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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?

3 answers

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 \( \boxed{90^\circ} \).

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!