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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)
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.
To find the values of x, y, and z in the parallelogram, we can use the fact that opposite angles in a parallelogram are congruent.
Given:
- Top left corner angle: 33°
- Top right corner angle: 109°
Step 1: Find the value of z.
Since opposite angles in a parallelogram are congruent, the bottom left corner angle also measures z°.
Therefore, z = 33°.
Step 2: Find the value of x.
Since opposite angles in a parallelogram are congruent, the bottom right corner angle that is one-half y° also measures y°.
Therefore, x = y/2.
However, we still need more information to determine the value of y.
Step 3: Find the value of y.
To find the value of y, we can use the fact that the sum of the angles in a triangle is 180°. Since the line dividing the parallelogram creates two triangles, we can find y by subtracting the known angles from 180°.
In Triangle 1 (top triangle):
- Top left corner angle: 33°
- Top right corner angle: 109°
- Bottom angle: 180° - (33° + 109°) = 38°
In Triangle 2 (bottom triangle):
- Bottom left corner angle: z° = 33°
- Bottom right corner angle: x° = y/2
The sum of the angles in Triangle 2 will also be 180°:
z + x + y/2 = 180°
33° + y/2 + y/2 = 180°
33° + y = 180° - 33°
y = 147°
Step 4: Determine the values of x and y.
Now that we know y = 147°, we can substitute it back into the equation for x:
x = y/2
x = 147°/2
x = 73.5°
Therefore,
- x = 73.5°
- y = 147°
- z = 33°
So the values of x, y, and z in the parallelogram are x = 73.5°, y = 147°, and z = 33°.
To find the values of x, y, and z in the parallelogram, we can use the properties of angles in parallelograms.
Step 1: Using the fact that opposite angles in a parallelogram are congruent, we can determine the measure of angle z.
Since the top left corner has a 33 degree angle, the bottom right corner must also have a 33 degree angle. Therefore, z = 33 degrees.
Step 2: To find the values of x and y, we will use the fact that the sum of the angles in a triangle is 180 degrees.
In the top right triangle, we know that angle A = 109 degrees. Let's call the other two angles B and C.
Step 3: Using the fact that the sum of the angles in a triangle is 180 degrees, we can write the equation:
A + B + C = 180
Substituting the values we know:
109 + B + C = 180
Step 4: Since the opposite angle in the parallelogram is congruent, we know that angle B in the top right triangle is equal to angle x in the bottom right triangle.
Therefore, B = x.
Step 5: Similarly, since the opposite angle in the parallelogram is congruent, we know that angle C in the top right triangle is equal to angle y in the bottom right triangle.
Therefore, C = y.
Step 6: Substituting these values into the equation from Step 3:
109 + x + y = 180
Step 7: We can now solve for x and y.
x + y = 180 - 109
x + y = 71
Step 8: Since the bottom right triangle is split in half by the diagonal, the sum of the angles x and y must equal 180 degrees.
Therefore:
x + y = 180
Step 9: We can now solve the system of equations from Step 7 and Step 9:
x + y = 71
x + y = 180
Subtracting the equations, we can eliminate y:
0 = 180 - 71
109 = x
Step 10: Now that we know x, we can find y by substituting this value back into one of the equations from Step 7 or Step 9:
109 + y = 71
Subtracting 109 from both sides:
y = 71 - 109
y = -38
Therefore, the values of the variables x, y, and z in the parallelogram are:
x = 109 degrees
y = -38 degrees
z = 33 degrees