Given that the measure of angle 𝐴 equals 𝑦 degrees, the measure of angle 𝐵 equals four 𝑥 minus three degrees, and the measure of angle 𝐶 equals five 𝑥 degrees, find the values of 𝑥 and 𝑦.
Let’s begin by adding the information given in the question onto the diagram. So first, we have that the measure of angle 𝐴 equals 𝑦 degrees, then the measure of angle 𝐵 equals four 𝑥 minus three degrees, and finally the measure of angle 𝐶 is five 𝑥 degrees.
We can see then that we’ve been given expressions or values for all four angles within this quadrilateral. And we need to determine the values of 𝑥 and 𝑦. Now, this is in fact a special type of quadrilateral as its four vertices all lie on the circumference of a circle. It’s what’s known as a cyclic quadrilateral.
There’s also a key property about the angles in a cyclic quadrilateral, which we need to recall. And it’s this: opposite angles in a cyclic quadrilateral sum to 180 degrees. So this means that the measure of angle 𝐵 plus the measure of angle 𝐷 equals 180 degrees and also that the measure of angle 𝐴 plus the measure of angle 𝐶 equals 180 degrees.
We can use this information to form two equations involving 𝑥 and 𝑦. Firstly, for angles 𝐵 and 𝐷, we have that four 𝑥 minus three plus 115 is equal to 180 as the measure of angle 𝐵 is four 𝑥 minus three degrees and the measure of angle 𝐷 is 115 degrees. This gives an equation only in terms of 𝑥, which we can solve directly.
First, we can simplify the left-hand side slightly. We have negative three plus 115, which simplifies to positive 112. Next, we can subtract 112 from each side of the equation, giving four 𝑥 is equal to 68. The final step is to divide both sides of the equation by four which gives 𝑥 equals 17.
Notice that 𝑥 is just a value; it’s not a value with a degree symbol because if we look at angle 𝐶, we can see that its measure is five 𝑥 degrees. So the degree symbol is already included. Therefore, 𝑥 is just 17, not 17 degrees.
Now, we can return to our second statement: the measure of angle 𝐴 plus the measure of angle 𝐶 equals 180 degrees. And substituting 𝑦 for the measure of angle 𝐴 and five 𝑥 for the measure of angle 𝐶, we have 𝑦 plus five 𝑥 equals 180.
We’ve already determined that the value of 𝑥 is 17. So we can substitute this into our equation. Five multiplied by 17 is 85. So we have 𝑦 plus 85 equals 180. And we can solve for 𝑦 in one step. We subtract 85 from each side of the equation, giving 𝑦 is equal to 95. Again, there is no degree symbol with this value.
So we found the values of 𝑥 and 𝑦. They’re 17 and 95, respectively.
It’s always good to check our answers where possible. So in this question, what we’ll do is go back to the original diagram and calculate the measures of all of the angles: 𝐴, 𝐵, 𝐶, and 𝐷. As these are the four internal angles in a quadrilateral, their sum should be 360 degrees.
Angle 𝐴, which was just 𝑦 degrees, will now be equal to 95 degrees, angle 𝐶, whose measure was five 𝑥 degrees, will now be equal to five multiplied by 17 which is 85 degrees. The measure of angle 𝐵 is four 𝑥 minus three degrees. So four multiplied by 17 is 68 and then subtracting three gives 65.
We can then perform a quick check. 95 plus 65 plus 85 plus 115 is indeed equal to 360. So this confirms that our values of 17 and 95 for 𝑥 and 𝑦, respectively, are correct.