Given that the measure of angle 𝐵𝐶𝐴 equals 61 degrees and the measure of angle 𝐷𝐴𝐵 equals 98 degrees, can a circle pass through the points 𝐴, 𝐵, 𝐶, and 𝐷?
In this problem, we’re given this figure and we’re asked if the circle can pass through the four points. Let’s consider if we draw in the line segment 𝐶𝐷, then we would have created a quadrilateral 𝐴𝐵𝐶𝐷. If we have a quadrilateral and there is a circle which passes through the four vertices, then that would be a cyclic quadrilateral. So let’s determine if this quadrilateral 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral. We can fill in the given information that the measure of angle 𝐵𝐶𝐴 is 61 degrees and the measure of angle 𝐷𝐴𝐵 is 98 degrees.
We can use the angle properties in a quadrilateral to help determine if that quadrilateral is cyclic or not. In this case, we’re given the diagonals. This might give us a clue that we can use the property that if an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side, then the quadrilateral is cyclic. We can note that this angle 𝐷𝐴𝐵 is not an angle created with a diagonal. It’s an angle created by two sides. However, this angle of 𝐵𝐶𝐴 is an angle created by a diagonal and side. The angle created by the other diagonal and opposite side would occur here at angle 𝐵𝐷𝐴.
If angle 𝐵𝐷𝐴 is equal in measure to angle 𝐵𝐶𝐴, then that would mean that 𝐴𝐵𝐶𝐷 is cyclic. We don’t know this angle measure, but let’s see if we can work it out. We can take this triangle of 𝐷𝐴𝐵 and observe that we’re given these two equal line segments, which means that triangle 𝐷𝐴𝐵 is an isosceles triangle. This means that the two angles at the base will be equal in length. We can even define them both as something like 𝑥 degrees.
We then use the fact that the interior angle measures in a triangle add up to 180 degrees. Therefore, the three angle measures of 𝑥 degrees, 𝑥 degrees, and 98 degrees must add to give 180 degrees. Simplifying this, we have two 𝑥 degrees equals 180 degrees minus 98 degrees, which is 82 degrees. When we divide through by two, we find that 𝑥 degrees is 41 degrees. And so, the two angle measures in this isosceles triangle must be 41 degrees.
But remember that we wanted to find this angle measure of 𝐵𝐷𝐴 because we wanted to check if it was the same as the angle measure of 𝐵𝐶𝐴. And, of course, 61 degrees is not equal to 41 degrees. And so we can say that the angle created by the diagonal and side is not equal to the angle created by the other diagonal and opposite side.
Therefore, we can give the answer no, there would not be a circle passing through the points 𝐴, 𝐵, 𝐶, and 𝐷.