Parallel and Perpendicular Lines with a Trasnversal Line worksheet ... - Free Printable

Let's solve each problem step by step using the properties of parallel lines and a transversal.

We are given that:
- Lines A and B are parallel.
- Line C is a transversal, cutting through both parallel lines.

Key angle relationships:
1. Corresponding angles are equal.
2. Alternate interior angles are equal.
3. Alternate exterior angles are equal.
4. Co-interior (consecutive) angles add up to 180°.
5. Vertically opposite angles are equal.
6. Angles on a straight line sum to 180°.

---

Problem 1

Given: Angle adjacent to ∠1 = 85°

```
1
/ \
/ \
A →-------→
\ /
\ /
↓
B →-------→
2
```

- The 85° angle and ∠1 are vertically opposite angles → so ∠1 = 85°
- ∠1 and ∠2 are corresponding angles (or alternate interior), but let's check:

Actually, ∠1 and the 85° angle are vertical angles, so ∠1 = 85°

Now, ∠2 is alternate interior to the 85° angle? Wait — let’s look carefully.

Wait: The 85° angle is above line A, on the right side of transversal C.

∠1 is below line A, on the left side → this is not vertical.

Wait — actually, in diagram 1:

- The 85° angle is at the top-right intersection.
- ∠1 is at the top-left intersection → it's adjacent to 85°, forming a straight line.

So ∠1 + 85° = 180° → ∠1 = 180° - 85° = 95°

But wait — let’s clarify:

Looking at the diagram:

- The 85° angle is formed between line A and transversal C.
- ∠1 is on the opposite side of the transversal, but same line — so they are supplementary if on a straight line.

Wait — no: the 85° and ∠1 are on the same side of the transversal, but one is above line A, one is below?

No — let's assume standard notation:

In diagram 1:

- Transversal C crosses line A at an angle of 85° (top-right).
- ∠1 is the angle below line A, on the left side of transversal → that would be vertical to the angle opposite 85°.

Actually, let's use geometry:

At the top intersection (line A and transversal):

- One angle is 85° (top-right)
- Then, vertically opposite is also 85° (bottom-left) → that’s ∠1?

Wait — ∠1 is labeled at the top-left position.

So:
- Top-right: 85°
- Top-left: ∠1
- Bottom-left: ?
- Bottom-right: ?

So:
- ∠1 and 85° are adjacent angles on a straight line → they form a straight line → sum to 180°

Therefore:
- ∠1 = 180° - 85° = 95°

Now, ∠2 is the angle at the bottom intersection, on the left side of transversal.

Since lines A and B are parallel, and transversal cuts them:

- ∠1 and ∠2 are corresponding angles? Let’s see:

∠1 is at the top-left of line A
∠2 is at the bottom-left of line B → not corresponding.

Wait — better: ∠2 is alternate interior to ∠1?

Actually, ∠1 (top-left of A) and ∠2 (bottom-left of B) are on the same side of transversal, but different lines.

But since A || B, then:

- The angle corresponding to ∠1 would be at the bottom-left of B → that’s ∠2

Yes! So ∠1 and ∠2 are corresponding angles → so ∠2 = ∠1 = 95°

Wait — but is that correct?

Wait: ∠1 is at top-left of A
∠2 is at bottom-left of B → yes, they are corresponding → so equal.

But we just found ∠1 = 95°, so ∠2 = 95°

But let’s double-check with another method.

The 85° angle is on the top-right of A.

Its alternate interior angle would be at bottom-left of B → that’s ∠2

So alternate interior angles: 85° and ∠2 → so ∠2 = 85°?

Wait — now we have a contradiction.

Let’s resolve this carefully.

Correct Approach for Problem 1:

- At the top intersection (line A):
- One angle is 85° (top-right)
- The angle opposite to it (bottom-left) is also 85° (vertically opposite)

- But ∠1 is labeled at the top-left → so it's adjacent to 85° → so ∠1 = 180° - 85° = 95°

- Now, ∠2 is at the bottom-left of line B.

- The angle at bottom-left of line A is 85° (vertically opposite to 85° top-right)

- Since A || B, the angle at bottom-left of B (∠2) is corresponding to the angle at bottom-left of A → which is 85°

So ∠2 = 85°

Wait — so ∠2 = 85°

But earlier I said ∠1 = 95°, and ∠2 = 85°

Is that consistent?

Yes.

Alternatively, ∠1 and ∠2 are co-interior angles? No — they are on the same side of transversal.

Wait — ∠1 (top-left of A) and ∠2 (bottom-left of B) — are they corresponding?

No — corresponding angles are in matching corners.

Let’s define:

- Top-right of A → corresponds to top-right of B
- Top-left of A → corresponds to top-left of B
- Bottom-left of A → corresponds to bottom-left of B
- Bottom-right of A → corresponds to bottom-right of B

But in this case, ∠1 is top-left of A → so its corresponding angle is top-left of B

But ∠2 is bottom-left of B → not the same.

So what is ∠2?

∠2 is at bottom-left of B → so it corresponds to bottom-left of A

And bottom-left of A is vertically opposite to 85° (top-right of A) → so bottom-left of A = 85°

Therefore, ∠2 = 85° (corresponding)

Now, ∠1 is top-left of A → which is adjacent to 85° → so ∠1 = 180° - 85° = 95°

So:
- ∠1 = 95°
- ∠2 = 85°

✔

---

Problem 2

Given: 122° at bottom-right of line B

Labelled:
- ∠1 at top-left of A
- ∠2 at bottom-left of B

Given: 122° is at bottom-right of B → so that's the angle between B and transversal, on the right side.

So:
- That 122° angle and ∠2 are adjacent → on a straight line → so ∠2 = 180° - 122° = 58°

Now, ∠2 is at bottom-left of B → so it's on the left side.

Its corresponding angle is at bottom-left of A → but we don’t know that yet.

But ∠1 is at top-left of A.

Let’s find ∠1.

The 122° angle is at bottom-right of B.

Its alternate interior angle would be at top-left of A → which is ∠1

Because:
- Interior angles: between the two parallel lines
- Alternate: on opposite sides of transversal

So:
- 122° is at bottom-right of B → inside, right side
- Its alternate interior is at top-left of A → inside, left side → that’s ∠1

So ∠1 = 122°

Then ∠2 = 180° - 122° = 58°

So:
- ∠1 = 122°
- ∠2 = 58°

✔

---

Problem 3

Given: 72° at top-right of A

Labelled:
- ∠1 at bottom-right of B
- ∠2 at bottom-left of B

So:
- 72° is at top-right of A → so its corresponding angle is at top-right of B → but not labeled
- But ∠1 is at bottom-right of B → which is vertically opposite to the angle at top-left of B

Better:

- 72° is at top-right of A
- Its corresponding angle is at top-right of B → but not labeled
- But ∠1 is at bottom-right of B → which is vertically opposite to the angle at top-left of B

Wait — perhaps easier:

- 72° and ∠1 are corresponding angles?

No — 72° is top-right of A
∠1 is bottom-right of B → not corresponding.

Wait — 72° is at top-right of A

Its alternate interior angle is at bottom-left of B → which is ∠2

So ∠2 = 72°

Also, ∠1 is at bottom-right of B → which is adjacent to ∠2 → so they form a straight line → ∠1 + ∠2 = 180°

So ∠1 = 180° - 72° = 108°

Alternatively, 72° and ∠1 are co-interior angles? Let’s see:

- 72° is at top-right of A
- ∠1 is at bottom-right of B → they are on the same side of transversal, and between the lines → co-interior angles

Yes! So co-interior angles add to 180° → ∠1 = 180° - 72° = 108°

And ∠2 = 72° (alternate interior)

So:
- ∠1 = 108°
- ∠2 = 72°

✔

---

Problem 4

Given: 140° at bottom-right of B

Labelled:
- ∠1 at top-right of A
- ∠2 at bottom-left of B

So:
- 140° is at bottom-right of B
- Its vertically opposite angle is at top-left of B → not labeled
- But ∠2 is at bottom-left of B → so it's adjacent to 140° → so ∠2 = 180° - 140° = 40°

Now, ∠2 is at bottom-left of B → its corresponding angle is at bottom-left of A → not labeled

But ∠1 is at top-right of A

What about ∠1?

The 140° angle is at bottom-right of B

Its alternate interior angle is at top-left of A → but ∠1 is at top-right of A

Wait — let’s think:

- 140° is at bottom-right of B → inside, right side
- Its alternate interior is at top-left of A → but ∠1 is at top-right of A → not the same

Wait — what is the corresponding angle?

- 140° is at bottom-right of B → corresponding angle is at bottom-right of A → not labeled

But ∠1 is at top-right of A → so it's vertically opposite to the angle at bottom-left of A

Wait — better:

Let’s find the angle at top-right of A (∠1)

It is corresponding to the angle at top-right of B

But we don’t have that.

But the angle at bottom-right of B is 140°

Its corresponding angle is at bottom-right of A → let's call it X

So X = 140°

Now, ∠1 is at top-right of A → adjacent to X → so ∠1 = 180° - 140° = 40°

Because they are on a straight line.

Alternatively, ∠1 and the 140° angle are co-interior angles? No — co-interior are between the lines.

Wait — ∠1 is at top-right of A → outside
140° is at bottom-right of B → inside

Better: ∠1 and the angle at top-right of B are vertically opposite?

No.

Wait — let’s use alternate interior.

The angle at bottom-right of B is 140°

Its alternate interior is at top-left of A

But ∠1 is at top-right of A

So not helpful.

But we can use:

- The angle at bottom-right of B is 140°
- Its vertically opposite angle is at top-left of B = 140°
- This 140° angle at top-left of B is corresponding to ∠1 at top-right of A? No — different positions.

Wait — corresponding angles:

- Top-left of B → corresponds to top-left of A
- Top-right of B → corresponds to top-right of A → that’s ∠1

But we don’t know top-right of B.

But top-right of B is adjacent to 140° → so top-right of B = 180° - 140° = 40°

Then, ∠1 = corresponding angle = 40°

Yes!

So:
- At B: bottom-right = 140° → so top-right = 180° - 140° = 40°
- ∠1 is at top-right of A → corresponding to top-right of B → so ∠1 = 40°

Now, ∠2 is at bottom-left of B

We already found ∠2 = 180° - 140° = 40° (adjacent angles)

So ∠2 = 40°

Alternatively, ∠2 and ∠1 are alternate interior? Let’s see:

- ∠1 = 40° at top-right of A
- ∠2 = 40° at bottom-left of B → not alternate

But they are both 40°

So final answer:

- ∠1 = 40°
- ∠2 = 40°

✔

---

✔ Final Answers:

#### 1
- Angle 1 = 95°
- Angle 2 = 85°

#### 2
- Angle 1 = 122°
- Angle 2 = 58°

#### 3
- Angle 1 = 108°
- Angle 2 = 72°

#### 4
- Angle 1 = 40°
- Angle 2 = 40°

---

Summary of Reasoning:

| Problem | Given | Key Rule Used | Answer |
|--------|-------|----------------|--------|
| 1 | 85° at top-right of A | Adjacent angles = 180°; Corresponding angles | ∠1=95°, ∠2=85° |
| 2 | 122° at bottom-right of B | Alternate interior angles; Straight line | ∠1=122°, ∠2=58° |
| 3 | 72° at top-right of A | Co-interior angles = 180°; Alternate interior | ∠1=108°, ∠2=72° |
| 4 | 140° at bottom-right of B | Adjacent angles; Corresponding angles | ∠1=40°, ∠2=40° |

Let me know if you'd like a visual explanation or diagram annotations!