Given: circle k(O), diameter US, measure of arc RU=50, measure of arc UT=30
Find: measure of angle RUS and measure of angle UST

Answer:

angle RUS = 65 degrees

angle STU = 90 degrees

Step-by-step explanation:

(a) find angle RUS

Given arc RU = 50 degrees, so angle RSU = 50/2 = 25 degrees

Since US is the diameter,

angle URS is a right-angle.

Consider triangle RUS,

sum of angles of a triangle = 25+90+RUS = 180  

solve for

RUS = 180 - 25 - 90 = 65 degrees.

(b) Find STU

All diameters span a right angle at the circumference,

since we're given US is a diameter, this means that

angle STU is a right angle.

The measure of the ∠RUS is 65°, while the measure of ∠UST is 15°.

The angle ABC is a right angle if A, B, and C are different points on a circle where the line AC is a diameter, according to Thales' theorem.

We know that when an angle is made by the diameter of a circle at its circumference, then the measure of that angle is 90°.

In ΔUST,

The angle made by the diameter is ∠UST, therefore, the measure of this angle will be 90°. Since the measure of the arcRU=50°, therefore,

arcRU = ∠ROU = 50°, Also,

[tex]\angle RSU=\dfrac{\angle ROU}2 =\dfrac{50^o}2 = 25^o[/tex]

Now, the sum of the angles of a triangle is 180°.

∠R+∠U+∠S=180°

90°+∠RUS+25°=180°

∠RUS = 65°

In ΔRUS,

The angle made by the diameter is ∠URS, therefore, the measure of this angle will be 90°. Since the measure of the arcUT=30°, therefore,

arcUT = ∠UOT = 30°, Also,

[tex]\angle UST=\dfrac{\angle UOT}2 =\dfrac{30^o}2 = 15^o[/tex]

Thus, the measure of the ∠RUS is 65°, while the measure of ∠UST is 15°.

Learn more about Thale's Theorem:

https://brainly.com/question/14417137

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