Answer:
m<RUS = 65°
m<UST = 15°
Explanation:
Hi there!
We are given circle O, with a diameter of US
The measure of arc RU is 50°, and the measure of arc UT is 30°
We want to find the measure of <RUS and <UST
First, let's start with <RUS
As stated before, we were given the diameter of a circle - that is segment US
Notice how ΔRUS (which contains <RUS) contains the diameter US - this means that the portion of the circle that contains ΔRUS is a semicircle.
If that portion is a semicircle, then that means that m<URS is 90°
Next, we know that the measure of arc RU is 50°
Notice how <RSU is an inscribed angle, meaning that it is created by 2 chords, and that its vertex is on the circle itself
Inscribed angles are half the measure of the arcs they intercept. The arc that <RSU intercepts is arc RU, which means that the measure of <RSU is 25°
Now, to find m<RUS, you can do m<URS - m<RSU, as the acute angles in a right triangle add up to 90°
In that case:
m<RUS = m<URS - m<RSU
via substitution, m<RUS = 90° - 25°
m<RUS = 65°
Now we need to find the measure of <UST
Notice how m<UST is also an inscribed angle
The arc it intercepts is arc UT, which we were given has a measure of 30 degrees
Therefore, m<UST = half of the measure of arc UT
Via substitution,
m<UST = 1/2 * 30
m<UST = 15°
Hope this helps!