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Question 1-Circle

I look this question up on the internet and found out that the answer was 58, but could someone please explain how to get to the answer? I know that the angle is 46 degrees, so I used the supplementary angle to figure out 134 degrees and vertical angles to figure out the other two angles. After that I have no clue what to do.

Chord Question

The answer is 10, but could someone please explain this to me? i thought that a chord has two of its endpoints on the circle, so how can it be tangent to the circle?

Quadrilateral inside a circle

Answer is 87, but could someone please explain this to me? I know that if a quadrilateral is inscribed inside of a circle, the opposite angles are supplementary.

Circle Angle

I don't know the answer for this one, could someone please explain this to me. First of all, I don't understand how BAC can be an angle. If it is an angle, I have no clue how to solve it.

Many thanks

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  • $\begingroup$ Welcome to StackExchange! Please provide an explanation of what you've tried and where you're stuck so we can help. $\endgroup$ – rb612 May 6 '18 at 7:06
  • $\begingroup$ "i thought that a chord has two of its endpoints on the circle, so how can it be tangent to the circle?" It's tangent to a different circle. Consider the segment $(-1,0)(1,0)$. It is a chord (the diameter actually) of the circle $x^2 + y^2 = 1$. But it is tangent to the circle $x^2 + (y-1)^2 = 1$ which is a different circle. $\endgroup$ – fleablood May 12 '18 at 16:21
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Question One:

Let us say that the arc opposite x has an arc measure of y.

Because the measure of the angle of two lines that intersect outside the circle is half the difference of the two intercepting arcs, you know that 1/2(x-y) = 12, leading to y = x - 24.

You also know that the measure of the angle of two lines that intersect inside the circle is the average of the intercepting arcs. This means that 1/2(x+y) = 46, or x + y = 92.

You have 2 variables and 2 equations, so when you solve for x, you get x = 58

Question Two:

The chord is tangent to the smaller circle and a chord of the larger circle.

If you take half of the chord, the radius of the small circle to the point of tangency, and the radius of the big circle to the point of intersection, you form a right triangle. Using Pythagorean Theorem, you find that half the chord is 5. Since it is half, if you multiply it by two, you get 10, which is the answer.

Question Three:

Because all arcs of a circle add up to 360, if you subtract major arc ABC, you get minor arc ADC to be 174 degrees. Because the measure of an inscribed angle is half the measure of its intercepted arc, the measure of angle ABC is half of 174, which is 87.

Question Four:

It means that if you draw segment AB, the measure of angle ABC.

To solve this problem, use the same theorem as above; The measure of an inscribed angle is half the measure of its intercepted arc. Because the intercepted arc is 100 degrees, measure of angle ABC is 50 degrees. (It's that simple)

If you have any questions, ask me!

That took forever

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