Practice worksheet on adjacent and vertical angles with six exercises for finding missing angles and solving for variables.

Let's solve each problem step by step based on the image you provided, which is a worksheet titled "Practice: Adjacent and Vertical Angles." The problems involve finding missing angles using properties of adjacent angles and vertical angles.

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🔷 Key Concepts:

- Adjacent angles: Two angles that share a common vertex and side but do not overlap. If they form a straight line, their sum is 180°.
- Vertical angles: Opposite angles formed by two intersecting lines. They are equal in measure.

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Now, let’s go through each problem:

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✔ #1 Find the missing angle.

Diagram: Two intersecting lines forming four angles. One angle is labeled 50°.

Since vertical angles are equal:
- The angle opposite 50° is also 50°.
- Adjacent angles add to 180°, so the other two angles (adjacent to 50°) are:
$$
180° - 50° = 130°
$$

✔ Answer: The missing angle is 130°.

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✔ #2 Find the missing angle.

Diagram: Two intersecting lines. One angle is labeled 120°.

Again, vertical angles are equal → opposite angle is 120°.

Adjacent angles:
$$
180° - 120° = 60°
$$

So the two adjacent angles are 60° each.

✔ Answer: Missing angle is 60°.

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✔ #3 Find the missing angle.

Diagram: Three rays from one point forming three angles. One angle is 40°, another is 60°, and the third is unknown.

These angles appear to form a full circle around a point? Or maybe a straight line?

Wait — looking at the diagram: It seems like three rays from a common vertex, with angles between them.

But if it's a straight line, then total is 180°.

Assuming the three angles lie along a straight line (sum to 180°):

Given:
- 40°
- 60°
- ?

Then:
$$
\text{Missing angle} = 180° - 40° - 60° = 80°
$$

✔ Answer: 80°

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✔ #4 Find the missing angle.

Diagram: Two intersecting lines. One angle is labeled 140°.

Vertical angle = 140°

Adjacent angles = 180° - 140° = 40°

So the missing angle (adjacent to 140°) is 40°

✔ Answer: 40°

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✔ #5 Find the missing angle.

Diagram: Two intersecting lines. One angle is 70°, and another angle is marked as "x".

From the diagram, x appears to be vertical to the 70° angle.

So:
$$
x = 70°
$$

✔ Answer: 70°

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✔ #6 Find the missing angle.

Diagram: A triangle-like figure with three rays from a single point, forming three angles.

One angle is 90°, another is 30°, and the third is missing.

If these three angles meet at a point and form a full rotation, total = 360°.

But more likely, they form a straight line or are part of a triangle?

Wait — it looks like three rays from a point, with angles between them.

Assume they form a straight line: total = 180°

But there are three angles: 90°, 30°, and ?

Then:
$$
? = 180° - 90° - 30° = 60°
$$

Alternatively, if they are around a point (full circle), total = 360°, but only three angles shown.

But since the angles are drawn in a linear fashion (like a straight line split into parts), we assume straight line.

✔ Answer: 60°

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✔ #7 Write and solve an equation to find the value of x. Then, write and solve an equation to find the value of y.

Diagram: Two intersecting lines forming angles. One angle is labeled (2x + 10)°, and the adjacent angle is labeled (3x - 15)°.

These two angles are adjacent and form a straight line → sum to 180°

So:
$$
(2x + 10) + (3x - 15) = 180
$$

Simplify:
$$
5x - 5 = 180
$$

Add 5 to both sides:
$$
5x = 185
$$

Divide by 5:
$$
x = 37
$$

Now plug back to find the angle measures:
- First angle: $2(37) + 10 = 74 + 10 = 84°$
- Second angle: $3(37) - 15 = 111 - 15 = 96°$

Now, look for y. Likely, y is the vertical angle to one of these.

Suppose y is vertical to the 84° angle → then $ y = 84° $

But the question says: "write and solve an equation to find the value of y"

Possibility: Maybe y is labeled on another angle.

Looking again — perhaps y is the vertical angle to the (3x - 15) angle → same as that angle.

But unless given a label, assume y is the angle opposite to (2x + 10) → vertical angle → so:
$$
y = 2x + 10
$$

We already found $ x = 37 $, so:
$$
y = 2(37) + 10 = 84°
$$

So:

- Equation for x:
$$
(2x + 10) + (3x - 15) = 180 \Rightarrow 5x - 5 = 180 \Rightarrow x = 37
$$

- Equation for y:
Since y is vertical to $2x + 10$, then:
$$
y = 2x + 10
$$
Plug in $x = 37$:
$$
y = 2(37) + 10 = 84
$$

✔ Final Answers:
- $ x = 37 $
- $ y = 84 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| #1 | 130° |
| #2 | 60° |
| #3 | 80° |
| #4 | 40° |
| #5 | 70° |
| #6 | 60° |
| #7 | $x = 37$, $y = 84$ |

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