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
Answers
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°.
What is Thales theorem?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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where
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