Angle Pair Relationships Pages - Free Printable

Let's go through each problem step by step, using the concept of vertical angles.

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🔷 Key Concept: Vertical Angles

- Vertical angles are the angles opposite each other when two lines cross.
- They are always congruent (equal in measure).
- They share the same vertex but are not adjacent.

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🔹 Problem 1: Find $ m\angle 2 $

We are given:
- One angle is $ 135^\circ $
- $ \angle 2 $ is vertical to this angle.

Since vertical angles are congruent:

$$
m\angle 2 = 135^\circ
$$

✔ Answer: $ m\angle 2 = 135^\circ $

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🔹 Problem 2: Find $ m\angle AOC $

We are given:
- $ \angle DOB = 75^\circ $
- $ \angle AOC $ is vertical to $ \angle DOB $

So, since vertical angles are equal:

$$
m\angle AOC = 75^\circ
$$

✔ Answer: $ m\angle AOC = 75^\circ $

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🔹 Problem 3: Find $ x $ and the measure of the angles

Given:
- One angle is $ 74^\circ $
- The other angle is $ 2x^\circ $
- These two angles are vertical angles, so they must be equal.

So set them equal:

$$
2x = 74
$$

Solve for $ x $:

$$
x = \frac{74}{2} = 37
$$

Now find the measure of the angles:
- $ 2x = 2(37) = 74^\circ $
- So both angles are $ 74^\circ $

✔ Answer:
- $ x = 37 $
- Measure of each angle = $ 74^\circ $

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🔹 Problem 4: Find $ x $ and the measure of the angles

Given:
- Two vertical angles: $ 3x + 20 $ and $ 10x - 15 $
- Since they are vertical angles, they are equal:

$$
3x + 20 = 10x - 15
$$

Solve for $ x $:

Subtract $ 3x $ from both sides:

$$
20 = 7x - 15
$$

Add 15 to both sides:

$$
35 = 7x
$$

Divide by 7:

$$
x = 5
$$

Now plug back to find the angle measures:

- $ 3x + 20 = 3(5) + 20 = 15 + 20 = 35^\circ $
- $ 10x - 15 = 10(5) - 15 = 50 - 15 = 35^\circ $

✔ Answer:
- $ x = 5 $
- Each angle measures $ 35^\circ $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ m\angle 2 = 135^\circ $ |
| 2 | $ m\angle AOC = 75^\circ $ |
| 3 | $ x = 37 $, angles = $ 74^\circ $ |
| 4 | $ x = 5 $, angles = $ 35^\circ $ |

All based on the property that vertical angles are congruent.