### Video Transcript

π΄, π΅, πΆ, and π· are points on a
circle with center π. π΄πΆ is a diameter. πΆπ is a tangent to the
circle. This figure is not drawn to
scale. Work out the size of angle π₯ and
the size of angle π¦.

Letβs start with the first
sentence. π΄, π΅, πΆ, and π· are points on a
circle with center π. Hereβs our center π. Points π΄, π΅, πΆ, and π· fall on
the circumference of the circle and cross through the center. That means that π΄π is a radius,
ππ΅ is a radius, ππΆ is a radius, and ππ· is a radius. This means that triangle ππ΅πΆ is
isosceles.

Angle ππ΅πΆ is equal to the angle
of ππΆπ΅. Angle π΅ππΆ equals 110
degrees. If we subtract 110 degrees from 180
degrees, we get 70 degrees. We know that this 70 degrees must
be divided evenly into the two remaining angles. 70 degrees divided by two equals 35
degrees. ππ΅πΆ equals 35 degrees and ππΆπ΅
equals 35 degrees.

Weβre told that line πΆπ is a
tangent to the circle. And that means that the angle
ππΆπ is 90 degrees. A radius meets a tangent line of a
circle at 90 degrees. The size of angle π₯ is 90 degrees
minus 35 degrees. The size of angle π₯ is 55
degrees.

Next, letβs look at the place that
line ππΆ and line π΅π· intersect. Weβll call this intersection point
πΈ. Angle ππΈπ΅ equals 58 degrees. This is because vertically opposite
angles are equivalent. Because π΅π· is a straight line,
angle ππΈπ· must be equal to 180 degrees minus 58 degrees. Angle ππΈπ· then measures 122
degrees.

We still have a few more angles we
need to find before we can say what π¦ is. Looking at the triangle created
from points ππΈπ΅ here in yellow, if we add 110 degrees plus 58 degrees and then
subtract that from 180, weβll find the angle ππ΅πΈ. 110 degrees plus 58 degrees equals
168 degrees. When we subtract that from 180, we
get 12 degrees. Angle ππ΅πΈ equals 12 degrees. We know that triangle π·ππ΅ is
isosceles. ππ΅ and ππ· are radii. And that means angle ππ·π΅ is
equal to angle ππ΅π·. And they both are 12 degrees.

Finally, letβs consider triangle
ππΈπ·. The three angles inside triangle
ππΈπ· must add up to 180 degrees. One of the angles is π¦. One of the angles is 122
degrees. And the last angle is 12
degrees. 180 degrees minus 134 degrees
equals π¦. π¦ equals 46 degrees.