Geometry worksheet on parallel lines cut by a transversal, featuring angle calculations and relationship identification.

We are given that lines y and z are parallel, and they are cut by two transversals: line b (on the left) and line a (on the right). We must find the measures of all missing angles and name the relationships between specific angle pairs.

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## Step 1: Understand the diagram

There are two horizontal parallel lines:
- Top line: y
- Bottom line: z

Two transversals cross them:
- Line b (left diagonal) intersects y and z.
- Line a (right diagonal) intersects y and z.

The angles are labeled ∠1 through ∠18.

We’re given:
- At the top left intersection (line b and line y): 74° is shown — this is the angle above line y and to the left of line b. Looking at the diagram, this is ∠2.
- At the bottom left intersection (line a and line z): 39° is shown — this is the angle below line z and to the left of line a. Looking at the diagram, this is ∠14.

So we know:
> ∠2 = 74°
> ∠14 = 39°

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## Step 2: Use angle relationships with parallel lines

A. Angles formed by transversal b cutting parallel lines y and z

At the top intersection (line b and line y):

- ∠2 = 74° (given)
- ∠1 and ∠2 are adjacent supplementary angles (they form a straight line), so:
> ∠1 = 180° - 74° = 106°

- ∠3 is vertically opposite to ∠1 → so ∠3 = ∠1 = 106°
- ∠2 and ∠3 are also adjacent supplementary → 74° + 106° = 180° ✔️

At the bottom intersection (line b and line z):

Since lines y and z are parallel, and line b is a transversal:

- ∠2 and ∠16 are corresponding angles → so ∠16 = ∠2 = 74°
- ∠3 and ∠15 are corresponding angles → so ∠15 = ∠3 = 106°
- ∠1 and ∠17 are corresponding angles → so ∠17 = ∠1 = 106°
- ∠16 and ∠17 are adjacent supplementary → 74° + 106° = 180° ✔️
- ∠15 and ∠16 are adjacent supplementary → 106° + 74° = 180° ✔️

Also, vertical angles at bottom intersection:

- ∠15 and ∠18 are vertical → ∠18 = ∠15 = 106°
- ∠16 and ∠17 are vertical? Wait — let’s check:

Actually, at the bottom intersection of line b and line z:

The four angles are:
- ∠15 (top-left)
- ∠16 (top-right)
- ∠17 (bottom-right)
- ∠18 (bottom-left)

So:
- ∠15 and ∠17 are vertical → ∠17 = ∠15 = 106°
- ∠16 and ∠18 are vertical → ∠18 = ∠16 = 74°

Wait — this contradicts our earlier assignment.

Let me reorient.

Looking at the diagram:

At the bottom intersection (line b and line z), the angles are labeled clockwise starting from top-left:

- ∠15 (above line z, left of line b)
- ∠16 (above line z, right of line b)
- ∠17 (below line z, right of line b)
- ∠18 (below line z, left of line b)

So:
- ∠15 and ∠17 are vertical → equal
- ∠16 and ∠18 are vertical → equal

And since line z is straight:
- ∠15 + ∠16 = 180°
- ∠17 + ∠18 = 180°

From corresponding angles with top:
- ∠15 corresponds to ∠3 → ∠15 = 106° → so ∠17 = 106°
- ∠16 corresponds to ∠2 → ∠16 = 74° → so ∠18 = 74°

✔️ Consistent.

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B. Angles formed by transversal a cutting parallel lines y and z

Given: ∠14 = 39°

At the bottom intersection (line a and line z):

- ∠14 = 39° (given) — this is the angle below line z and to the left of line a → so it's the bottom-left angle → ∠14
- ∠13 is adjacent to ∠14 on the straight line → so ∠13 = 180° - 39° = 141°
- ∠12 is vertically opposite to ∠14 → ∠12 = 39°
- ∠11 is vertically opposite to ∠13 → ∠11 = 141°

Now, use parallel lines:

Transversal a cuts parallel lines y and z.

Corresponding angles:

- ∠14 corresponds to ∠7 → ∠7 = ∠14 = 39°
- ∠13 corresponds to ∠6 → ∠6 = ∠13 = 141°
- ∠12 corresponds to ∠5 → ∠5 = ∠12 = 39°
- ∠11 corresponds to ∠4 → ∠4 = ∠11 = 141°

Check top intersection (line a and line y):

Angles:
- ∠4 (top-left)
- ∠5 (top-right)
- ∠6 (bottom-right)
- ∠7 (bottom-left)

Vertical angles:
- ∠4 and ∠6 are vertical → should be equal → 141° = 141° ✔️
- ∠5 and ∠7 are vertical → 39° = 39° ✔️

Adjacent angles:
- ∠4 + ∠5 = 141° + 39° = 180° ✔️

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C. Angles at the center intersection (where lines a and b cross)

This is where transversals a and b intersect each other, between lines y and z.

The angles here are ∠8, ∠9, ∠10, ∠11.

We already found ∠11 = 141° (from above).

Note: ∠11 is part of the intersection of transversals a and b.

At the center intersection:

- ∠9 and ∠11 are vertical angles → so ∠9 = ∠11 = 141°
- ∠8 and ∠10 are vertical angles
- Also, ∠8 and ∠9 are adjacent → sum to 180° → so ∠8 = 180° - 141° = 39°
- Thus, ∠10 = ∠8 = 39°

✔️ Check: ∠9 + ∠10 = 141° + 39° = 180° ✔️

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## Step 3: Compile all angle measures

Let’s list them in order:

Top row (line y):

- ∠1 = 106°
- ∠2 = 74°
- ∠3 = 106°
- ∠4 = 141°
- ∠5 = 39°
- ∠6 = 141°
- ∠7 = 39°

Center intersection (transversals a and b):

- ∠8 = 39°
- ∠9 = 141°
- ∠10 = 39°
- ∠11 = 141°

Bottom row (line z):

- ∠12 = 39°
- ∠13 = 141°
- ∠14 = 39°
- ∠15 = 106°
- ∠16 = 74°
- ∠17 = 106°
- ∠18 = 74°

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## Step 4: Name the relationships for the angle pairs

a. ∠9 & ∠11

These are vertical angles — they are opposite each other at the intersection of two lines (transversals a and b).

✔ Answer: Vertical Angles

b. ∠15 & ∠3

∠15 is at bottom left (line b and z), ∠3 is at top left (line b and y).

They are on the same side of transversal b, and both are between the parallel lines y and z → so they are alternate interior angles.

Alternate interior angles are equal when lines are parallel → indeed, both are 106°.

✔ Answer: Alternate Interior Angles

c. ∠14 & ∠5

∠14 is at bottom left (line a and z), ∠5 is at top right (line a and y).

They are on opposite sides of transversal a, and both are outside the parallel lines → so they are alternate exterior angles.

Alternate exterior angles are equal when lines are parallel → both are 39°.

✔ Answer: Alternate Exterior Angles

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## ✔ Final Answers:

Angle Measures:

- ∠1 = 106°
- ∠2 = 74°
- ∠3 = 106°
- ∠4 = 141°
- ∠5 = 39°
- ∠6 = 141°
- ∠7 = 39°
- ∠8 = 39°
- ∠9 = 141°
- ∠10 = 39°
- ∠11 = 141°
- ∠12 = 39°
- ∠13 = 141°
- ∠14 = 39°
- ∠15 = 106°
- ∠16 = 74°
- ∠17 = 106°
- ∠18 = 74°

Angle Pair Relationships:

a. ∠9 & ∠11 → Vertical Angles
b. ∠15 & ∠3 → Alternate Interior Angles
c. ∠14 & ∠5 → Alternate Exterior Angles

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✔ All solved with clear reasoning using properties of parallel lines and transversals.