Parallel Lines Cut by a Transversal Practice Worksheet

Let’s solve each part step by step.

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Part A: Find the measure of each specified angle

We are dealing with parallel lines cut by a transversal. That means we can use rules like:

- Vertical angles are equal.
- Corresponding angles are equal.
- Alternate interior/exterior angles are equal.
- Consecutive (same-side) interior angles add to 180°.
- Linear pairs (angles on a straight line) add to 180°.

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Problem 1)

Given: ∠DBF = 75° (angle between line AD and transversal CG at point B)

We need:
→ ∠HFC
→ ∠HFG

Look at the diagram:

- Line AD || EH, cut by transversal CG.
- ∠DBF and ∠HFC are corresponding angles → so they are equal.
→ So, ∠HFC = 75°

Now, ∠HFG is next to ∠HFC on the straight line EH → they form a linear pair → sum to 180°
→ ∠HFG = 180° - 75° = 105°

✔ Answer for 1):
∠HFC = 75°, ∠HFG = 105°

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Problem 2)

Given: ∠EFB = 38° (angle between line EH and transversal CG at point F)

We need:
→ ∠HFC
→ ∠DBC

First, ∠HFC and ∠EFB are vertical angles → they are opposite each other at intersection point F → so they are equal.
→ ∠HFC = 38°

Next, ∠DBC and ∠EFB are corresponding angles (both above the parallel lines, same side of transversal) → so they are equal.
→ ∠DBC = 38°

Wait — let me double-check.

Actually, looking again: ∠EFB is below line EH, and ∠DBC is above line AD. Since AD || EH, and transversal cuts them, then yes — corresponding angles should be equal if they’re in matching positions.

But wait — ∠EFB is on the lower left, ∠DBC is on the upper right? Let me think differently.

Actually, ∠EFB and ∠ABG would be corresponding? Maybe better to use vertical or alternate.

Alternative approach:

∠EFB = 38° → its vertical angle is ∠GFH = 38° (but that’s not asked).

∠HFC is adjacent to ∠EFB on the straight line → no, actually ∠EFB and ∠HFC are vertical? Wait — points E-F-H are on one line, C-F-G on another.

At point F: angles around it.

∠EFB and ∠HFC are vertically opposite? Let's see:

Line EH and line CG cross at F.

So angles:

- ∠EFB and ∠HFC are vertical → YES → so ∠HFC = 38°

Then ∠DBC: look at point B.

Line AD and CG cross at B.

∠DBC is the angle between DB and BC.

Since AD || EH, and CG is transversal, then ∠DBC and ∠EFB are alternate exterior angles? Or corresponding?

Actually, ∠DBC and ∠EFB are both on the “outside” but on opposite sides — that makes them alternate exterior angles, which are equal when lines are parallel.

Yes! So ∠DBC = ∠EFB = 38°

✔ Answer for 2):
∠HFC = 38°, ∠DBC = 38°

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Problem 3)

Given: ∠EBF = 36° (angle between line EH and transversal CG at point B? Wait — label says 36° near point B, between EB and FB? Actually, looking at diagram: it’s angle between line AB (which is part of AD?) and transversal CG at point B — labeled as 36°.

Actually, the 36° is marked at point B, between line segment EB (which is part of line EH?) and transversal CG.

Wait — let’s clarify labels.

In problem 3:

Lines: AD and EH are parallel? Points: A-B-D on top line, E-F-H on bottom line. Transversal is C-G crossing both.

Angle given: 36° at point B, between line EB and transversal CB? Actually, it’s labeled as angle between EB and FB? No — probably angle between line AB (top line) and transversal CG at point B — but written as 36° inside triangle-like shape.

Actually, from diagram: the 36° is ∠EBF — meaning at point B, between points E, B, F. But E is on bottom line, B on top — so EB is part of transversal? Confusing.

Better interpretation: The 36° is the angle between the transversal CG and the lower parallel line EH at point F? No — it’s drawn at point B.

Wait — perhaps it’s ∠ABE = 36°? But labeled as 36° near B between the two lines.

Actually, standard way: if 36° is shown at intersection of transversal and top line, say ∠ABC = 36°, but here it’s labeled between E, B, F — which suggests it’s the angle between the transversal and the bottom line extended? This is messy.

Alternative: In many such diagrams, the angle marked at B between the transversal and the top line is given. Let’s assume the 36° is ∠ABG or something.

Looking again: the 36° is placed at point B, between line segment EB (which goes down to E on bottom line) and FB? That doesn’t make sense.

Perhaps it’s ∠EBF = 36°, where E is on bottom line, B on top, F on bottom — so EB and FB are parts of different lines.

I think there might be a mislabel, but based on common problems, likely the 36° is the angle between the transversal and the top line at point B — so ∠CBD = 36° or similar.

But the question asks for ∠DBC and EFG.

Assume: the 36° is ∠ABF or ∠EBF — let’s take it as the acute angle formed at B between the transversal and the top line.

Actually, in diagram 3, the 36° is likely ∠FBE — which is an alternate interior angle to something.

Let’s try this:

If ∠FBE = 36°, and since AD || EH, then ∠FBE and ∠DBC are vertical angles? At point B, if FBE is one angle, DBC might be opposite.

Points: at B, lines AD and CG intersect.

So angles at B: ∠ABG, ∠GBD, etc.

Perhaps ∠FBE is not at B — wait, point F is on bottom line, B on top — so EB is a segment from E to B, which is part of the transversal? Then FB is from F to B — also transversal? That can’t be.

I think I need to reinterpret.

Standard setup: two horizontal parallel lines, one slanted transversal.

In diagram 3: top line has points A, B, D; bottom line has E, F, H; transversal is C-G passing through B and F.

The 36° is marked at point B, between the transversal and the top line — specifically, between segment AB and segment CB? Or between EB and CB?

Label says "36°" near B, and lines going to E and F — but E and F are on bottom line.

Perhaps it’s ∠EBF = 36°, which is the angle at B between points E, B, F — but E and F are both on the bottom line, so EB and FB are two segments from B to the bottom line — that would mean B is connected to two points on the bottom line, which isn't typical.

This is confusing. Let me look for symmetry with other problems.

In problem 4, 62° is clearly at point G between AG and EG — so likely in problem 3, 36° is at point B between AB and CB or something.

Perhaps the 36° is ∠ABF = 36°, where F is on the bottom line, so BF is part of the transversal.

Assume that the 36° is the angle between the top line AD and the transversal CG at point B — so ∠ABG = 36° or ∠DBG = 36°.

The question asks for ∠DBC and EFG.

∠DBC is at point B, between D, B, C — so if C is on the transversal above, then ∠DBC is the angle between BD and BC.

If the 36° is the adjacent angle, then ∠DBC might be 180° - 36° = 144°, but that seems large.

Another idea: perhaps the 36° is ∠EBF, and since E and F are on the bottom line, and B on top, then EB and FB are not both present — unless it's a typo.

Let’s consider that in many textbooks, when they mark an angle at the intersection, it's between the transversal and the parallel line.

So for problem 3, let's assume the 36° is the acute angle between the transversal and the top line at B — so ∠ABG = 36°.

Then ∠DBC is the vertical angle to that? At point B, if ∠ABG = 36°, then the vertical angle is ∠DBH or something — but C is on the other side.

Perhaps ∠DBC is the same as the given angle if it's on the other side.

I recall that in some diagrams, the angle marked is the one inside the "Z" for alternate interior.

Let’s try this: if the 36° is ∠FBE, and F and E are on the bottom line, then perhaps it's the angle at B for triangle, but that doesn't help.

Perhaps it's ∠BFE = 36°, but it's labeled at B.

I think there might be a mistake in my reading. Let me search for standard interpretation.

Upon second thought, in diagram 3, the 36° is likely the angle between the transversal and the bottom line at point F, but it's drawn near B — no, it's at B.

Another approach: use the fact that ∠DBC and the 36° angle are related by parallel lines.

Suppose the 36° is ∠ABF = 36°, where F is on the bottom line, so BF is the transversal.

Then ∠ABF and ∠EFB are alternate interior angles? If AD || EH, then yes, ∠ABF = ∠EFB = 36°.

Then ∠EFG is the supplement if it's on the straight line.

At point F, on line EH, angles on a straight line sum to 180°.

If ∠EFB = 36°, then ∠BFH = 180° - 36° = 144°, but ∠EFG is probably the angle between EF and FG, which is the same as ∠EFB if G is on the extension.

Transversal is C-G, so from C to G, passing through B and F.

So at F, the transversal is FG or FC.

∠EFG is the angle between EF and FG.

If ∠EFB = 36°, and B and G are on the same line from F, then if G is on the opposite side of B from F, then ∠EFG might be the same as ∠EFB if it's the same ray, but usually G is beyond F.

Typically, the transversal is named from C to G, with C above, G below, so at F, the ray towards G is downward.

So if ∠EFB = 36°, and B is above F, then the angle between EF and the upward ray FB is 36°, so the angle between EF and the downward ray FG would be 180° - 36° = 144°, because FB and FG are opposite rays.

Yes! Because B-F-G are colinear on the transversal, with F between B and G.

So at point F, ray FB and ray FG are opposite directions.

Therefore, ∠EFB and ∠EFG are adjacent angles on a straight line → sum to 180°.

So if ∠EFB = 36°, then ∠EFG = 180° - 36° = 144°.

But what is ∠EFB? In the diagram, the 36° is marked at B, not at F.

Unless the 36° is ∠FBE, which is the same as ∠EFB if it's the angle at B in triangle, but it's not a triangle.

I think I found the issue: in diagram 3, the 36° is likely the angle at B between the top line and the transversal, so let's call it ∠ABG = 36°.

Then, since AD || EH, the corresponding angle at F would be ∠EFB = 36°.

Then, as above, ∠EFG = 180° - 36° = 144°.

Now, ∠DBC: at point B, between D, B, C.

If C is on the transversal above B, and D is on the right on the top line, then ∠DBC is the angle between BD and BC.

If ∠ABG = 36°, and A-B-D is straight, then ∠ABG and ∠DBG are adjacent, summing to 180° if G is on the other side, but typically, if G is below, then at B, the angle between AB and BG is 36°, so the angle between DB and BG would be 180° - 36° = 144°, since A-B-D is straight.

And ∠DBC is the same as ∠DBG if C is on BG.

So ∠DBC = 144°.

But let's confirm with the name: ∠DBC — points D, B, C. If C is on the transversal above B, then yes.

In many diagrams, C is the top end of the transversal, so from C to G, C above, G below, passing through B and F.

So at B, ray BC is upward, ray BD is to the right on the top line.

So if the acute angle between AB and BC is 36°, then since AB and BD are opposite, the angle between BD and BC is 180° - 36° = 144°.

Yes.

So ∠DBC = 144°.

And ∠EFG = 144°, as calculated.

But is that consistent? Let's see.

If ∠DBC = 144°, and it's on the top line, then the corresponding angle on the bottom line would be ∠HFG or something.

At F, if ∠EFB = 36° (corresponding to ∠ABG = 36°), then ∠EFG = 180° - 36° = 144°, which matches.

Also, ∠DBC and ∠EFG are both 144°, and they are corresponding angles? Let's see: ∠DBC is at B, between D and C; ∠EFG is at F, between E and G. Since D and E are on the left? No, in standard labeling, if A-B-D left to right, E-F-H left to right, then D is right, E is left, so not corresponding.

Actually, ∠DBC and HFG might be corresponding, but anyway, our calculation seems consistent.

So for problem 3:

∠DBC = 144°, ∠EFG = 144°

But let's write it.

Perhaps the 36° is intended to be the angle at B for the alternate interior.

Another way: sometimes the angle marked is the one that is alternate interior to another.

But I think 144° is correct for both.

Let me check online or standard, but since I can't, I'll go with this.

So answer for 3): ∠DBC = 144°, ∠EFG = 144°

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Problem 4)

Given: ∠AGE = 62° (at point G, between A, G, E)

We need: ∠AEH and ∠DFG

First, ∠AGE = 62° — this is at point G, between points A, G, E.

A is on top line, E on bottom line, G is on the transversal.

So likely, G is the intersection point on the top line? But in diagram, G is probably on the transversal above.

Typically, in such diagrams, the transversal intersects the top line at E? No, in problem 4, points: top line has A, E, B? Bottom line has C, F, D.

Transversal is G-H, with G above, H below, intersecting top line at E, bottom line at F.

Given ∠AGE = 62° — so at point G, but G is not on the parallel lines; it's on the transversal extension.

Probably, G is the point where the transversal starts, and it intersects the top line at E.

So ∠AGE is the angle at G between points A, G, E — but A is on the top line, E is on the top line, so if G is not on the line, then AGE is a triangle, but that doesn't make sense for parallel lines cut by transversal.

Perhaps it's a typo, and it's ∠AEG = 62° or something.

Looking at the diagram description: "62°" is marked at G, between AG and EG.

But if A and E are both on the top line, and G is off the line, then AG and EG are two lines from G to the top line, which is unusual.

Perhaps G is the intersection point on the top line. In many diagrams, the intersection points are labeled.

In problem 4, likely the transversal intersects the top line at E, and bottom line at F, and G is a point on the transversal above E, so ray GE is part of the transversal.

Then ∠AGE is the angle at G between points A, G, E — but A is on the top line, so if G is above E, and A is on the line, then GA is a line from G to A, which is not the parallel line.

This is confusing.

Perhaps "G" is the intersection point on the top line. Let's assume that in diagram 4, the transversal intersects the top line at G, and bottom line at F.

Then ∠AGE = 62° — but A and E are on the top line, so if G is on the top line, then A-G-E are colinear, so angle at G would be 180°, not 62°.

That can't be.

Unless E is not on the top line. In the label, it's "∠AGE", and in the diagram, probably E is on the transversal or something.

I think there's a standard interpretation: in such problems, when they say ∠AGE = 62°, and G is on the transversal, A on top line, E on bottom line, but then it's not at the intersection.

Perhaps it's the angle between the transversal and the top line at the intersection point.

Let's look at the name: ∠AEH and ∠DFG are asked.

∠AEH is at E, between A, E, H — so E is on the top line, H is on the transversal below.

Similarly, ∠DFG at F, between D, F, G.

So likely, the transversal intersects top line at E, bottom line at F.

Then the 62° is at G, which is probably a point on the transversal above E, so ray EG is upward.

Then ∠AGE = 62° — this is the angle at G between points A, G, E.

Since A is on the top line, and G is on the transversal above E, then GA is a line from G to A, which is not necessarily related.

This is problematic.

Perhaps "G" is a typo, and it's ∠AEG = 62°, which would be the angle at E between A, E, G.

That makes more sense.

In many diagrams, the angle at the intersection is given.

So let's assume that ∠AEG = 62°, which is the angle between the top line AE and the transversal EG at point E.

Then, since the lines are parallel, we can find other angles.

So assume ∠AEG = 62°.

Then, ∠AEH: H is on the transversal below, so if G is above, H is below, then at E, the ray EH is downward along the transversal.

So ∠AEH is the angle between AE and EH.

Since EG and EH are opposite rays (if G-E-H are colinear), then ∠AEG and ∠AEH are adjacent angles on a straight line, so they sum to 180°.

Thus, ∠AEH = 180° - 62° = 118°.

Now, ∠DFG: at F, between D, F, G.

D is on the bottom line, F on bottom line, G on transversal above.

So likely, ∠DFG is the angle between DF and FG.

Since the lines are parallel, and transversal, then ∠AEG and ∠DFG are corresponding angles? Let's see.

∠AEG is at E, between top line and transversal, on the left side.

∠DFG is at F, between bottom line and transversal, on the right side? D is probably on the right, so if the transversal is slanting, it might be alternate or corresponding.

If the transversal is cutting from top-left to bottom-right, then at E, ∠AEG is the angle on the left between top line and transversal.

At F, ∠DFG would be the angle on the right between bottom line and transversal, which might be alternate exterior or something.

Actually, ∠AEG and ∠CFG or something.

Note that ∠AEG and the angle at F on the same side are consecutive interior or something.

Better: the angle corresponding to ∠AEG at F would be ∠CFE or ∠DFH, depending on labeling.

Assume that at F, the angle between the bottom line and the transversal on the same side as ∠AEG is equal if corresponding.

But ∠AEG is on the "upper" side, while at F, the corresponding angle would be on the "lower" side.

Specifically, if ∠AEG = 62°, and it's the acute angle, then the corresponding angle at F would be ∠EFH or something.

Let's define: at E, the angle between the top line and the transversal is 62°, say on the left side.

Then at F, the corresponding angle would be the angle between the bottom line and the transversal on the left side, which might be ∠CFE.

But the question asks for ∠DFG.

If D is on the right of F on the bottom line, and G is on the transversal above, then ∠DFG is the angle between FD and FG.

FD is to the right, FG is upward along the transversal.

So if the transversal is coming down to the right, then at F, the angle between the bottom line to the right and the transversal upward might be the supplement or something.

Perhaps ∠DFG is the vertical angle or alternate.

Another way: the angle between the bottom line and the transversal at F on the same side as the 62° at E.

Since the lines are parallel, the alternate interior angles are equal.

For example, the angle at E between the top line and transversal on the lower side is equal to the angle at F between the bottom line and transversal on the upper side.

At E, if ∠AEG = 62°, and assuming A is left, E on line, G above, then the angle on the other side, between BE and EG, would be 180° - 62° = 118°, if B is right.

Then the alternate interior angle at F would be the angle between CF and FE or something.

Let's calculate ∠DFG directly.

Suppose at F, the angle between the bottom line and the transversal.

Since the lines are parallel, the corresponding angle to ∠AEG is the angle at F on the same relative position.

If ∠AEG is the angle from the top line to the transversal, measured from the left, then at F, the corresponding angle would be from the bottom line to the transversal, measured from the left, which might be ∠CFE.

But the question asks for ∠DFG, which is from D to F to G.

If D is on the right, then ∠DFG is the angle on the right side.

So if the transversal is slanting down to the right, then at F, the angle between the bottom line to the right (FD) and the transversal upward (FG) would be the same as the angle at E between the top line to the left (EA) and the transversal upward (EG), because they are alternate exterior angles or something.

Let's think: ∠AEG and ∠DFG.

Point A and D are on opposite ends, G and F on transversal.

Actually, ∠AEG and ∠DFG are not directly related, but we can find the angle at F.

From the parallel lines, the consecutive interior angles sum to 180°.

Or, the alternate interior angles are equal.

Let me denote the angle at F between the bottom line and the transversal on the side towards C as x.

Then, since AD || CH (assuming), then the alternate interior angle to ∠AEG is the angle at F between the bottom line and the transversal on the opposite side.

Specifically, if ∠AEG = 62°, and it's on the "north-west" side, then the alternate interior angle would be on the "south-east" side at F, which might be ∠DFH or something.

Perhaps ∠DFG is the vertical angle to that.

To simplify, in many such problems, if the angle at the top is 62°, then the corresponding angle at the bottom is also 62°, but on the same side.

For ∠DFG, if G is above, and D is on the right, then if the transversal is coming from top-left to bottom-right, then at F, the angle between FD (right) and FG (up-left) would be the supplement of the acute angle.

Assume that the acute angle between the bottom line and the transversal at F is 62°, same as at E, because corresponding.

Then, if ∠DFG is the angle between FD and FG, and if FG is the ray towards G, which is up, and if the transversal is slanting, then the angle between the bottom line to the right and the transversal up might be 180° - 62° = 118°, if 62° is the acute angle on the other side.

Let's calculate.

Suppose at F, the angle between the bottom line and the transversal on the left side is 62° (corresponding to ∠AEG = 62°).

Then, since the bottom line is straight, the angle on the right side is 180° - 62° = 118°.

Now, ∠DFG: if D is on the right, and G is on the transversal above, then if the transversal is coming from G to F to H, with G above, H below, then at F, ray FG is upward, ray FD is to the right.

So the angle between them is the angle on the right side, which is 118°.

Yes.

So ∠DFG = 118°.

And earlier, ∠AEH = 118°.

So both are 118°.

Is that possible? Let's see.

∠AEH = 118°, as calculated from 180° - 62°.

∠DFG = 118°, as the supplementary angle at F.

And they are both obtuse, which makes sense.

So for problem 4: ∠AEH = 118°, ∠DFG = 118°

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Now, Part B: True or False statements.

Diagram: two parallel lines cut by a transversal, with angles numbered 1 to 8.

Standard numbering: at the top intersection, angles 1,2,3,4 with 1 and 3 vertical, 2 and 4 vertical, 1 and 2 adjacent, etc.

Usually, angle 1 is top-right, 2 top-left, 3 bottom-left, 4 bottom-right at the top intersection.

At the bottom intersection, angle 5 is top-right, 6 top-left, 7 bottom-left, 8 bottom-right.

So:

- Angles 1,2,3,4 at top line-transversal intersection.
- Angles 5,6,7,8 at bottom line-transversal intersection.

With 1 and 5 corresponding, 2 and 6, 3 and 7, 4 and 8.

Vertical angles: 1 and 3, 2 and 4, 5 and 7, 6 and 8.

Alternate interior: 3 and 6, 4 and 5.

Alternate exterior: 1 and 8, 2 and 7.

Consecutive interior: 3 and 5, 4 and 6.

Now, statements:

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

Vertically opposite means vertical angles, which are opposite each other at the intersection.

At the top intersection, ∠1 and ∠2 are adjacent, not opposite. Opposite would be ∠1 and 3, or ∠2 and ∠4.

So ∠1 and ∠2 are adjacent, form a linear pair, not vertical.

So FALSE.

2) ∠1 and ∠5 are corresponding angles.

Yes, both are on the top-right of their respective intersections. So corresponding. TRUE.

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

Alternate exterior: outside the parallel lines, on opposite sides of the transversal.

∠2 is at top-left, which is exterior if we consider the region between the lines as interior.

Typically, for two parallel lines, the angles outside are exterior.

∠2 is at the top intersection, on the left, so if the transversal is slanting, ∠2 is on the exterior side.

∠5 is at the bottom intersection, on the right, so also exterior.

Are they on opposite sides of the transversal? ∠2 is on the left side of the transversal, ∠5 is on the right side, so yes, opposite sides.

And both are exterior, so they are alternate exterior angles. TRUE.

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

Alternate interior: inside the parallel lines, on opposite sides of the transversal.

∠4 is at top intersection, bottom-right, so between the lines, on the right side.

∠6 is at bottom intersection, top-left, so between the lines, on the left side.

So yes, both interior, opposite sides, so alternate interior. TRUE.

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

Consecutive interior (also called same-side interior) are on the same side of the transversal, both interior.

∠3 is at top intersection, bottom-left, so interior, left side.

∠6 is at bottom intersection, top-left, so interior, left side.

Same side (left), both interior, so yes, consecutive interior. TRUE.

6) ∠3 is congruent to ∠8.

∠3 and ∠8: ∠3 is at top, bottom-left; ∠8 is at bottom, bottom-right.

Are they related? ∠3 and 7 are corresponding (both bottom-left), so if lines parallel, ∠3 = ∠7.

∠7 and ∠8 are adjacent, sum to 180°, so not equal unless 90°.

∠3 and ∠8: ∠3 corresponds to ∠7, ∠8 is vertical to ∠6, etc.

Actually, ∠3 and ∠8 are not necessarily equal. For example, if the transversal is not perpendicular, they are different.

Specifically, ∠3 and 8 are alternate exterior or something? ∠3 is interior, ∠8 is exterior.

∠3 and ∠8 are on opposite sides, but not both exterior or both interior.

In fact, ∠3 and 8 are supplementary if the lines are parallel, because ∠3 + ∠6 = 180° (consecutive interior), and ∠6 = ∠8 (vertical angles), so ∠3 + ∠8 = 180°, so they are supplementary, not congruent, unless 90°.

So generally, not congruent. FALSE.

So summary for B:

1) F

2) T

3) T

4) T

5) T

6) F

Now, let's compile all answers.

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Final Answers:

Part A:

1) ∠HFC = 75°, ∠HFG = 105°

2) ∠HFC = 38°, ∠DBC = 38°

3) ∠DBC = 144°, ∠EFG = 144°

4) ∠AEH = 118°, ∠DFG = 118°

Part B:

1) F

2) T

3) T

4) T

5) T

6) F

I think that's it.