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. Find the size of the angle DYC
To find the measures of angles in a quadrilateral, we can use the property that the sum of the measures of the angles in any quadrilateral is always 360 degrees.
In this case, we are given the measures of angles A and B, but we need to find the measure of angle DYC. To do this, we first find the measure of angle C.
Since the sum of the measures of all angles in a quadrilateral is 360 degrees, we can set up the equation:
angle A + angle B + angle C + angle D = 360 degrees
Given that angle A = 120 degrees and angle B = 140 degrees, we can substitute these values into the equation:
120 degrees + 140 degrees + angle C + angle D = 360 degrees
Simplifying the equation gives us:
angle C + angle D = 100 degrees
Now, we are also given that the bisectors of angles C and D meet at the point Y. This means that angle CYD is divided into two equal angles: angle CYA and angle DYB.
So, angle CYD = angle CYA + angle DYB.
Since the bisectors of angles C and D meet at the point Y, we can assume that angle CYA and angle DYB are equal.
Let's represent the measure of angle DYC as x degrees.
Therefore, we can set up the equation:
angle CYA + angle DYB + angle DYC = 180 degrees
Substituting the equal angles with x, we have:
x + x + x = 180 degrees
Simplifying the equation gives us:
3x = 180 degrees
Dividing both sides by 3, we get:
x = 60 degrees
Therefore, the measure of angle DYC is 60 degrees.