Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.

Question

A parallelogram is shown with the diagonal drawn from the upper left vertex to the bottom right vertex.

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

Answer 1

Since the diagonal of the parallelogram divides it into two congruent triangles, we can use the properties of triangles to find the values of x, y, and z.

Let's call the upper left vertex of the parallelogram A, the upper right vertex B, the bottom left vertex C, and the bottom right vertex D. Let's also label the intersection point of the diagonal and side AD as E.

Now, let's focus on triangle AED. Since AD is a diagonal of the parallelogram, it bisects angle A, so angle DAE is congruent to angle EAD. This means that triangle AED is an isosceles triangle with angles DAE, DEA, and AED all congruent.

Since angle DAE and angle DEA are congruent, we can set up the following equation:
DEA + DEA + AED = 180 degrees

Simplifying the equation:
2DEA + AED = 180 degrees

Since triangle AED is isosceles, we know that angle AED is congruent to angle EAD. Let's call these angles x.

Therefore, we can rewrite the equation as:
2x + x = 180 degrees

Combining like terms:
3x = 180 degrees

To solve for x, divide both sides of the equation by 3:
x = 180 degrees / 3
x = 60 degrees

So the value of x is 60 degrees.

Now, let's move on to triangle ABC. Since ABC is a parallelogram, opposite angles are congruent. So angle A is congruent to angle C. Let's call these angles y.

Therefore, we can set up the following equation:
2y = 180 degrees

To solve for y, divide both sides of the equation by 2:
y = 180 degrees / 2
y = 90 degrees

So the value of y is 90 degrees.

Now, let's look at triangle BCD. Since BCD is a parallelogram, opposite angles are congruent. So angle C is congruent to angle D. Let's call these angles z.

Therefore, we can set up the following equation:
2z = 180 degrees

To solve for z, divide both sides of the equation by 2:
z = 180 degrees / 2
z = 90 degrees

So the value of z is 90 degrees.

Therefore, the values of the variables x, y, and z in the parallelogram are x = 60 degrees, y = 90 degrees, and z = 90 degrees.