Find the unknown angles in intersecting lines worksheet.

Let's solve each problem step by step using the properties of intersecting lines.

When two lines intersect, they form four angles. The key properties are:

1. Vertical angles are equal (opposite angles).
2. Adjacent angles are supplementary (they add up to 180°).

We'll use these rules to find the unknown angles.

---

Problem 1

Given:
- $ m\angle 2 = 45^\circ $

From the diagram:
- $ \angle 1 $ and $ \angle 3 $ are vertical angles with $ \angle 2 $ and $ \angle 4 $, respectively.
- $ \angle 1 $ is adjacent to $ \angle 2 $, so $ \angle 1 + \angle 2 = 180^\circ $

So:
- $ m\angle 1 = 180^\circ - 45^\circ = 135^\circ $
- $ m\angle 3 = m\angle 1 = 135^\circ $ (vertical angle)
- $ m\angle 4 = m\angle 2 = 45^\circ $ (vertical angle)

✔ Answers:
- $ m\angle 1 = 135^\circ $
- $ m\angle 2 = 45^\circ $
- $ m\angle 3 = 135^\circ $
- $ m\angle 4 = 45^\circ $

---

Problem 2

Given:
- $ m\angle 3 = 83^\circ $

From the diagram:
- $ \angle 1 $ and $ \angle 3 $ are vertical angles → $ m\angle 1 = 83^\circ $
- $ \angle 2 $ is adjacent to $ \angle 1 $ → $ m\angle 2 = 180^\circ - 83^\circ = 97^\circ $
- $ \angle 4 $ is vertical to $ \angle 2 $ → $ m\angle 4 = 97^\circ $

✔ Answers:
- $ m\angle 1 = 83^\circ $
- $ m\angle 2 = 97^\circ $
- $ m\angle 3 = 83^\circ $
- $ m\angle 4 = 97^\circ $

---

Problem 3

Given:
- $ m\angle 1 = 157^\circ $

From the diagram:
- $ \angle 1 $ and $ \angle 3 $ are vertical angles → $ m\angle 3 = 157^\circ $
- $ \angle 2 $ is adjacent to $ \angle 1 $ → $ m\angle 2 = 180^\circ - 157^\circ = 23^\circ $
- $ \angle 4 $ is vertical to $ \angle 2 $ → $ m\angle 4 = 23^\circ $

✔ Answers:
- $ m\angle 1 = 157^\circ $
- $ m\angle 2 = 23^\circ $
- $ m\angle 3 = 157^\circ $
- $ m\angle 4 = 23^\circ $

---

Problem 4

Given:
- $ m\angle 4 = 82^\circ $

From the diagram:
- $ \angle 4 $ and $ \angle 2 $ are vertical angles → $ m\angle 2 = 82^\circ $
- $ \angle 1 $ is adjacent to $ \angle 2 $ → $ m\angle 1 = 180^\circ - 82^\circ = 98^\circ $
- $ \angle 3 $ is vertical to $ \angle 1 $ → $ m\angle 3 = 98^\circ $

✔ Answers:
- $ m\angle 1 = 98^\circ $
- $ m\angle 2 = 82^\circ $
- $ m\angle 3 = 98^\circ $
- $ m\angle 4 = 82^\circ $

---

Problem 5

Given:
- $ m\angle 2 = 55^\circ $

From the diagram:
- $ \angle 2 $ and $ \angle 4 $ are vertical angles → $ m\angle 4 = 55^\circ $
- $ \angle 1 $ is adjacent to $ \angle 2 $ → $ m\angle 1 = 180^\circ - 55^\circ = 125^\circ $
- $ \angle 3 $ is vertical to $ \angle 1 $ → $ m\angle 3 = 125^\circ $

✔ Answers:
- $ m\angle 1 = 125^\circ $
- $ m\angle 2 = 55^\circ $
- $ m\angle 3 = 125^\circ $
- $ m\angle 4 = 55^\circ $

---

Problem 6

Given:
- $ m\angle 3 = 113^\circ $

From the diagram:
- $ \angle 3 $ and $ \angle 1 $ are vertical angles → $ m\angle 1 = 113^\circ $
- $ \angle 2 $ is adjacent to $ \angle 1 $ → $ m\angle 2 = 180^\circ - 113^\circ = 67^\circ $
- $ \angle 4 $ is vertical to $ \angle 2 $ → $ m\angle 4 = 67^\circ $

✔ Answers:
- $ m\angle 1 = 113^\circ $
- $ m\angle 2 = 67^\circ $
- $ m\angle 3 = 113^\circ $
- $ m\angle 4 = 67^\circ $

---

✔ Final Answers Summary:

| Problem | $ m\angle 1 $ | $ m\angle 2 $ | $ m\angle 3 $ | $ m\angle 4 $ |
|--------|------------------|------------------|------------------|------------------|
| 1 | 135° | 45° | 135° | 45° |
| 2 | 83° | 97° | 83° | 97° |
| 3 | 157° | 23° | 157° | 23° |
| 4 | 98° | 82° | 98° | 82° |
| 5 | 125° | 55° | 125° | 55° |
| 6 | 113° | 67° | 113° | 67° |

Let me know if you'd like this formatted for printing or need a visual explanation!