In a quadrilateral ABCD, Angle A = 120 degrees, Angle B = 140 degrees.
The bisectors of the angles C and D meet at the point Y.
What is the size of the angle DYC?
The sum of angles C and D must be 100 degrees, so that angles A, B, C and D add up to 360 degrees for the quadrilateral.
Half of the sum of 100 degrees (which is 50 degrees) must be the sum of the two bisected acute angles C and D. Two of those bisected angles form triangle DYC. That angle must therefore be 180-50 = 130 degrees at point Y.
It will be clearer if you draw yourself a picture. The angles C and D can be arbitratily drawn, but their sum has to be 100 degrees.
bottom...he he he
To find the size of angle DYC, we need to use the property that the sum of the angles in a quadrilateral is always 360 degrees.
We can start by finding the measure of angle C. Since the opposite angles in a quadrilateral are supplementary, we can subtract the given angles A and B from 180 degrees:
Angle C = 180 degrees - Angle A - Angle B
= 180 degrees - 120 degrees - 140 degrees
= - 80 degrees.
Since angle C is negative, it means that the given information is inconsistent or there is an error. A quadrilateral cannot have a negative angle. Therefore, we cannot determine the size of angle DYC based on the given information.
To determine the size of the angle DYC, we need to use the properties of the angle bisectors.
First, let's draw the quadrilateral ABCD with the given information.
Now, we know that the bisectors of angles C and D meet at the point Y. Let's label the intersection of the angle bisectors as point Y.
To find the size of angle DYC, we need to find the angles at points C and D. We can use the fact that the sum of the angles of a quadrilateral is always 360 degrees.
Let's use this fact to find the size of angle C.
In quadrilateral ABCD, we know angle A is 120 degrees and angle B is 140 degrees. We also know that the sum of angles C and D is 360 - (angle A + angle B). Let's calculate that.
360 - (120 + 140) = 360 - 260 = 100 degrees
So, angle C measures 100 degrees.
Now, since Y is the intersection of the angle bisectors of angles C and D, angle CYD is divided equally by the bisectors. Therefore, angle DYC is half the size of angle CYD.
Since angle CYD is 100 degrees, angle DYC is 100/2 = 50 degrees.
Therefore, the size of angle DYC is 50 degrees.