Parallel Lines Cut by a Transversal | tothesquareinch - Free Printable

Let’s go step by step to solve each problem. We’re working with two parallel lines cut by a transversal (a line that crosses both). The angles formed have special relationships.

We’ll use these key facts:
- Vertical angles are opposite each other when two lines cross — they are always equal.
- Corresponding angles are in the same position at each intersection — if lines are parallel, they are equal.
- Alternate interior angles are on opposite sides of the transversal and inside the parallel lines — if lines are parallel, they are equal.
- Same-side interior angles are on the same side of the transversal and inside the parallel lines — if lines are parallel, they add up to 180°.
- If an angle is given as 112°, we can find others using these rules.

Now let’s look at the diagram mentally (even though we don’t describe it):

Angles around point where transversal meets top line:
Top left = ∠1, Top right = ∠2, Bottom right = ∠3, Bottom left = ∠4

Angles around point where transversal meets bottom line:
Top left = ∠5, Top right = ∠6, Bottom right = ∠7, Bottom left = ∠8

(Standard labeling for such diagrams.)

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Part A: True or False?

1. ∠1 and ∠8 are vertical angles.
→ Vertical angles are directly across from each other at one intersection. ∠1 and 8 are not at the same intersection — they’re on different lines. So FALSE.

2. ∠1 and ∠5 are corresponding angles.
→ Yes! Both are top-left corners of their intersections. Since lines are parallel, corresponding angles are equal → TRUE.

3. ∠4 and ∠5 are alternate exterior angles.
→ Alternate exterior angles would be outside the parallel lines and on opposite sides of the transversal. ∠4 is bottom-left (inside), ∠5 is top-left (inside) — so they’re actually alternate *interior*. Also, “exterior” means outside — neither is exterior here. FALSE.

4. ∠4 and 6 are alternate interior angles.
→ ∠4 is bottom-left (inside), ∠6 is top-right (inside) — yes, they’re on opposite sides of the transversal and inside the parallel lines → TRUE.

5. ∠4 and ∠5 are corresponding angles.
→ Corresponding angles must be in same relative position. ∠4 is bottom-left at top line; ∠5 is top-left at bottom line — not same position. They’re actually consecutive interior (same-side interior). FALSE.

6. ∠3 and ∠6 are linear angles on the same side of the transversal.
→ “Linear angles” usually means adjacent angles forming a straight line (sum to 180°). But ∠3 and ∠6 are not adjacent — they’re on different lines. Also, “linear pair” requires sharing a side. This statement is confusing/misworded. Probably meant “same-side interior”? But even then, ∠3 and 6 are not both interior — ∠3 is interior, ∠6 is interior? Wait — ∠3 is bottom-right (interior), ∠6 is top-right (interior) — same side? Yes, both on right side. But “linear angles” isn’t standard term here. Likely FALSE because terminology is wrong. Let’s say FALSE.

Wait — maybe they mean “consecutive interior” or “same-side interior”. But the phrase “linear angles on the same side” is incorrect. Linear pairs are adjacent. So FALSE.

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Part B: Fill in the blank

7. ___ ∠3 is congruent to ∠8
→ Are ∠3 and 8 congruent? ∠3 is bottom-right (top line), ∠8 is bottom-left (bottom line). Not corresponding, not vertical, not alternate. Actually, ∠3 and 7 are corresponding. ∠8 and ∠4 are corresponding. ∠3 and 8? Let’s think: ∠3 + ∠4 = 180° (linear pair), ∠4 = ∠8 (corresponding), so ∠3 + ∠8 = 180° → supplementary, not congruent. So answer is NOT — but the blank probably expects “If” or something? Wait, looking back: “___ ∠3 is congruent to ∠8” — likely asking for condition? Or maybe it’s “Is ∠3 congruent to ∠8?” and you write T/F? But instruction says “fill in the blank”.

Actually, re-reading: “Decide whether each statement is TRUE or FALSE... Then fill in the blanks.”

Looking at #7: “___ ∠3 is congruent to ∠8” — this seems like it might be missing a word. Perhaps it’s “If ___ , then ∠3 ≅ ∠8”? But no context.

Wait — perhaps it’s just asking to complete the sentence logically. Maybe it’s “∠3 is congruent to ∠8” — which is false, so maybe leave blank? That doesn’t make sense.

Alternative interpretation: Maybe the blank is for “True” or “False”? But earlier part was T/F, now it says “Fill in the blank”.

Looking at #9: “___ ∠8 and 2 are alternate exterior angles.” — again, likely expecting “True” or “False” in the blank? But that would duplicate Part A.

Perhaps the blanks are for words like “always”, “sometimes”, “never”? Or maybe it’s a typo and should be T/F.

Given the structure, I think for #7–10, we are to state whether the statement is true or false, and put that in the blank.

So:

7. False — ∠3 is NOT congruent to ∠8 (they are supplementary)

8. True — ∠4 and ∠7 are corresponding angles? ∠4 is bottom-left (top line), ∠7 is bottom-right (bottom line)? No — corresponding would be ∠4 and 8 (both bottom-left). ∠4 and ∠7: ∠4 is interior, ∠7 is interior, but on same side? ∠4 left, ∠7 right — so alternate interior? Yes! ∠4 and 7 are alternate interior angles → and since lines are parallel, they are congruent. So statement “∠4 and ∠7 are corresponding angles” — is that true? No! Corresponding would be same position. ∠4 is bottom-left, ∠7 is bottom-right — not corresponding. So FALSE.

Wait — let's clarify:

Corresponding angles:
- ∠1 & ∠5 (top-left)
- ∠2 & ∠6 (top-right)
- ∠3 & ∠7 (bottom-right)
- ∠4 & 8 (bottom-left)

So ∠4 and ∠7 are NOT corresponding — they are alternate interior? ∠4 is bottom-left, ∠7 is bottom-right — same side? No, opposite sides. And both interior — yes, so alternate interior → congruent, but not corresponding. So statement “∠4 and ∠7 are corresponding angles” is FALSE.

9. ∠8 and ∠2 are alternate exterior angles.
→ Alternate exterior: outside the parallel lines, opposite sides of transversal. ∠2 is top-right (exterior), ∠8 is bottom-left (exterior) — yes, opposite sides, both exterior → TRUE.

10. If the measure of ∠8 is 112°, find the missing angle measures.
→ Given ∠8 = 112°

Since lines are parallel:

- ∠4 = ∠8 (corresponding) → ∠4 = 112°
- ∠1 = ∠4 (vertical) → ∠1 = 112°
- ∠5 = ∠1 (corresponding) → ∠5 = 112°
- ∠8 and ∠7 are linear pair → ∠7 = 180° - 112° = 68°
- ∠3 = ∠7 (corresponding) → ∠3 = 68°
- ∠2 = ∠3 (vertical) → ∠2 = 68°
- ∠6 = ∠2 (corresponding) → ∠6 = 68°

So all angles:
∠1=112°, ∠2=68°, ∠3=68°, ∠4=112°, ∠5=112°, ∠6=68°, ∠7=68°, ∠8=112°

But the question says “find the missing angle measures” — probably list them all or specify which ones? Since it’s open-ended, we can state the values.

But for the blank, perhaps just note that we can find all using the rules.

However, since it’s a fill-in, maybe write the values? But space is limited.

Perhaps for #10, the blank is for the value of another angle? But it says “find the missing angle measures” — plural.

Given the format, I think for #7-9, we put True/False in the blank, and for #10, we calculate.

But to match the response, let’s assume:

For #7: False
#8: False (because ∠4 and ∠7 are not corresponding)
#9: True
#10: We need to provide the measures.

But the instruction says “fill in the blank”, so for #10, perhaps write the measures.

To simplify, let’s do:

After careful analysis:

Part A:
1. False
2. True
3. False
4. True
5. False
6. False

Part B:
7. False (since ∠3 and ∠8 are not congruent)
8. False (∠4 and ∠7 are alternate interior, not corresponding)
9. True (∠8 and ∠2 are alternate exterior)
10. If ∠8 = 112°, then:
∠1 = 112°, ∠2 = 68°, ∠3 = 68°, ∠4 = 112°, ∠5 = 112°, ∠6 = 68°, ∠7 = 68°

But since it’s a blank, perhaps for #10, we can say "All angles can be found: four are 112°, four are 68°" or list them.

Given the constraints, I'll provide the answers as per standard expectation.

Final decision:

For #7-9, the blank is for True/False.

For #10, since it says "find the missing angle measures", and ∠8 is given, we find others.

But in the answer format, we need to box the final answer.

Perhaps the entire worksheet answers are expected.

Let me compile:

──────────────────────────────────────

Final Answer:
1. False
2. True
3. False
4. True
5. False
6. False
7. False
8. False
9. True
10. ∠1 = 112°, ∠2 = 68°, ∠3 = 68°, ∠4 = 112°, ∠5 = 112°, ∠6 = 68°, ∠7 = 68°

(Note: ∠8 is given as 112°, so not missing.)