Parallel Lines cut by a Transversal interactive worksheet | Live ... - Free Printable

Let’s solve this step by step.

We are given two tests about parallel lines cut by a transversal. We’ll handle each test separately.

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Test I: Given l₁ ∥ l₂

We need to tell whether each pair of angles is congruent (C) or supplementary (S).

First, let’s label the diagram mentally:

- Line l₁ (top) and line l₂ (bottom) are parallel.
- A transversal cuts them, creating 8 angles:
- At top intersection (l₁): angles 5,6,7,8 — with 5 and 6 on bottom side of l₁, 7 and 8 on top; 5 and 8 left, 6 and 7 right.
- At bottom intersection (l₂): angles 1,2,3,4 — with 1 and 2 on bottom side of l₂, 3 and 4 on top; 1 and 4 left, 2 and 3 right.

Standard angle relationships when lines are parallel:

- Corresponding angles: congruent → e.g., ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8
- Alternate interior angles: congruent → e.g., ∠3 & ∠5, ∠4 & ∠6
- Alternate exterior angles: congruent → e.g., ∠1 & ∠7, ∠2 & ∠8
- Consecutive interior angles (same-side interior): supplementary → e.g., ∠3 & ∠6, ∠4 & ∠5
- Vertical angles: always congruent → e.g., ∠1 & ∠3, ∠2 & ∠4, etc.
- Linear pairs: always supplementary → e.g., ∠1 & ∠2, ∠5 & ∠6, etc.

Now go one by one:

1. ∠3 and ∠5 → alternate interior → C
2. ∠2 and ∠6 → corresponding → C
3. ∠4 and ∠5 → consecutive interior → S
4. ∠1 and ∠7 → alternate exterior → C
5. ∠2 and ∠7 → not standard pair? Let’s see: ∠2 is bottom-right, ∠7 is top-right → they are on same side but one inside, one outside → actually, ∠2 and 7 are not corresponding, not alternate... Wait — ∠2 and ∠7: if you look, ∠7 corresponds to ∠3, and ∠2 is adjacent to ∠3 → so ∠2 + ∠3 = 180°, and ∠3 = 7 → so ∠2 + ∠7 = 180° → S
Alternatively: ∠2 and ∠7 are “consecutive exterior”? Not standard term, but geometrically: since ∠7 = ∠3 (corresponding), and ∠2 + ∠3 = 180° (linear pair at bottom), then yes, ∠2 + ∠7 = 180° → S
6. ∠4 and ∠6 → alternate interior → C
7. ∠1 and ∠5 → corresponding → C
8. ∠3 and ∠6 → consecutive interior → S
9. ∠2 and ∠8 → alternate exterior → C (∠2 is bottom-right, ∠8 is top-left? Wait no — let's check positions again.)

Wait — let me double-check numbering from diagram description:

From the diagram in Test I:

At top line l₁:
- Left side: above = ∠8, below = ∠5
- Right side: above = ∠7, below = ∠6

At bottom line l₂:
- Left side: above = ∠4, below = ∠1
- Right side: above = ∠3, below = ∠2

So:

∠1: bottom-left, below l₂
∠2: bottom-right, below l₂
∠3: bottom-right, above l₂
∠4: bottom-left, above l₂
∠5: top-left, below l₁
∠6: top-right, below l₁
∠7: top-right, above l₁
∠8: top-left, above l₁

Now recheck #9: ∠2 and 8

∠2: bottom-right, below l₂
∠8: top-left, above l₁

These are alternate exterior angles? Alternate exterior would be: one on left outside, one on right outside.

Outside means not between the parallel lines.

So for l₁ and l₂ parallel:

- Exterior angles: ∠1, 2 (below l₂), ∠7, ∠8 (above l₁)
- Interior: ∠3, ∠4, ∠5, ∠6

Alternate exterior: ∠1 & ∠7, ∠2 & ∠8 → YES! So ∠2 and ∠8 are alternate exterior → C

#10: ∠1 and 8

∠1: bottom-left, below l₂
∠8: top-left, above l₁

Same side (left), both exterior → these are consecutive exterior angles → should be supplementary.

Check: ∠1 and ∠8 — are they related?

Note: ∠1 and ∠5 are corresponding → ∠1 = ∠5
∠5 and ∠8 are vertical? No — ∠5 and ∠8 are adjacent on top line? Actually, ∠5 and ∠8 form a linear pair? No — at top intersection, ∠5 and ∠8 are on same side (left) but one above, one below → actually, ∠5 and 8 are adjacent and form a straight line? Yes — because they are on opposite sides of the transversal but same side of the line? Wait no.

Actually, at the top intersection (line l₁ crossed by transversal):

Angles around point: ∠5, ∠6, ∠7, ∠8

∠5 and ∠8 are adjacent? If we go clockwise: say ∠8 (top-left), then ∠7 (top-right), then ∠6 (bottom-right), then ∠5 (bottom-left). So ∠8 and ∠5 are NOT adjacent — they are separated by ∠7 and ∠6? No — actually, ∠8 and ∠5 are on the same side of the transversal (left side) but one above l₁, one below l₁ → so they are vertical? No.

Actually, ∠8 and ∠5 are adjacent angles forming a linear pair? Let’s think: the transversal creates four angles at each intersection. At top intersection, the four angles are:

- Top-left: ∠8
- Top-right: ∠7
- Bottom-right: ∠6
- Bottom-left: ∠5

So ∠8 and ∠5 are on the same side of the transversal (left) but on opposite sides of line l₁ → so they are vertical angles? No — vertical angles are opposite each other.

Actually, ∠8 and ∠6 are vertical? No — ∠8 and 6 are diagonal? Let’s define:

Vertical angles at top intersection:
- ∠8 and ∠6 are vertical? No — ∠8 (top-left) and ∠6 (bottom-right) are vertical → yes!
Similarly, ∠7 (top-right) and ∠5 (bottom-left) are vertical.

So ∠8 and ∠6 are vertical → congruent.

But back to ∠1 and ∠8.

∠1 is at bottom-left, below l₂.

∠8 is at top-left, above l₁.

They are on the same side of the transversal (left), and both are exterior (outside the parallel lines).

In parallel lines, consecutive exterior angles are supplementary.

Confirm: ∠1 and ∠8 — what is their relationship?

Since l₁ ∥ l₂, then:

∠1 corresponds to ∠5 → ∠1 = 5

∠5 and 8: are they supplementary? At the top intersection, ∠5 and 8 are adjacent angles that form a straight line? Let’s see: from the transversal, going down-left is ∠5, up-left is ∠8 — together they make the angle along the transversal? No.

Actually, at the top intersection, the angles around the point sum to 360°. Adjacent angles on a straight line sum to 180°.

Specifically, ∠5 and ∠6 are adjacent on the bottom side of l₁ → they form a linear pair? No — ∠5 and 6 are on the same side of the transversal? I think I'm confusing myself.

Better approach: use known pairs.

We know:

- ∠1 and ∠3 are vertical → ∠1 = ∠3
- ∠3 and ∠5 are alternate interior → ∠3 = ∠5 → so ∠1 = 5
- ∠5 and ∠8: at the top intersection, ∠5 and ∠8 are on the same side of the transversal (left) but on opposite sides of line l₁. The angle between them along the transversal is 180°? Actually, ∠5 and ∠8 are adjacent angles that together with the transversal form a straight line? No.

Let’s consider the straight line l₁. On line l₁, the angles on one side of the transversal: for example, on the left side of the transversal, above l₁ is ∠8, below l₁ is ∠5. These two angles (∠8 and ∠5) are adjacent and together form the angle between the transversal and the line l₁ on the left side — but actually, they are on a straight line only if we consider the transversal.

I recall that any two angles that are on the same side of the transversal and both exterior are called consecutive exterior angles, and they are supplementary when lines are parallel.

Yes, standard theorem: if two parallel lines are cut by a transversal, then consecutive exterior angles are supplementary.

So ∠1 and 8 are consecutive exterior angles (both on left side, both outside the parallel lines) → S

Similarly, ∠2 and ∠7 would be consecutive exterior on the right side → also supplementary, which matches our earlier conclusion for #5.

So #10: ∠1 and 8 → S

Now list all answers for Test I:

1. ∠3 and 5 → C
2. 2 and ∠6 → C
3. ∠4 and 5 → S
4. 1 and ∠7 → C
5. ∠2 and 7 → S
6. 4 and ∠6 → C
7. ∠1 and 5 → C
8. 3 and ∠6 → S
9. ∠2 and 8 → C
10. ∠1 and ∠8 → S

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Test II: Given m ∥ n, and one angle is 70°

Diagram: line m (top), line n (bottom), transversal t.

At top intersection (line m):

- Angles: 1,2,7, and the 70° angle.

The 70° angle is labeled next to angle 2? Looking at the diagram description:

It says: "1 / 70°" and then "7 2" — probably meaning:

At top intersection:

- Above line m, left of transversal: ∠1
- Above line m, right of transversal: 70° (so this is not numbered, but given as 70°)
- Below line m, left of transversal: ∠7
- Below line m, right of transversal: ∠2

Similarly, at bottom intersection (line n):

- Above line n, left: ∠6
- Above line n, right: ∠3
- Below line n, left: ∠5
- Below line n, right: ∠4

And it's given that the angle above line m on the right is 70°. So that angle is adjacent to ∠2 and ∠1.

Specifically, the 70° angle and ∠2 are vertical? Or adjacent?

If the 70° is above line m on the right, and ∠2 is below line m on the right, then they are vertical angles? No — vertical angles are opposite.

Actually, at the top intersection, the four angles are:

- North-West: ∠1
- North-East: 70°
- South-East: ∠2
- South-West: ∠7

So:

- ∠1 and 2 are not directly related, but ∠1 and the 70° angle are adjacent on the top side → they form a linear pair? Yes, because they are on a straight line (line m).

Line m is straight, so angles on one side of the transversal on line m should add to 180°.

Specifically, ∠1 and the 70° angle are adjacent and on the straight line m → so ∠1 + 70° = 180° → ∠1 = 110°

Similarly, ∠2 and the 70° angle are vertical? No — ∠2 is south-east, 70° is north-east → they are adjacent vertically? Actually, ∠2 and the 70° angle are on the same side of the transversal (right) but on opposite sides of line m → so they are vertical angles? Let's see: the transversal crosses line m, so the angle above and below on the same side are not vertical.

Vertical angles are opposite each other across the intersection.

So at top intersection:

- Vertical pairs:
- ∠1 and ∠2? No — ∠1 is NW, ∠2 is SE → yes, they are vertical! Because NW and SE are opposite.
- Similarly, 70° (NE) and ∠7 (SW) are vertical.

Is that correct? In standard labeling, if you have two lines crossing, vertical angles are opposite.

So if the transversal and line m cross, then:

- Angle in quadrant I (NE) and quadrant III (SW) are vertical.
- Quadrant II (NW) and IV (SE) are vertical.

So:

- NE: 70°
- SW: ∠7 → so ∠7 = 70° (vertical angles)
- NW: ∠1
- SE: ∠2 → so ∠1 and 2 are vertical → 1 = ∠2

But also, ∠1 and 70° are adjacent on the straight line m → so ∠1 + 70° = 180° → ∠1 = 110°

Then since ∠1 and ∠2 are vertical, ∠2 = 1 = 110°

But that can't be because if ∠1 = 110° and ∠2 = 110°, and they are vertical, that's fine, but then the adjacent angles: ∠1 and 70° are adjacent and should sum to 180° — 110 + 70 = 180, good.

∠2 and 70° are also adjacent? ∠2 is SE, 70° is NE — they share the ray along the transversal upward, but on line m, they are on different sides.

Actually, ∠2 and the 70° angle are adjacent angles that form a linear pair along the transversal? No.

Let's clarify:

At the intersection of line m and transversal t:

The four angles are:

- Between line m (left part) and transversal (up): ∠1
- Between line m (right part) and transversal (up): 70°
- Between line m (right part) and transversal (down): ∠2
- Between line m (left part) and transversal (down): ∠7

So, the straight line m means that the angles on one side of the transversal along line m sum to 180°.

For example, on the upper side of line m: ∠1 and 70° are adjacent and together make the straight angle along line m → so ∠1 + 70° = 180° → ∠1 = 110°

On the lower side of line m: ∠7 and 2 are adjacent and should sum to 180° → ∠7 + ∠2 = 180°

Also, vertical angles:

- ∠1 and 2 are vertical? 1 is upper-left, ∠2 is lower-right — yes, they are vertical angles → so ∠1 = ∠2 = 110°

Then from ∠7 + ∠2 = 180°, ∠7 + 110° = 180° → ∠7 = 70°

Which matches the other vertical pair: 70° (upper-right) and ∠7 (lower-left) are vertical → so ∠7 = 70°, consistent.

So summary for top intersection:

- ∠1 = 110°
- ∠2 = 110° (vertical to ∠1)
- ∠7 = 70° (vertical to the given 70°)
- Given angle = 70°

Now, since m ∥ n, we can find angles at bottom intersection using corresponding, alternate, etc.

Given angle at top is 70° (upper-right). This corresponds to which angle at bottom?

Corresponding angles: same relative position.

The 70° angle is at top, above line m, right of transversal.

Corresponding angle at bottom would be above line n, right of transversal → that is ∠3.

So ∠3 = 70° (corresponding angles)

Then, at bottom intersection:

- ∠3 = 70°
- ∠4 is vertical to ∠3? ∠3 is upper-right, ∠4 is lower-right → not vertical.

At bottom intersection:

Angles:

- Upper-left: ∠6
- Upper-right: ∠3
- Lower-right: ∠4
- Lower-left: ∠5

Vertical angles:

- ∠3 and ∠5 are vertical? ∠3 is UR, ∠5 is LL → yes, vertical → so ∠5 = ∠3 = 70°
- ∠6 and ∠4 are vertical → ∠6 = ∠4

Also, linear pairs:

- ∠3 and 4 are adjacent on the right side → ∠3 + ∠4 = 180° → 70° + ∠4 = 180° → ∠4 = 110°
- Then ∠6 = ∠4 = 110° (vertical)
- ∠5 = 70° (as above)
- ∠6 and ∠5 are adjacent on the left side → ∠6 + 5 = 110° + 70° = 180°, good.

Now, we can answer all questions.

Recall:

Top intersection (line m):

- ∠1 = 110°
- ∠2 = 110°
- ∠7 = 70°
- Given 70° (not numbered, but we don't need it for answers)

Bottom intersection (line n):

- ∠3 = 70° (corresponding to given 70°)
- ∠4 = 110° (linear pair with ∠3)
- ∠5 = 70° (vertical to ∠3)
- ∠6 = 110° (vertical to ∠4)

Now answer Test II:

11. m∠1 = 110°
12. m∠2 = 110°
13. m∠3 = 70°
14. m∠4 = 110°
15. m∠5 = 70°
16. m∠6 = 110°
17. m∠7 = 70°
18. m∠7 + m∠2 = 70° + 110° = 180°
19. m∠2 + m∠3 = 110° + 70° = 180°
20. m∠1 + m∠5 = 110° + 70° = 180°

All seem consistent.

Now, compile final answers.

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Final Answer:
Test I:
1. C
2. C
3. S
4. C
5. S
6. C
7. C
8. S
9. C
10. S

Test II:
11. 110°
12. 110°
13. 70°
14. 110°
15. 70°
16. 110°
17. 70°
18. 180°
19. 180°
20. 180°