In the figure below, 𝐷𝐵𝐶 is an
isosceles triangle, angle 𝐶𝐵𝐷 has measure 40 degrees, angle 𝐷𝐶𝐸 has measure 15
degrees, angle 𝐵𝐷𝐶 has measure eight 𝑥 plus 20, and angle 𝐸𝐵𝐶 has measure 11
times 𝑦 minus two. Find the values of 𝑥 and 𝑦.
Let’s find 𝑥 first. Since 𝐷𝐵𝐶 is an isosceles
triangle, we know that angles 𝐷𝐶𝐵 and 𝐶𝐵𝐷 are equal. Therefore, angle 𝐵𝐷𝐶 has measure
180 minus 40 minus 40, which is 100 degrees. The question tells us that the
measure of angle 𝐵𝐷𝐶 is equal to eight 𝑥 plus 20 degrees. Substituting in our value of 100
for the measure of 𝐵𝐷𝐶 and simplifying, we find that 𝑥 equals 10.
Since angles 𝐷𝐵𝐸 and 𝐷𝐶𝐸 are
subtended by the same arc, they must be equal. We are told that the measure of
𝐷𝐶𝐸 is 15 degrees. Therefore, the measure of 𝐷𝐵𝐸 is
15 degrees too. Therefore, angle 𝐸𝐵𝐶, which is
the sum of angles 𝐶𝐵𝐷 and 𝐷𝐵𝐸, has measure 40 plus 15, which is 55. We are told that the measure of
𝐸𝐵𝐶 is equal to 11 times 𝑦 minus two degrees. We can now substitute in our value
of 55 for the measure of 𝐸𝐵𝐶 and simplify, giving us 𝑦 equals seven. The answer to the question is 𝑥
equals 10 and 𝑦 equals seven.