Question Video: Using Properties of Cyclic Quadrilaterals to Find Two Unknowns | Nagwa Question Video: Using Properties of Cyclic Quadrilaterals to Find Two Unknowns | Nagwa

# Question Video: Using Properties of Cyclic Quadrilaterals to Find Two Unknowns Mathematics • Third Year of Preparatory School

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Find the values of π₯ and π¦.

03:26

### Video Transcript

Find the values of π₯ and π¦.

In the figure below, weβre given a quadrilateral π΄π΅πΆπ· and we note that all four vertices of this quadrilateral are inscribed on the circle. This means that π΄π΅πΆπ· is a cyclic quadrilateral. An important property which we might then be able to apply is that opposite angles in a cyclic quadrilateral are supplementary. They add to 180 degrees. The first thing we might observe then is that the angle at π΄ is opposite to the angle at πΆ. We could therefore write the equation that four π₯ degrees plus 68 degrees is equal to 180 degrees. Do be careful because a very common mistake when weβre working with cyclic quadrilaterals is to forget that opposite angles are supplementary and mistakenly think that opposite angles are equal.

So because four π₯ plus 68 degrees equals 180 degrees, then four π₯ degrees must be equal to 180 degrees subtract 68 degrees. And thatβs 112 degrees. When we divide through by four, we get that π₯ degrees is equal to 28 degrees, or more simply π₯ is equal to 28. Now that we have found π₯, letβs see how we can find π¦. The angle which is opposite to the angle at π΅, which is π¦ degrees, is the angle at π·. But weβre not given a measurement for it. Even if we worked out that the measurement of angle π΄ is 112 degrees, that isnβt enough to allow us to find the measurement of angle π΅.

So letβs use the other piece of information that weβre given in the diagram. And that is that these line segments of π΄π΅ and π΄π· are congruent. Both of these line segments are in fact chords of the circle. Letβs imagine that we create this triangle π΄π΅π and another triangle here at π΄π·π. π΅π and π΄π are both radii of the circle, and so theyβll be the same length. π·π is also a radius, and so it will be congruent with the other two radii. So what can we say about this triangle π΄π΅π? Well, we know that itβs going to be an isosceles triangle because itβs got two sides that are the same length. And therefore, if angle π΄π΅π is π¦ degrees, then angle π΅π΄π must also be π¦ degrees.

In the same way, we can also say that this triangle π΄ππ· is also an isosceles triangle. More importantly, however, we can also say that triangle π΄π΅π is congruent to triangle π΄π·π. We can say this by applying the SSS congruency criterion. There are three corresponding sides which are congruent. And therefore, we can say that the measure of angle ππ΄π· must also be π¦ degrees and so must the measure of angle ππ·π΄. We were given that this angle π΅π΄π· is four π₯ degrees, and we know that four π₯ degrees is 112 degrees. Because thatβs equal to two π¦, then we can divide through by two to find that π¦ degrees is equal to 56 degrees. We can then give the answer that π₯ is equal to 28 and π¦ is equal to 56.

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