Angles in parallel lines worksheet with diagrams and questions.

Let's solve Question 3 from the worksheet: Find the angle $ x $ in each diagram, using properties of parallel lines and angles.

We'll go through each part step by step. The key angle rules we'll use are:

- Corresponding angles are equal (when two parallel lines are cut by a transversal).
- Alternate angles are equal.
- Co-interior (or consecutive) angles add up to $ 180^\circ $.
- Vertically opposite angles are equal.
- Angles on a straight line sum to $ 180^\circ $.

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(a)

Given:
- Line BD ∥ EG
- Angle at F is $ 125^\circ $
- We need to find angle $ x $ at point C.

Looking at the diagram:
- The $ 125^\circ $ angle is on line EG, between EF and HF.
- Since BD ∥ EG and AH is a transversal, the angle $ x $ at point C is corresponding to the angle adjacent to $ 125^\circ $ on the same side.

But wait — $ 125^\circ $ and its adjacent angle on the straight line make $ 180^\circ $. So the adjacent angle at F is:
$$
180^\circ - 125^\circ = 55^\circ
$$
This $ 55^\circ $ angle is corresponding to angle $ x $ at point C (since both are on the same side of the transversal and between the parallel lines).

So:
$$
x = 55^\circ \quad \text{(Corresponding angles)}
$$

✔ Answer: $ x = 55^\circ $

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(b)

Given:
- BD ∥ EG
- Angle at C is $ 57^\circ $
- Need to find $ x $ at F

The $ 57^\circ $ angle is at point C on line BD. It’s formed between BC and AC.

Since BD ∥ EG and AH is the transversal, the angle $ x $ at F is alternate to the $ 57^\circ $ angle.

Alternate angles are equal when lines are parallel.

So:
$$
x = 57^\circ \quad \text{(Alternate angles)}
$$

✔ Answer: $ x = 57^\circ $

---

(c)

Given:
- BD ∥ EG
- Angle at H is $ 70^\circ $
- Need to find $ x $ at F

The $ 70^\circ $ angle is at point H, below EG. It's between EH and FH.

Now, look at the angle at F — it's above EG, and on the same side as the $ 70^\circ $ angle.

But this is co-interior (same-side interior) angle with $ 70^\circ $. Co-interior angles add up to $ 180^\circ $ when lines are parallel.

So:
$$
x + 70^\circ = 180^\circ \\
x = 180^\circ - 70^\circ = 110^\circ
$$

✔ Answer: $ x = 110^\circ $
*(Reason: Co-interior angles add to $ 180^\circ $)*

---

(d)

Given:
- BD ∥ EG
- Angle at C is $ 105^\circ $
- Need to find $ x $ at F

The $ 105^\circ $ angle is at C, on line BD. It's between BC and AC.

Now, $ x $ is at F, on line EG, and is on the same side of the transversal.

But note: This $ 105^\circ $ angle is not adjacent to the transversal in the same way. Let's check what kind of angle $ x $ is.

Actually, $ x $ is alternate to the angle that is vertically opposite to $ 105^\circ $.

Wait — better: The $ 105^\circ $ angle is on the upper line, and $ x $ is on the lower line, but on the opposite side of the transversal? No — let’s clarify.

Actually, $ x $ is corresponding to the angle adjacent to $ 105^\circ $?

No — simpler: The $ 105^\circ $ angle and $ x $ are co-interior angles? Wait.

Wait: $ 105^\circ $ is on the top line, and $ x $ is on the bottom line, and they are on the same side of the transversal → so they are co-interior angles.

So:
$$
x + 105^\circ = 180^\circ \Rightarrow x = 75^\circ
$$

✔ Answer: $ x = 75^\circ $
*(Reason: Co-interior angles add to $ 180^\circ $)*

---

(e)

Given:
- BD ∥ EG
- Angle at C is $ 53^\circ $
- Need to find $ x $ at F

The $ 53^\circ $ angle is at C, between BC and AC.

Now, $ x $ is at F, on line EG, and appears to be below the transversal.

But $ x $ is vertically opposite to the angle at F that corresponds to $ 53^\circ $? Let’s think.

Actually, $ 53^\circ $ and $ x $ are alternate interior angles.

Yes! Because:
- The transversal cuts the two parallel lines.
- $ 53^\circ $ is inside the parallel lines, on one side.
- $ x $ is inside, on the other side, but same relative position.

So $ x $ is alternate to $ 53^\circ $ → they are equal.

✔ Answer: $ x = 53^\circ $
*(Reason: Alternate angles are equal)*

---

(f)

Given:
- BD ∥ EG
- Angle at F is $ 133^\circ $
- Need to find $ x $ at C

$ 133^\circ $ is at F, on line EG, between EF and HF.

This is co-interior with the angle at C on the top line.

So:
$$
x + 133^\circ = 180^\circ \Rightarrow x = 47^\circ
$$

✔ Answer: $ x = 47^\circ $
*(Reason: Co-interior angles add to $ 180^\circ $)*

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## ✔ Summary for Question 3:

| Part | Answer | Reason |
|------|--------|--------|
| (a) | $ x = 55^\circ $ | Corresponding angles; $ 180^\circ - 125^\circ = 55^\circ $ |
| (b) | $ x = 57^\circ $ | Alternate angles |
| (c) | $ x = 110^\circ $ | Co-interior angles: $ 180^\circ - 70^\circ $ |
| (d) | $ x = 75^\circ $ | Co-interior angles: $ 180^\circ - 105^\circ $ |
| (e) | $ x = 53^\circ $ | Alternate angles |
| (f) | $ x = 47^\circ $ | Co-interior angles: $ 180^\circ - 133^\circ $ |

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Now, moving to Question 4 — however, the diagrams are blurry or not clearly visible. But I can still help if you describe them or provide clearer images.

Would you like me to attempt Question 4 based on your description or try to interpret the diagrams?