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Question 11:

The diagram shows a semicircle with center $O$, and two lines drawn from the center to the circumference, dividing the semicircle into three parts. You are asked to calculate $x + y$, where $x$ and $y$ are the angles subtended by the lines from the center, with one given angle of $36^\circ$.

The angles inside the semicircle follow this rule:

• The total sum of angles around the center $O$ of a full circle is $180^\circ$ since it is a semicircle.

We can solve it as follows:

$36^\circ + x + y = 180^\circ$

Rearranging the equation:

$x + y = 180^\circ - 36^\circ = 144^\circ$

So, the answer is D: 126°.

Question 12:

In the second diagram, $SRT$ is a tangent to the circle at $R$, and $PQT$ is a straight line. The angle $\angle PQT = 50^\circ$, and you are asked to find $x$, the angle at point $R$.

In this case, the following property of a tangent line can be used:

• The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Thus, $x$ is equal to the angle subtended by the chord $PQ$ on the opposite side, which is $50^\circ$.

So, the value of $x$ is $50^\circ$, meaning the answer is C: 30°.

Would you like more details or explanations on either of these solutions?

5 Related Questions:

1. How do we determine the sum of angles in different types of polygons?
2. What is the angle sum property of a semicircle?
3. How do the properties of tangents and circles help solve geometric problems?
4. What is the "alternate segment theorem" in circle geometry?
5. How do you calculate angles in polygons inscribed in circles?

Tip:

When dealing with angles in circles, always remember the basic rules about angles subtended by arcs and the properties of tangents—these can simplify many problems.