Solved Two Parallel Lines Cut by a Transversal t 1/2 4/3 5/6 ... - Free Printable

Let's solve each problem step by step using geometry principles such as vertical angles, supplementary angles, corresponding angles, alternate interior angles, and parallel lines.

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1. Find m∠1

We have two intersecting lines forming four angles. One angle is labeled $67^\circ$, and ∠1 is vertically opposite to it.

- Vertical angles are equal.
- So, $m\angle 1 = 67^\circ$

✔ Answer: $67^\circ$

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2. Find $m\angle HJK$

We are given:
- A transversal crossing two lines.
- Angle at point K is $142^\circ$
- We need to find $m\angle HJK = x^\circ$

Looking at the diagram:
- The $142^\circ$ angle and $x^\circ$ appear to be on a straight line, meaning they are supplementary (they add up to $180^\circ$).

So:
$$
x + 142^\circ = 180^\circ \\
x = 180^\circ - 142^\circ = 38^\circ
$$

✔ Answer: $38^\circ$

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3. Find $m\angle 3$

We are given:
- Two parallel lines cut by a transversal.
- One angle is $111^\circ$
- ∠3 is adjacent to this angle on a straight line.

So, ∠3 and $111^\circ$ are supplementary (they form a straight line):
$$
m\angle 3 + 111^\circ = 180^\circ \\
m\angle 3 = 180^\circ - 111^\circ = 69^\circ
$$

✔ Answer: $69^\circ$

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4. Find $m\angle 4$

We have two lines intersecting, and one angle is marked with a right angle symbol (⊥), so it's $90^\circ$.
∠4 is adjacent to this right angle and forms a straight line.

So:
$$
m\angle 4 + 90^\circ = 180^\circ \\
m\angle 4 = 90^\circ
$$

Wait — actually, if one angle is $90^\circ$, then all angles formed by perpendicular lines are $90^\circ$. Since the lines are perpendicular (indicated by the square), every angle at the intersection is $90^\circ$.

✔ Answer: $90^\circ$

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5. Find $m\angle BST$

We have:
- Two parallel lines cut by a transversal.
- An angle of $138^\circ$ is shown above the top line.
- Point S is on the transversal, and we're looking for $\angle BST = x^\circ$

From the diagram:
- The $138^\circ$ angle and $x^\circ$ are same-side interior angles (consecutive interior angles) because they lie between the parallel lines and on the same side of the transversal.

Same-side interior angles are supplementary:
$$
x + 138^\circ = 180^\circ \\
x = 180^\circ - 138^\circ = 42^\circ
$$

✔ Answer: $42^\circ$

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6. Lines p and q are parallel. $ \angle 3 = 126^\circ $. Find all angles.

We have:
- Parallel lines $p$ and $q$
- Transversal $t$
- Angles labeled 1 through 8

Given: $ \angle 3 = 126^\circ $

Let’s label the angles:

```
t
/ \
/ \
/ \
p-------q
| |
| |
| |
4 1 5
3 2 6
8 7 ?
```

Actually, standard labeling:
- Above line p: ∠1 and ∠2
- Below line p: ∠3 and ∠4
- Above line q: ∠5 and ∠6
- Below line q: ∠7 and ∠8

But from the diagram:
- ∠3 is below line p, left side → adjacent to ∠1
- ∠1 is above p, left
- ∠2 is above p, right
- ∠4 is below p, right

So:
- ∠1 and ∠3 are vertical angles? No — wait, let's clarify.

Standard:
- At the top intersection (with line p):
- Top-left: ∠1
- Top-right: ∠2
- Bottom-left: ∠3
- Bottom-right: ∠4

At bottom intersection (with line q):
- Top-left: ∠5
- Top-right: ∠6
- Bottom-left: ∠7
- Bottom-right: ∠8

Now, since $p \parallel q$, and $t$ is transversal.

Given: $ \angle 3 = 126^\circ $

#### Step 1: Find related angles at the same vertex
- ∠3 and ∠1 are vertical angles → equal
- But wait: ∠1 is opposite ∠3? Actually, yes:
- ∠1 and ∠3 are vertical angles → so $ \angle 1 = 126^\circ $
- ∠3 and ∠4 are supplementary (straight line)
- $ \angle 4 = 180^\circ - 126^\circ = 54^\circ $
- ∠1 and ∠2 are supplementary → $ \angle 2 = 180^\circ - 126^\circ = 54^\circ $

So far:
- ∠1 = 126°
- ∠2 = 54°
- ∠3 = 126°
- ∠4 = 54°

Now use parallel lines properties.

#### Corresponding angles:
- ∠1 corresponds to ∠5 → $ \angle 5 = 126^\circ $
- ∠2 corresponds to ∠6 → $ \angle 6 = 54^\circ $
- ∠3 corresponds to ∠7 → $ \angle 7 = 126^\circ $
- ∠4 corresponds to ∠8 → $ \angle 8 = 54^\circ $

Alternatively, check alternate interior angles:
- ∠3 and ∠6 are alternate interior → should be equal?
- But ∠3 = 126°, ∠6 = 54° → not equal. Wait — no!

Wait! Let's double-check.

Actually:
- Alternate interior angles:
- ∠3 and ∠6 → are they alternate interior?

Yes:
- ∠3 is on the left, below p
- ∠6 is on the right, above q → that’s not alternate interior.

Wait — correct pairs:
- Alternate interior angles:
- ∠3 and ∠6 → no, wrong side
- Actually, ∠3 and ∠5 are not alternate.

Let’s define:
- Interior angles: between the two parallel lines → ∠2, ∠3, ∠5, ∠6
- Exterior: ∠1, ∠4, ∠7, ∠8

Alternate interior angles:
- ∠2 and ∠6 → both inside, on opposite sides of transversal → yes
- ∠3 and ∠5 → both inside, on opposite sides → yes

So:
- ∠2 = 54° → ∠6 = 54°
- ∠3 = 126° → ∠5 = 126°

Then:
- ∠7 is vertically opposite to ∠5 → ∠7 = 126°
- ∠8 is vertically opposite to ∠6 → ∠8 = 54°

Also:
- ∠1 = 126° (vertical to ∠3)
- ∠4 = 54° (vertical to ∠2)

So final values:

| Angle | Measure |
|-------|--------|
| ∠1 | 126° |
| ∠2 | 54° |
| ∠3 | 126° |
| ∠4 | 54° |
| ∠5 | 126° |
| ∠6 | 54° |
| ∠7 | 126° |
| ∠8 | 54° |

✔ All angles determined.

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✔ Final Answers:

1. $m\angle 1 = \boxed{67^\circ}$
2. $m\angle HJK = \boxed{38^\circ}$
3. $m\angle 3 = \boxed{69^\circ}$
4. $m\angle 4 = \boxed{90^\circ}$
5. $m\angle BST = \boxed{42^\circ}$
6. All angles:
- ∠1 = 126°, ∠2 = 54°, ∠3 = 126°, ∠4 = 54°
- ∠5 = 126°, ∠6 = 54°, ∠7 = 126°, ∠8 = 54°

Let me know if you'd like a diagram sketch or further explanation!