Parallel Line Transversal Worksheets - Free Printable

Problem Statement:

The image describes a geometric problem involving parallel lines and transversals. The task is to determine the measure of angle $ x $.

#### Given:
1. Lines $ k $ and $ m $ are parallel.
2. Line $ t $ is a transversal intersecting both $ k $ and $ m $.
3. Angle measures are provided:
- $\angle 1 = 70^\circ$
- $\angle 2 = 65^\circ$

#### Goal:
Find the measure of angle $ x $.

---

Solution:

#### Step 1: Understand the setup
- Lines $ k $ and $ m $ are parallel, and line $ t $ is a transversal.
- Angles formed by a transversal intersecting parallel lines have specific relationships:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Same-side interior angles are supplementary (sum to $ 180^\circ $).
- Vertical angles are equal.

#### Step 2: Identify key angles
From the diagram:
- $\angle 1$ and $\angle 2$ are given.
- $\angle x$ is the angle we need to find.
- Since $ k $ and $ m $ are parallel, the angles formed by the transversal $ t $ will follow the properties of parallel lines.

#### Step 3: Use the properties of parallel lines
- $\angle 1$ and $\angle 2$ are on the same side of the transversal $ t $ and between the parallel lines $ k $ and $ m $. These are same-side interior angles.
- The property of same-side interior angles states that they are supplementary. Therefore:
$$
\angle 1 + \angle 2 = 180^\circ
$$

#### Step 4: Verify the given angles
Substitute the given values:
$$
\angle 1 = 70^\circ, \quad \angle 2 = 65^\circ
$$
$$
\angle 1 + \angle 2 = 70^\circ + 65^\circ = 135^\circ
$$
This confirms that $\angle 1$ and $\angle 2$ are not same-side interior angles but are instead part of a larger configuration involving $\angle x$.

#### Step 5: Relate $\angle x$ to the given angles
- From the diagram, $\angle x$ is an exterior angle relative to the triangle formed by the intersection of the transversal $ t $ with lines $ k $ and $ m $.
- The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

#### Step 6: Calculate $\angle x$
- The two non-adjacent interior angles corresponding to $\angle x$ are $\angle 1$ and $\angle 2$.
- Therefore:
$$
\angle x = \angle 1 + \angle 2
$$
- Substitute the given values:
$$
\angle x = 70^\circ + 65^\circ = 135^\circ
$$

---

Final Answer:

$$
\boxed{135^\circ}
$$