Question

Use the figure to answer the question.

An illustration shows a right inclined transversal passing through two parallel lines labeled upper M and upper N. Four angles are formed at the intersections of the transversal with upper M and upper N.

If M
and N
are parallel and angle 1 is 135°
, what is the sum of angle 6 and angle 8?

Answer 1

wrong

Answer 2

To determine the sum of angle 6 and angle 8, we need to analyze their relationships given that \( M \) and \( N \) are parallel lines and angle 1 is \( 135^\circ \).

In this scenario, we can use the properties of angles formed by a transversal intersecting parallel lines:

1. **Corresponding Angles:** Corresponding angles are equal.
2. **Alternate Interior Angles:** Alternate interior angles are equal.
3. **Consecutive Interior Angles (Same-Side Interior Angles):** These angles are supplementary (they add up to \( 180^\circ \)).

If we designate the angles as follows based on their positions:
- Angle 1 is given as \( 135^\circ \).

Here's the relationship between the angles:
- Angle 1 (which is \( 135^\circ \)) and angle 8 are on the same side of the transversal and are supplementary.
- Therefore, angle 1 and angle 8 satisfy the relationship:
\[
\text{angle 1} + \text{angle 8} = 180^\circ
\]
Substituting \( 135^\circ \) for angle 1:
\[
135^\circ + \text{angle 8} = 180^\circ
\]
Solving for angle 8 gives:
\[
\text{angle 8} = 180^\circ - 135^\circ = 45^\circ
\]

Since angle 6 is vertically opposite to angle 8 (because of the transversal crossings), then:
- Angle 6 is equal to angle 8:
\[
\text{angle 6} = 45^\circ
\]

Now, we can find the sum of angle 6 and angle 8:
\[
\text{Sum of angle 6 and angle 8} = \text{angle 6} + \text{angle 8} = 45^\circ + 45^\circ = 90^\circ.
\]

Thus, the sum of angle 6 and angle 8 is \( \boxed{90^\circ} \).

Answer 3

I apologize for the confusion in my previous response. Let's reanalyze the problem step-by-step given that angle 1 is \(135^\circ\).

### Given:
- Transversal intersects two parallel lines \( M \) and \( N \).
- Angle 1 is \( 135^\circ \).

### Understand the angles:
1. **Angle 1** is one of the angles formed by the transversal intersecting line \( M \).
2. Since lines \( M \) and \( N \) are parallel and angle 1 is at the intersection of the transversal with line \( M \), we know:
- **Angle 2** (which is vertically opposite to angle 1) is also \( 135^\circ \).
- **Angle 3** (corresponding angle to angle 1 at line \( N \)) is also \( 135^\circ \).
- **Angle 4** (the angle that is supplementary to angle 1) gives:
\[
\text{Angle 4} = 180^\circ - 135^\circ = 45^\circ.
\]

Next, let’s place the angles:
- If angle 1 is \( 135^\circ \):
- \( \text{Angle 2} \) (vertically opposite to angle 1) is \( 135^\circ \).
- \( \text{Angle 3} \) (corresponding to angle 1 at line \( N \)) is \( 135^\circ \).
- \( \text{Angle 4} = 45^\circ \) (same side interior angle).

### Relating with angle 6 and angle 8:
- **Angle 8** (which is on the same side as angle 1, and interior to line \( N \)) is supplementary to angle 3:
- Therefore,
\[
\text{Angle 8} = 180^\circ - \text{Angle 3} = 180^\circ - 135^\circ = 45^\circ.
\]

- **Angle 6**, which is vertically opposite to angle 4 (and therefore also equals angle 4), is again:
- \( \text{Angle 6} = 45^\circ \).

### Final Calculation:
Now, the sum of angle 6 and angle 8 is:
\[
\text{Angle 6} + \text{Angle 8} = 45^\circ + 45^\circ = 90^\circ.
\]

So, the correct answer is still \( \boxed{90^\circ} \). If I've misunderstood any part of your setup or the diagram, please provide further clarification!