Practice worksheet for identifying and calculating angles formed by parallel lines cut by a transversal.

Let's solve this step-by-step.

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Part A: Find the measure of each of the specified angles

We'll use properties of parallel lines cut by a transversal, including:

- Corresponding angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Consecutive (same-side) interior angles are supplementary (add to 180°).
- Vertical angles are equal.
- Linear pairs add up to 180°.

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#### Problem 1)

Given:
- Lines AD and EH are parallel (assumed from diagram).
- Transversal CF cuts them.
- ∠BFC = 75°

We need to find:
- ∠HFC
- ∠HFG

##### Step 1: ∠HFC

∠BFC and ∠HFC are vertical angles (they are opposite each other at point F).

So,
> ∠HFC = ∠BFC = 75°

##### Step 2: ∠HFG

∠HFC and ∠HFG form a linear pair (they are adjacent and on a straight line), so they sum to 180°.

> ∠HFG = 180° – ∠HFC = 180° – 75° = 105°

✔ Answer:
- ∠HFC = 75°
- ∠HFG = 105°

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#### Problem 2)

Given:
- Lines AD and EH are parallel.
- Transversal CG cuts them.
- ∠DBC = 38°

Wait — let’s clarify the notation.

Point B is on AD, and point F is on EH. The angle given is ∠DBC = 38°.

But D–B–C is not a straight line? Wait — actually, looking at the diagram:

- Line AD is horizontal.
- Line CG is the transversal going through B and F.
- ∠DBC is the angle at B between D–B and B–C.

But since AD is a straight line, and CB is going upward, ∠DBC is an angle formed at point B between line AD and transversal BC.

Wait — but we're told ∠DBC = 38°, which is likely the angle at B between DB and BC.

Now, since AD || EH, and transversal is CG, then:

- ∠DBC = 38° → this is an angle between the top line and the transversal.

We need:
- ∠HFC
- ∠DBC

Wait — ∠DBC is already given as 38°, so we just write that.

But let's double-check.

Actually, in the diagram:
- ∠DBC is the angle at B between D–B and B–C → this is interior angle on the right side.
- Since AD || EH, and transversal is CF, then ∠DBC and ∠HFC are corresponding angles?

Wait — let’s analyze.

From the diagram:
- ∠DBC is above the top line AD, on the right side of transversal.
- ∠HFC is below the bottom line EH, on the right side of transversal.

So they are on the same side of the transversal, one above, one below — that makes them corresponding angles only if they are in the same relative position.

Wait — actually, ∠DBC and ∠HFC are not corresponding because they are on different sides.

Let’s look carefully.

Actually, ∠DBC is formed by line AD and transversal BC, and ∠HFC is formed by line EH and transversal FG.

Since AD || EH, and transversal is CF, then:

- ∠DBC and ∠HFC are corresponding angles if they are in the same relative position.

But ∠DBC is above AD and to the right of the transversal.
- ∠HFC is below EH and to the right of the transversal.

So yes — both are on the right side of the transversal, and one above, one below → these are corresponding angles?

No — corresponding angles are in the same relative position — e.g., both upper-right or both lower-left.

But here, ∠DBC is upper-right, and ∠HFC is lower-right → they are not corresponding.

Wait — perhaps better to think about alternate interior or something else.

Alternatively, notice that ∠DBC and ∠BFE are alternate interior angles?

Wait — maybe it's easier to use vertical angles and linear pairs.

Let’s assume:
- ∠DBC = 38° (given)
- Then, since AD is a straight line, and ∠DBC is at B, then the angle adjacent to it (on the other side of the transversal) would be supplementary?

Wait — actually, ∠DBC is part of triangle or just an angle at B.

But looking at the figure: point B is where transversal CF intersects AD.

So ∠DBC is the angle between AD and CF at point B, on the right side.

So ∠DBC = 38° → this is the angle between AD and CF.

Then, since AD || EH, and CF is the transversal, then:

- ∠DBC and ∠HFC are corresponding angles? Let’s see:

- ∠DBC is on the top line, right side of transversal.
- ∠HFC is on the bottom line, right side of transversal.

Yes! So they are corresponding angles → therefore:

> ∠HFC = ∠DBC = 38°

And ∠DBC is given as 38°, so:

> ∠DBC = 38°

✔ Answer:
- ∠HFC = 38°
- ∠DBC = 38°

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#### Problem 3)

Given:
- Lines AD and EH are parallel.
- Transversal BG cuts them.
- ∠ABF = 36° (at point B)

We need:
- ∠DBC
- ∠EFG

First, ∠ABF = 36° → this is the angle between AB and BF.

Since AD is a straight line, and ∠ABF = 36°, then the adjacent angle ∠FBD (which is ∠DBC) is:

> ∠DBC = 180° – 36° = 144°

Because AB and BD are a straight line (AD), so angles at B on a straight line sum to 180°.

So ∠DBC = 144°

Now, ∠EFG — this is the angle at F between E–F and F–G.

Since AD || EH, and transversal is BG, then:

- ∠ABF and ∠EFG are corresponding angles?

Let’s see:
- ∠ABF is at the top line, left side of transversal.
- ∠EFG is at the bottom line, left side of transversal.

Yes — both are on the left side of the transversal, and above and below respectively.

So they are corresponding angles → therefore:

> ∠EFG = ∠ABF = 36°

✔ Answer:
- ∠DBC = 144°
- ∠EFG = 36°

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#### Problem 4)

Given:
- Lines AB and CD are parallel.
- Transversal GH cuts them.
- ∠GEB = 62° (angle at E between GE and EB)

We need:
- ∠AEH
- ∠DFG

First, ∠GEB = 62° → this is the angle between transversal GH and line AB at point E.

Since AB || CD, and GH is transversal.

Now, ∠AEH is the angle at E between AE and EH.

Note: AE is part of AB, and EH is part of GH.

So ∠AEH is adjacent to ∠GEB.

Since AB is a straight line, and GH crosses it at E, then:

- ∠AEH and ∠GEB are adjacent angles forming a linear pair.

So:
> ∠AEH = 180° – ∠GEB = 180° – 62° = 118°

Now, ∠DFG — this is at point F on line CD.

We want to find ∠DFG.

Note: ∠GEB = 62° → this is the angle between AB and GH.

Since AB || CD, then corresponding angles are equal.

∠GEB and ∠DFG are both on the right side of the transversal GH, and one is on top (AB), one on bottom (CD).

So they are corresponding angles → thus:

> ∠DFG = ∠GEB = 62°

✔ Answer:
- ∠AEH = 118°
- ∠DFG = 62°

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Part B: True or False

We have two parallel lines cut by a transversal, with angles labeled 1 through 8.

Labeling convention:

- Top line: angles 1, 2, 3, 4 (clockwise)
- Bottom line: angles 5, 6, 7, 8 (clockwise)

Standard labeling:
- At top intersection: ∠1 (top-right), ∠2 (top-left), ∠3 (bottom-left), ∠4 (bottom-right)
- At bottom intersection: ∠5 (top-right), ∠6 (top-left), ∠7 (bottom-left), ∠8 (bottom-right)

Wait — actually, standard is:

- At top: ∠1 (top-right), ∠2 (top-left), ∠3 (bottom-left), ∠4 (bottom-right)
- At bottom: ∠5 (top-right), ∠6 (top-left), ∠7 (bottom-left), ∠8 (bottom-right)

But in the diagram, the labels are:

- Top: ∠1 (top-right), ∠2 (top-left), ∠3 (bottom-left), ∠4 (bottom-right)
- Bottom: ∠5 (top-right), ∠6 (top-left), ∠7 (bottom-left), ∠8 (bottom-right)

So:

| Angle | Position |
|-------|----------|
| ∠1 | Top-right |
| ∠2 | Top-left |
| ∠3 | Bottom-left |
| ∠4 | Bottom-right |
| ∠5 | Bottom-right (on lower line) |
| ∠6 | Bottom-left (on lower line) |
| ∠7 | Top-left (on lower line) |
| ∠8 | Top-right (on lower line) |

Wait — no — actually, when two lines cross, the four angles around a point are labeled clockwise.

So at the top intersection:
- ∠1: top-right
- ∠2: top-left
- ∠3: bottom-left
- ∠4: bottom-right

At the bottom intersection:
- ∠5: top-right
- ∠6: top-left
- ∠7: bottom-left
- ∠8: bottom-right

But the diagram shows:
- At top: ∠1, ∠2, ∠3, ∠4 (around the top intersection)
- At bottom: ∠5, ∠6, ∠7, ∠8 (around the bottom intersection)

So:
- ∠1 and ∠3 are vertical angles
- ∠2 and ∠4 are vertical angles
- ∠5 and ∠7 are vertical angles
- ∠6 and ∠8 are vertical angles

Also:
- ∠1 and ∠5 are corresponding
- ∠2 and ∠6 are corresponding
- ∠3 and ∠7 are corresponding
- ∠4 and ∠8 are corresponding

Alternate interior:
- ∠3 and ∠6 are alternate interior (both inside, on opposite sides)
- ∠4 and ∠5 are alternate interior

Alternate exterior:
- ∠1 and ∠8
- ∠2 and ∠7

Consecutive interior:
- ∠3 and ∠5
- ∠4 and ∠6

Now evaluate each statement:

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1) ∠1 and ∠2 are vertically opposite angles.

- No. They are adjacent angles forming a linear pair.
- Vertically opposite would be ∠1 and ∠3, or ∠2 and ∠4.

→ False (F)

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2) ∠1 and ∠5 are corresponding angles.

- Yes. Both are on the right side of the transversal, and above and below the parallel lines → corresponding angles.

→ True (T)

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3) ∠2 and ∠5 are alternate exterior angles.

- ∠2 is top-left, ∠5 is bottom-right → not on the same side.

Alternate exterior angles: one on top, one on bottom, on opposite sides of transversal.

- ∠2 (top-left) and ∠8 (bottom-right) → are alternate exterior?
- ∠2 is left side, ∠8 is right side → no.

Wait: alternate exterior angles are:
- ∠2 and ∠7 (both on left side, one top, one bottom) → yes!
- ∠1 and ∠8 (both on right side, one top, one bottom) → yes!

So ∠2 and ∠5: ∠2 is top-left, ∠5 is bottom-right → they are on opposite sides of transversal, but ∠5 is on the right, while ∠2 is on the left.

So not alternate exterior.

Alternate exterior should be:
- ∠2 and ∠7 (both on left side, one top, one bottom)
- ∠1 and ∠8 (both on right side, one top, one bottom)

So ∠2 and ∠5 are not alternate exterior.

→ False (F)

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4) ∠4 and ∠6 are alternate interior angles.

- ∠4 is bottom-right (top line), ∠6 is top-left (bottom line)
- Both are inside the parallel lines.
- ∠4 is on the right, ∠6 is on the left → opposite sides of transversal → yes, alternate interior.

Wait: alternate interior angles are:
- ∠3 and ∠6 (both inside, left side)
- ∠4 and ∠5 (both inside, right side)

So ∠4 and ∠6 are on opposite sides of transversal, but both are inside.

But ∠4 is on the right, ∠6 is on the left → so they are on opposite sides → yes, alternate interior.

Wait — let's check:

- Alternate interior: inside, opposite sides.

- ∠4: bottom-right → inside, right side
- ∠6: top-left → inside, left side

Yes → opposite sides, both inside → alternate interior

So yes, they are alternate interior angles.

→ True (T)

Wait — but standard pairs are:
- ∠3 and ∠6: both on left, inside, opposite sides → yes
- ∠4 and ∠5: both on right, inside, opposite sides → yes

But ∠4 and ∠6 are not on the same side — one on right, one on left → so they are not alternate interior.

Wait — alternate interior means on opposite sides of the transversal, and between the lines.

So:
- ∠3 and ∠6: both inside, one on left, one on right → yes → alternate interior
- ∠4 and ∠5: both inside, one on right, one on left → yes → alternate interior

But ∠4 and ∠6: ∠4 is on right, ∠6 is on left → but they are not opposite sides? Actually, they are on opposite sides.

But are they alternate interior?

Wait — ∠4 and ∠6 are not adjacent to the same transversal segment.

Let’s list:

- ∠4: bottom-right at top intersection → inside, right side
- ∠6: top-left at bottom intersection → inside, left side

So they are on opposite sides of the transversal, and both inside → yes, they are alternate interior angles.

But wait — no: alternate interior angles are pairs that are on opposite sides and between the lines.

So:
- ∠3 (bottom-left, top) and ∠6 (top-left, bottom) → both on left side? No.

Wait — confusion.

Standard definition:

- Alternate interior angles: between the two lines, on opposite sides of the transversal.

So:
- ∠3 and ∠6: ∠3 is on left side, ∠6 is on left side → same side → not alternate.

Wait — no:

At top intersection:
- ∠3: bottom-left → inside, left side
- ∠4: bottom-right → inside, right side

At bottom intersection:
- ∠5: top-right → inside, right side
- ∠6: top-left → inside, left side

So:
- ∠3 and ∠6: both on left side of transversal → same side → consecutive interior
- ∠4 and ∠5: both on right side → same side → consecutive interior

Alternate interior:
- ∠3 and ∠5: ∠3 is left, ∠5 is right → opposite sides, both inside → yes
- ∠4 and ∠6: ∠4 is right, ∠6 is left → opposite sides, both inside → yes

So yes, ∠4 and ∠6 are alternate interior angles.

Wait — no: ∠4 is at top, right; ∠6 is at bottom, left.

So they are on opposite sides of transversal, both inside → yes, alternate interior.

But typically, alternate interior angles are named as:
- ∠3 and ∠5
- ∠4 and ∠6

Yes — so ∠4 and ∠6 are alternate interior angles.

So statement 4: True (T)

Wait — but ∠4 and ∠6 are not adjacent — but that’s fine.

Yes, they are alternate interior.

→ True (T)

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5) ∠3 and ∠6 are consecutive interior angles.

- Consecutive interior angles are same side, inside, and supplementary.

- ∠3: bottom-left at top → inside, left side
- ∠6: top-left at bottom → inside, left side

So both on left side of transversal, and inside → yes, they are consecutive interior angles.

→ True (T)

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6) ∠3 is congruent to ∠8.

- ∠3: bottom-left at top → inside, left side
- ∠8: top-right at bottom → outside, right side

Are they congruent?

Check relationships:

- ∠3 and ∠8: are they corresponding? No.
- Alternate? ∠3 and ∠8: ∠3 is left, ∠8 is right → opposite sides.
- But ∠3 is inside, ∠8 is outside.

Wait — ∠3 and ∠8: are they alternate exterior? No.

But ∠3 and ∠8: are they vertically opposite? No.

But let’s see:

- ∠3 and ∠6 are vertical angles? No — ∠3 and ∠1 are vertical.

Wait — at top: ∠1 and ∠3 are vertical? No — ∠1 and ∠3 are not adjacent.

Wait — angles around a point:

At top intersection:
- ∠1 and ∠3 are opposite → vertical angles? No — ∠1 and ∠3 are not directly opposite.

Wait — actually:
- ∠1 and ∠3 are vertical angles? No — ∠1 and ∠3 are adjacent.

Wait — correct:

- ∠1 and ∠3 are not vertical — they are adjacent.

Vertical angles are:
- ∠1 and ∠3? No — ∠1 and ∠3 are not opposite.

Actually, in a crossing:
- ∠1 and ∠3 are not vertical.

Wait — the angles are labeled clockwise:

- ∠1 (top-right), ∠2 (top-left), ∠3 (bottom-left), ∠4 (bottom-right)

So:
- ∠1 and ∠3 are not opposite — they are diagonal? No — ∠1 and ∠3 are separated.

Actually, vertical angles are:
- ∠1 and ∠3 → no
- ∠1 and ∠3 are not opposite.

Wait — the correct vertical pairs are:
- ∠1 and ∠3 → no
- ∠1 and ∠3 are not adjacent.

Wait — in a straight line:

At the top intersection:
- ∠1 and ∠2 are adjacent
- ∠2 and ∠3 are adjacent
- ∠3 and ∠4 are adjacent
- ∠4 and ∠1 are adjacent

So vertical angles:
- ∠1 and ∠3 → no — they are not opposite
- Actually, ∠1 and ∠3 are not vertical.

Wait — the diagonally opposite angles are:
- ∠1 and ∠3 → no — ∠1 is top-right, ∠3 is bottom-left → yes, they are diagonally opposite → so vertical angles

Similarly, ∠2 and ∠4 are vertical.

So:
- ∠1 and ∠3 are vertical → equal
- ∠2 and ∠4 are vertical → equal

At bottom:
- ∠5 and ∠7 are vertical
- ∠6 and ∠8 are vertical

Now, back to statement 6: ∠3 ≅ ∠8

- ∠3 and ∠8: are they related?

- ∠3 is at top, bottom-left
- ∠8 is at bottom, top-right

They are not vertical, not corresponding, not alternate.

But let’s see: are they alternate exterior?

- ∠3: inside, left side
- ∠8: outside, right side → no

But ∠3 and ∠8: could they be alternate interior?

No — ∠3 is inside, ∠8 is outside.

Wait — ∠8 is at bottom, top-right → outside.

So ∠3 is inside, ∠8 is outside → not same type.

But are they congruent?

Only if they are corresponding or alternate.

But ∠3 and ∠8 are not corresponding.

Corresponding angles:
- ∠1 and ∠5
- ∠2 and ∠6
- ∠3 and ∠7
- ∠4 and ∠8

Ah! So ∠3 and ∠7 are corresponding → so ∠3 = ∠7

∠4 and ∠8 are corresponding → ∠4 = ∠8

So ∠3 and ∠8 are not corresponding.

But ∠3 and ∠8 are not equal unless specific.

But the question is: is ∠3 congruent to ∠8?

Only if they are corresponding or alternate or vertical.

But they are not.

So generally, no.

But let’s see: ∠3 and ∠8 are not related by any standard rule.

So unless the diagram implies otherwise, we assume general case.

So ∠3 and ∠8 are not necessarily congruent.

But are they?

Wait — ∠3 and ∠8: are they alternate exterior?

- ∠3: inside, left
- ∠8: outside, right → no

No.

But ∠3 and ∠8: are they alternate interior? No.

So no, they are not congruent.

But wait — ∠3 and ∠7 are corresponding → so ∠3 = ∠7

∠8 and ∠6 are vertical → ∠8 = ∠6

But ∠3 and ∠6: are they equal?

Only if the lines are parallel and angles are equal.

But in general, without more info, we can't say ∠3 = ∠8.

But the statement is asking whether ∠3 is congruent to ∠8.

In general, no.

But let’s check: ∠3 and ∠8 are not corresponding, not alternate, not vertical.

So not necessarily congruent.

But the statement says "is congruent" — meaning always?

So answer is False (F)

Wait — but let’s see: is there any relation?

Actually, ∠3 and ∠8: are they alternate exterior?

No.

But ∠3 and ∠8 are not related.

So unless the lines are parallel and angles are set, we cannot assume equality.

But in the context of parallel lines, corresponding angles are equal.

But ∠3 and ∠8 are not corresponding.

So False (F)

Wait — earlier I said ∠4 and ∠8 are corresponding → so ∠4 = ∠8

And ∠3 and ∠4 are supplementary (linear pair), so ∠3 + ∠4 = 180°

So ∠3 + ∠8 = 180° → not equal unless 90°.

So ∠3 ≠ ∠8 in general.

So False (F)

✔ Answer: F

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✔ Final Answers

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Part A

1) ∠HFC = 75°, ∠HFG = 105°
2) ∠HFC = 38°, ∠DBC = 38°
3) ∠DBC = 144°, ∠EFG = 36°
4) ∠AEH = 118°, ∠DFG = 62°

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Part B

1) ∠1 and ∠2 are vertically opposite angles. → F
2) ∠1 and ∠5 are corresponding angles. → T
3) ∠2 and ∠5 are alternate exterior angles. → F
4) ∠4 and ∠6 are alternate interior angles. → T
5) ∠3 and ∠6 are consecutive interior angles. → T
6) ∠3 is congruent to ∠8. → F

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✔ Boxed Final Answer:

A)
1) ∠HFC = 75°, ∠HFG = 105°
2) ∠HFC = 38°, ∠DBC = 38°
3) ∠DBC = 144°, ∠EFG = 36°
4) ∠AEH = 118°, ∠DFG = 62°

B)
1) F
2) T
3) F
4) T
5) T
6) F