Answer:
m∠SRC = 90°
m∠YRC = 50°
m∠RYC = 50°
m∠YCR = 80°
m arcRZY = 80°
Step-by-step explanation:
A tangent to a circle is always perpendicular to the radius at the point of tangency. Hence ∠SRC = 90°.
∠YRC is the difference between ∠SRC and ∠SRY, which is given as 40°. Hence ∠YRC is 90°-40° = 50°.
ΔYCR is isosceles, to ∠RYC = ∠YRC = 50°.
The sum of angles in a triangle is 180°, so the central angle YCR can be found from ...
∠YCR + 50° + 50° = 180°
∠YCR = 80°
The problem statement tells you the arc measure is the same as the angle you just found, so is 80°.
_____
Comment on the question
In the end, the arc measure (and the corresponding central angle) are double the angle made by the tangent and the chord RY. You can use a variable for the value of ∠SRY and work through this again to verify that will always be the case.
