### Video Transcript

Find π₯.

Considering our image, we have a circle and we have two segments that intersect at a point exterior to the circle. Letβs consider what other information we can determine based on what weβre given. We see that angle π΅πΈπ· measures 30 degrees and the intercepted arc of this angle is π΅π·. Additionally, we should notice that angle π΅πΆπ· also shares the intercepted arc π΅π·. And if two angles have the same intercepted arc, theyβll have the same angle measure. Therefore, the measure of angle π΅πΆπ· is also 30 degrees.

From there, we can look at triangle π΄πΆπ·. In this triangle, we have an angle that measures 30 degrees and an angle that measures 40 degrees. And therefore the measure of the missing angle πΆπ·π΄ will be equal to 180 degrees minus 30 degrees plus 40 degrees. We know the sum of these three angles must be 180 degrees since they form a triangle. Therefore, the measure of angle πΆπ·π΄ must be 110 degrees.

If we then consider triangle π΄π΅πΈ, we know the measures of two of the angles in the triangle, one being 30 degrees and the other being 40 degrees. Then the missing third angle, the measure of angle π΄π΅πΈ, will also be equal to 180 degrees minus 30 degrees plus 40 degrees, which is 110 degrees.

If we let the intersection of segments πΆπ· and π΅πΈ be π, we can consider the quadrilateral ππ·π΄π΅. And the measure of angle π΅ππ· will be equal to 360 degrees minus 110 degrees plus 110 degrees plus 40 degrees. This is because in a quadrilateral the interior angles must sum to 360 degrees. The measure of angle π΅ππ· equals 100 degrees.

For intersecting lines, opposite angles are congruent. The measure of angle πΆππΈ in the diagram is 100 degrees, which makes π₯ equal to 100.