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Question

A parallelogram is shown with the diagonal drawn from the upper left vertex to the bottom right vertex. Top left corner is 33 degrees with a line spliting the square into 2 triangles. Top right corner is 109 degrees. Bottom left corner z degrees. Bottom right corner has one half being y degrees amd the other half is x degrees.

Find the values of the variables x,y, and z in the parallelogram. The diagram is not drawn to scale.

(3 points)

Answer 1

To solve this problem, we can use the properties of parallelograms and the fact that the sum of the angles in a triangle is 180 degrees.

First, let's label the angles in the parallelogram:

Top left corner angle = 33 degrees
Top right corner angle = 109 degrees
Bottom left corner angle = z degrees
Bottom right corner angle:
One half = y degrees
Other half = x degrees

In a parallelogram, opposite angles are congruent. Since the top left and bottom right angles are opposite angles, we have:

Top left corner angle = Bottom right corner angle

Therefore, we can say that:

33 degrees = y + x

Next, let's look at the triangle formed by the top left corner, top right corner, and bottom right corner. The sum of the angles in a triangle is 180 degrees. So, we have:

Top left corner angle + Top right corner angle + Bottom right corner angle = 180 degrees

Substituting the given values, we have:

33 degrees + 109 degrees + (y + x) degrees = 180 degrees

Simplifying, we get:

142 degrees + (y + x) degrees = 180 degrees

Removing the parentheses, we have:

142 degrees + y degrees + x degrees = 180 degrees

Rearranging the equation, we get:

y + x = 180 degrees - 142 degrees

y + x = 38 degrees

Now we have two equations:

33 degrees = y + x
y + x = 38 degrees

We can solve this system of equations by substitution or elimination. Let's use substitution.

From the first equation, we have:

x = 33 degrees - y

Substituting this into the second equation, we get:

y + (33 degrees - y) = 38 degrees

Simplifying, we have:

33 degrees = 38 degrees

This is not possible because the degrees cannot be equal. Therefore, there is no solution for this system of equations.

Therefore, there are no valid values for x, y, and z in the parallelogram.