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A parallelogram is shown with the diagonal drawn from the upper left vertex to the bottom right vertex. the upper left angle is 33 degrees the upper right angle is 109 degrees the bottom left is z degrees, and the bottom right is y degrees and x degrees.
Find the values of the variables x,y, and z in the parallelogram. The diagram is not drawn to scale.
Let's first find the value of z.
In a parallelogram, opposite angles are congruent. Since the angle in the upper left vertex is 33 degrees, the angle in the bottom right vertex must also be 33 degrees. Therefore,
z = 33 degrees.
Now, let's find the values of x and y.
The sum of the angles in a parallelogram is 360 degrees. Since we already know the angles in the upper left (33 degrees) and upper right (109 degrees) vertices, we can find the sum of the angles in the bottom left and bottom right vertices:
Angle in bottom left vertex + Angle in bottom right vertex = 360 degrees - (33 degrees + 109 degrees)
Angle in bottom left vertex + Angle in bottom right vertex = 218 degrees.
Since opposite angles in a parallelogram are congruent, the angle in the bottom left vertex must be equal to the angle in the bottom right vertex. Therefore,
2 * (Angle in bottom left vertex) = 218 degrees.
Simplifying,
Angle in bottom left vertex = 109 degrees.
Therefore,
y = x = 109 degrees. Answer: x = 109 degrees, y = 109 degrees, z = 33 degrees.
To find the values of x, y, and z, we can use the properties of parallelograms. The opposite angles of a parallelogram are congruent.
Given:
Upper left angle = 33 degrees
Upper right angle = 109 degrees
Step 1: Find the value of z:
Since the opposite angles of a parallelogram are congruent, the upper left angle is congruent to the bottom right angle. Therefore, z = 33 degrees.
Step 2: Find the value of y:
The sum of the measures of the angles of a parallelogram is always 360 degrees.
Since y is the bottom right angle, we can find its value by subtracting the known angles from 360 degrees.
360 degrees - 33 degrees - 109 degrees = 218 degrees.
Therefore, y = 218 degrees.
Step 3: Find the value of x:
Similarly, the sum of the measures of the angles of a parallelogram is always 360 degrees.
Since x is the bottom left angle, we can find its value by subtracting the known angles from 360 degrees.
360 degrees - 33 degrees - 109 degrees - 218 degrees = 0 degrees.
Therefore, x = 0 degrees.
Final solution:
x = 0 degrees, y = 218 degrees, z = 33 degrees.
To find the values of x, y, and z in the parallelogram, we can use the fact that opposite angles in a parallelogram are equal.
Given information:
- Upper left angle = 33 degrees
- Upper right angle = 109 degrees
Since the opposite angles in a parallelogram are equal, we can conclude that the lower left angle is also equal to 109 degrees.
So, z = 109 degrees.
Now, let's find the value of y.
We know that the sum of the angles in a parallelogram is 360 degrees.
Since a parallelogram has two pairs of opposite angles, we can use the equation:
Sum of opposite angles = 360 degrees.
Upper left angle + bottom right angle = 360 degrees.
33 degrees + y degrees = 360 degrees.
Simplifying the equation,
y = 360 degrees - 33 degrees.
y = 327 degrees.
So, y = 327 degrees.
Finally, the value of x can be found by using the fact that opposite angles in a parallelogram are equal.
The opposite angle to x is the lower left angle, which is z degrees (109 degrees).
Therefore, x = 109 degrees.
In conclusion:
x = 109 degrees
y = 327 degrees
z = 109 degrees.