Question Video: Determining If a Circle Can Pass through Four Given Points Using the Properties of Cyclic Quadrilaterals | Nagwa Question Video: Determining If a Circle Can Pass through Four Given Points Using the Properties of Cyclic Quadrilaterals | Nagwa

# Question Video: Determining If a Circle Can Pass through Four Given Points Using the Properties of Cyclic Quadrilaterals Mathematics • Third Year of Preparatory School

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Given that πβ π΅πΆπ΄ = 61Β°, and πβ π·π΄π΅ = 98Β°, can a circle pass through the points π΄, π΅, πΆ, and π·?

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### Video Transcript

Given that the measure of angle π΅πΆπ΄ equals 61 degrees and the measure of angle π·π΄π΅ equals 98 degrees, can a circle pass through the points π΄, π΅, πΆ, and π·?

Remember, if there are a pair of congruent angles subtended by the same line segment and on the same side of it, then their vertices and the segmentβs endpoints lie on a circle in which that segment is a chord. Well, we have a line segment π΅π΄, from which angle π΅πΆπ΄ and π΅π·π΄ are subtended. The angles lie on the same side of that line segment. So if angle π΅πΆπ΄ is equal to angle π΅π·π΄, then all four of our points must lie on the circumference of a circle. Now, weβre given that the measure of angle π΅πΆπ΄ is 61 degrees and the measure of angle π΅π΄π· is 98.

Since triangle π΅π·π΄ is isosceles, we can calculate the measure of angle π΅π·π΄ by subtracting 98 from 180 and then dividing by two. And that gives us that the measure of angle π΅π·π΄ is 41 degrees. So we see that the measure of angle π΅πΆπ΄ is not equal to the measure of angle π΅π·π΄. Since these angles are not equal, we observe that a circle cannot pass through the points, and the answer is no.

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