# Video: Using the Properties of Cyclic Quadrilaterals to Find the Values of Unknown Symbols

Given that πβ π΄ = π¦Β°, πβ π΅ = (4π₯ β 3)Β°, and πβ πΆ = 5π₯Β°, find the values of π₯ and π¦.

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### Video Transcript

Given that the measure of angle π΄ equals π¦ degrees, the measure of angle π΅ equals four π₯ minus three degrees, and the measure of angle πΆ equals five π₯ degrees, find the values of π₯ and π¦.

Letβs begin by adding the information given in the question onto the diagram. So first, we have that the measure of angle π΄ equals π¦ degrees, then the measure of angle π΅ equals four π₯ minus three degrees, and finally the measure of angle πΆ is five π₯ degrees.

We can see then that weβve been given expressions or values for all four angles within this quadrilateral. And we need to determine the values of π₯ and π¦. Now, this is in fact a special type of quadrilateral as its four vertices all lie on the circumference of a circle. Itβs whatβs known as a cyclic quadrilateral.

Thereβs also a key property about the angles in a cyclic quadrilateral, which we need to recall. And itβs this: opposite angles in a cyclic quadrilateral sum to 180 degrees. So this means that the measure of angle π΅ plus the measure of angle π· equals 180 degrees and also that the measure of angle π΄ plus the measure of angle πΆ equals 180 degrees.

We can use this information to form two equations involving π₯ and π¦. Firstly, for angles π΅ and π·, we have that four π₯ minus three plus 115 is equal to 180 as the measure of angle π΅ is four π₯ minus three degrees and the measure of angle π· is 115 degrees. This gives an equation only in terms of π₯, 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 π₯ is equal to 68. The final step is to divide both sides of the equation by four which gives π₯ equals 17.

Notice that π₯ is just a value; itβs not a value with a degree symbol because if we look at angle πΆ, we can see that its measure is five π₯ degrees. So the degree symbol is already included. Therefore, π₯ is just 17, not 17 degrees.

Now, we can return to our second statement: the measure of angle π΄ plus the measure of angle πΆ equals 180 degrees. And substituting π¦ for the measure of angle π΄ and five π₯ for the measure of angle πΆ, we have π¦ plus five π₯ equals 180.

Weβve already determined that the value of π₯ is 17. So we can substitute this into our equation. Five multiplied by 17 is 85. So we have π¦ plus 85 equals 180. And we can solve for π¦ in one step. We subtract 85 from each side of the equation, giving π¦ is equal to 95. Again, there is no degree symbol with this value.

So we found the values of π₯ and π¦. Theyβre 17 and 95, respectively.

Itβs always good to check our answers where possible. So in this question, what weβll do is go back to the original diagram and calculate the measures of all of the angles: π΄, π΅, πΆ, and π·. As these are the four internal angles in a quadrilateral, their sum should be 360 degrees.

Angle π΄, which was just π¦ degrees, will now be equal to 95 degrees, angle πΆ, whose measure was five π₯ degrees, will now be equal to five multiplied by 17 which is 85 degrees. The measure of angle π΅ is four π₯ 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 π₯ and π¦, respectively, are correct.