The image shows two intersecting lines forming four angles around a central point. The provided information includes angle measures: 75 degrees and 45 degrees. The problem is to find the value of $z$. We are given $655$ as an irrelevant value.

1. Problem Description

The image shows two intersecting lines forming four angles around a central point. The provided information includes angle measures: 75 degrees and 45 degrees. The problem is to find the value of $z$. We are given $655$ as an irrelevant value.

2. Solution Steps

The intersecting lines form two pairs of vertically opposite angles, which are equal.

Also, the angles on a straight line add up to 180 degrees.

Let the angle adjacent to 75 degrees be $x$.

$x + 75 = 180$

$x = 180 - 75$

$x = 105$

Let the angle vertically opposite to 45 degrees be $y$. Then $y = 45$

Considering the angles around the point where the lines intersect, the sum is 360 degrees. So, $75+45+x+z = 360$. We already know that $x = 180 - 75 = 105$. So

$75+45+105+z = 360$

$225+z = 360$

$z = 360-225$

$z = 135$

We could have also argued that the angles on either side of the lines sum to

1

8

0. The angle opposite the 75 degree angle is also

7

5. And the angle opposite the 45 degree angle is also

4

5. Then $z$ is next to 45 and

7

5. Since the sum of angles around the point is 360:

$75 + 45 +z + unknown = 360$

We know the angle adjacent to 45 is $180 - 45 = 135$. The angle adjacent to 75 is $180-75=105$

Also $75 + z + 45 + z = 360 - 180 -180$ No

$75+z=180$. So $z = 105$ No

$45 + z = 180$. So $z = 135$.

We know $x+75=180$, and $y+45=180$

The angle diagonally across from 75 is 75 and the angle diagonally across from 45 is

4

5. Sum of angles in a circle is

3

6

0. So if z is opposite to the angle, x is opposite to 45 degrees, it is 45 degrees.

Therefore we can't say z is opposite anything.

Since the angle z appears to be adjacent to the angle 45, $z = 180-45 = 135$.

Therefore angle z should be 135 degrees.

Alternatively, the angle is adjacent to the angle with 75 degrees. Then $z = 180-75 = 105$. So this doesn't work.

3. Final Answer

135