select all the pairs of alternate interior angles.
∠dcf and ∠efc ∠gfc and ∠bca
∠dca and ∠bca ∠gfc and ∠dcf

Question

select all the pairs of alternate interior angles.
∠dcf and ∠efc  ∠gfc and ∠bca
∠dca and ∠bca  ∠gfc and ∠dcf

Accepted Answer

# Explanation:
## Step1: Recall Alternate Interior Angles Definition
Alternate interior angles are formed when a transversal crosses two parallel lines. They lie between the two lines and on opposite sides of the transversal.

## Step2: Analyze Each Pair
- **∠DCF and ∠EFC**: Transversal is $ HF $ (or $ HA $), lines are $ EG $ and $ BD $. ∠DCF and ∠EFC are between $ EG $ and $ BD $, on opposite sides of transversal. So they are alternate interior angles.
- **∠GFC and ∠BCA**: ∠GFC is on one side, ∠BCA is not between the two lines (or wrong sides), so not alternate interior.
- **∠DCA and ∠BCA**: These are adjacent angles, not formed by transversal crossing two lines (they share a common side and vertex, not alternate interior).
- **∠GFC and ∠DCF**: Transversal is $ HF $ (or $ HA $), lines are $ EG $ and $ BD $. ∠GFC and ∠DCF are between $ EG $ and $ BD $, on opposite sides of transversal. So they are alternate interior angles. Wait, wait, let's re - check. Wait, maybe I made a mistake earlier. Wait, the two lines are $ EG $ (with points $ E, F, G $) and $ BD $ (with points $ B, C, D $), and transversal is $ HF A $ (the line with $ H, F, C, A $).

Wait, let's re - evaluate each option:

1. **∠DCF and ∠EFC**: Line $ EG $ ( $ E - F - G $) and line $ BD $ ( $ B - C - D $), transversal $ H - F - C - A $. ∠DCF is at $ C $ (between $ DC $ and $ CF $), ∠EFC is at $ F $ (between $ EF $ and $ FC $). They are between the two lines ($ EG $ and $ BD $) and on opposite sides of transversal $ FC $. So this is a valid pair.

2. **∠GFC and ∠BCA**: ∠GFC is at $ F $ (between $ GF $ and $ FC $), ∠BCA is at $ C $ (between $ BC $ and $ CA $). $ CA $ is not part of the transversal in the same way, and ∠BCA is not between the two lines $ EG $ and $ BD $. So not alternate interior.

3. **∠DCA and ∠BCA**: These two angles share the side $ CA $ and vertex $ C $, they are adjacent angles forming a linear pair? No, they are adjacent, not alternate interior.

4. **∠GFC and ∠DCF**: Line $ EG $ ( $ G - F - E $) and line $ BD $ ( $ D - C - B $), transversal $ H - F - C - A $. ∠GFC is at $ F $ (between $ GF $ and $ FC $), ∠DCF is at $ C $ (between $ DC $ and $ CF $). They are between the two lines ($ EG $ and $ BD $) and on opposite sides of transversal $ FC $. Wait, but earlier I thought maybe, but let's check the definition again. Alternate interior angles: when two parallel lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines.

Wait, maybe the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF? Wait, no, maybe I messed up. Wait, let's look at the diagram again (mentally). The two lines are $ EG $ (going from $ E $ down to $ G $ through $ F $) and $ BD $ (going from $ B $ down to $ D $ through $ C $). The transversal is $ H - F - C - A $.

For ∠DCF and ∠EFC: $ EF $ is part of $ EG $, $ DC $ is part of $ BD $, transversal $ FC $. So between $ EG $ and $ BD $, alternate sides of $ FC $. So this is good.

For ∠GFC and ∠DCF: $ GF $ is part of $ EG $, $ DC $ is part of $ BD $, transversal $ FC $. Between $ EG $ and $ BD $, alternate sides of $ FC $. Wait, but maybe the original problem's correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF? Wait, no, let's check the options again. Wait, maybe I made a mistake in the first analysis. Let's re - check the options:

The options are:

- ∠DCF and ∠EFC

- ∠GFC and ∠BCA

- ∠DCA and ∠BCA

- ∠GFC and ∠DCF

Wait, maybe the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF? Wait, no, let's think again. The definition of alternate interior angles: two angles that lie between the two lines (the "interior") and on opposite sides of the transversal (the "alternate").

So for the two lines $ EG $ ( $ E, F, G $) and $ BD $ ( $ B, C, D $), and transversal $ H, F, C, A $:

- ∠DCF: between $ DC $ (part of $ BD $) and $ CF $ (transversal), at $ C $

- ∠EFC: between $ EF $ (part of $ EG $) and $ FC $ (transversal), at $ F $

These are between $ EG $ and $ BD $, alternate sides of transversal $ FC $. So this is a pair.

- ∠GFC: between $ GF $ (part of $ EG $) and $ FC $ (transversal), at $ F $

- ∠DCF: between $ DC $ (part of $ BD $) and $ CF $ (transversal), at $ C $

These are also between $ EG $ and $ BD $, alternate sides of transversal $ FC $. Wait, but the option ∠GFC and ∠DCF is there. But wait, maybe the lines are $ EG $ and $ BD $, transversal $ HFCA $. So ∠GFC and ∠DCF: ∠GFC is on the left - hand side of transversal (at $ F $), ∠DCF is on the right - hand side of transversal (at $ C $), between the two lines. So yes, alternate interior.

But wait, the first option ∠DCF and ∠EFC: ∠EFC is at $ F $, between $ EF $ and $ FC $, ∠DCF is at $ C $, between $ DC $ and $ FC $. So between the two lines, alternate sides.

Wait, but maybe I made a mistake with ∠GFC and ∠BCA. ∠BCA is at $ C $, between $ BC $ and $ CA $, $ CA $ is not part of the transversal in the same way, and ∠BCA is not between the two lines $ EG $ and $ BD $.

And ∠DCA and ∠BCA: they are adjacent angles, not alternate interior.

So the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF? Wait, but let's check the original problem again. Wait, maybe the two lines are $ EB $ and $ EG $? No, $ EB $ and $ BD $? Wait, the diagram has $ E, B $ as two parallel lines (since they have the same direction arrows), $ G, D $ as the other ends. So the two parallel lines are $ EB $ ( $ E - B $) and $ GD $ ( $ G - D $)? Wait, no, the lines are $ EG $ ( $ E - F - G $) and $ BD $ ( $ B - C - D $), with $ EB $ and $ GD $ as the "top" and "bottom" parts.

Wait, maybe I misidentified the two lines. Let's assume the two parallel lines are $ EG $ (passing through $ E, F, G $) and $ BD $ (passing through $ B, C, D $), and transversal is $ HFCA $ (passing through $ H, F, C, A $).

So alternate interior angles are pairs of angles that:

- Lie between the two parallel lines ( $ EG $ and $ BD $)

- Lie on opposite sides of the transversal ( $ HFCA $)

So:

1. ∠DCF (at $ C $, between $ DC $ and $ CF $) and ∠EFC (at $ F $, between $ EF $ and $ FC $): between $ EG $ and $ BD $, opposite sides of $ CF $ (transversal segment), so alternate interior.

2. ∠GFC (at $ F $, between $ GF $ and $ FC $) and ∠DCF (at $ C $, between $ DC $ and $ FC $): between $ EG $ and $ BD $, opposite sides of $ CF $, so alternate interior.

3. ∠GFC and ∠BCA: ∠BCA is not between $ EG $ and $ BD $, so no.

4. ∠DCA and ∠BCA: adjacent angles, not alternate interior.

But the options are:

- ∠DCF and ∠EFC

- ∠GFC and ∠BCA

- ∠DCA and ∠BCA

- ∠GFC and ∠DCF

So the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF? But maybe the answer is ∠DCF and ∠EFC, and ∠GFC and ∠DCF. Wait, but let's check again. Wait, maybe the two lines are $ EB $ and $ GD $, with transversal $ HFCA $. Then ∠EFC (at $ F $, between $ EF $ and $ FC $) and ∠DCF (at $ C $, between $ DC $ and $ FC $): between $ EB $ and $ GD $, opposite sides of transversal, so alternate interior. ∠GFC (at $ F $, between $ GF $ and $ FC $) and ∠DCF (at $ C $, between $ DC $ and $ FC $): also between $ EB $ and $ GD $, opposite sides of transversal.

But maybe the first option ∠DCF and ∠EFC is correct, and ∠GFC and ∠DCF is correct. But let's see the options again. Wait, maybe I made a mistake with ∠GFC and ∠DCF. Let's check the angles:

∠GFC: formed by $ GF $ and $ FC $

∠DCF: formed by $ DC $ and $ FC $

The transversal is $ FC $, and the two lines are $ GF $ (part of $ EG $) and $ DC $ (part of $ BD $). So yes, alternate interior.

∠DCF and ∠EFC: formed by $ DC $ and $ FC $ (at $ C $) and $ EF $ and $ FC $ (at $ F $), transversal $ FC $, two lines $ EF $ (part of $ EG $) and $ DC $ (part of $ BD $), so alternate interior.

So the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF. But wait, the option ∠GFC and ∠BCA is wrong, ∠DCA and ∠BCA is wrong.

Wait, maybe the initial analysis was wrong. Let's use the definition strictly: Alternate interior angles are two angles that lie between the two lines (the "interior") and on opposite sides of the transversal (the "alternate").

So for two parallel lines $ l $ and $ m $, cut by transversal $ t $, alternate interior angles are ∠1 and ∠2 where ∠1 is between $ l $ and $ m $, on one side of $ t $, and ∠2 is between $ l $ and $ m $, on the other side of $ t $.

So in the diagram, let's assume the two parallel lines are $ EG $ ( $ E - F - G $) and $ BD $ ( $ B - C - D $), transversal $ HFCA $ ( $ H - F - C - A $).

- ∠DCF: between $ BD $ ( $ D - C - B $) and transversal $ HFCA $, at $ C $

- ∠EFC: between $ EG $ ( $ E - F - G $) and transversal $ HFCA $, at $ F $

- ∠GFC: between $ EG $ ( $ G - F - E $) and transversal $ HFCA $, at $ F $

- ∠DCF: between $ BD $ ( $ D - C - B $) and transversal $ HFCA $, at $ C $

So ∠DCF (at $ C $) and ∠EFC (at $ F $): between the two lines, opposite sides of transversal → alternate interior.

∠GFC (at $ F $) and ∠DCF (at $ C $): between the two lines, opposite sides of transversal → alternate interior.

∠GFC and ∠BCA: ∠BCA is not between the two lines, so no.

∠DCA and ∠BCA: adjacent, not alternate interior.

So the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF. But let's check the options again. The options are:

1. ∠DCF and ∠EFC

2. ∠GFC and ∠BCA

3. ∠DCA and ∠BCA

4. ∠GFC and ∠DCF

So the correct ones are 1 (∠DCF and ∠EFC) and 4 (∠GFC and ∠DCF). But maybe I made a mistake with ∠GFC and ∠DCF. Wait, ∠GFC is at $ F $, between $ GF $ and $ FC $, ∠DCF is at $ C $, between $ DC $ and $ FC $. So the two angles are on opposite sides of the transversal ( $ FC $) and between the two lines ( $ EG $ and $ BD $). So yes, alternate interior.

But let's confirm with a simple example. If we have two parallel lines and a transversal, alternate interior angles are congruent. So ∠DCF and ∠EFC: if $ EG \parallel BD $, these should be congruent. ∠GFC and ∠DCF: also congruent if $ EG \parallel BD $.

So the answer should be ∠DCF and ∠EFC, and ∠GFC and ∠DCF. But let's check the original problem's options again. Wait, maybe the lines are $ EB $ and $ GD $, with $ EB $ being $ E - B $ and $ GD $ being $ G - D $, and transversal $ HFCA $. Then ∠EFC (at $ F $, between $ EF $ and $ FC $) and ∠DCF (at $ C $, between $ DC $ and $ FC $): between $ EB $ and $ GD $, opposite sides of transversal. ∠GFC (at $ F $, between $ GF $ and $ FC $) and ∠DCF (at $ C $, between $ DC $ and $ FC $): between $ EB $ and $ GD $, opposite sides of transversal.

So the correct pairs are ∠DCF and ∠EFC, and ∠GFC and ∠DCF.
# Answer:
∠DCF and ∠EFC, ∠GFC and ∠DCF

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Alternate Interior Angles Definition

Step2: Analyze Each Pair

Answer: