QuestionFind the value of $x+y+z+w$ given $\angle C = 140^\circ$ and angles $A$ , $B$ , $D$ , $E$ at intersection point $C$ .

Studdy Solution

STEP 1

Assumptions1. Line segments $\overline{AD}$ and $\overline{BE}$ intersect at point $C$ . . The measures of the angles are as follows $\angle A = x$ , $\angle B = y$ , $\angle C =140$ , $\angle D = z$ , and $\angle = w$ .
3. We need to find the value of $x+y+z+w$ .

STEP 2

We know that the sum of the angles at a point (here, point $C$ ) is $360$ degrees. Therefore, we can write $x + y +140 + z + w =360$

STEP 3

Rearrange the equation to find the value of $x+y+z+w$ .
$x + y + z + w =360 -140$

STEP 4

Calculate the value of $x+y+z+w$ .
$x + y + z + w =220$ So, the value of $x+y+z+w$ is $220$ degrees. However, this option is not available in the given options. This means there might be a mistake in the problem or in our solution. Let's check our steps again.

STEP 5

Upon reviewing, we realize that we made a mistake in2. The angles $x$ , $y$ , $z$ , and $w$ are not all at the same point. Only the angles $x$ , $140$ , and $z$ are at point $C$ , and only the angles $y$ , $140$ , and $w$ are at point $C$ . So, we should have two separate equations $x +140 + z =360$ and $y +140 + w =360$ .

STEP 6

Rearrange both equations to find the values of $x+z$ and $y+w$ .
$x + z =360 -140$ $y + w =360 -140$

STEP 7

Calculate the values of $x+z$ and $y+w$ .
$x + z =220$ $y + w =220$

STEP 8

Now, add these two equations to find the value of $x+y+z+w$ .
$x + y + z + w = (x + z) + (y + w) =220 +220$

STEP 9

Calculate the value of $x+y+z+w$ .
$x + y + z + w =220 +220 =440$ So, the value of $x+y+z+w$ is $440$ degrees. However, this option is also not available in the given options. This means there's still a mistake in our solution. Let's check our steps again.

STEP 10

Upon reviewing again, we realize that we made a mistake in5. The angles $x$ , $140$ , and $z$ form a straight line, and the angles $y$ , $140$ , and $w$ also form a straight line. Therefore, the sum of these angles should be $180$ degrees, not $360$ degrees. So, the correct equations should be $x +140 + z =180$ and $y +140 + w =180$ .

STEP 11

Rearrange both equations to find the values of $x+z$ and $y+w$ .
$x + z =180 -140$ $y + w =180 -140$

STEP 12

Calculate the values of $x+z$ and $y+w$ .
$x + z =40$ $y + w =40$

STEP 13

Now, add these two equations to find the value of $x+y+z+w$ .
$x + y + z + w = (x + z) + (y + w) =40 +40$

STEP 14

Calculate the value of $x+y+z+w$ .
$x + y + z + w =40 +40 =80$ So, the correct value of $x+y+z+w$ is $80$ degrees. This matches with option A.
The correct answer is A.80.

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