Through vertex $A$ of $\triangle ABC$, draw a line $\ell$ parallel to $BC$. The angles $\angle B$ and one angle at $A$ are alternate (equal), the angles $\angle C$ and another angle at $A$ are alternate (equal), and the three angles at $A$ make up a straight line $= 180^{\circ}$. So $\angle A + \angle B + \angle C = 180^{\circ}$.

Quick Review

Transversal

A line crossing two (or more) lines.

Alt. angles (Z)

EQUAL; opposite sides, between lines.

Co-int. (C)

SUM $180^{\circ}$; same side, between lines.

Corr. (F)

EQUAL; same position at each crossing.

Need parallel

All three rules need $AB \parallel CD$.

Reason

"(alt./co-int./corr. angles, $AB \parallel CD$)"

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