Questions

1) An angle inscribed in a circle is an angle with its __________ on the circle and its rays are __________ of the circle.

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2) An intercepted arc is an arc that lies in the __________ of an inscribed angle and is formed by the intersection of the chords of an inscribed angle with the circle.

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3) Identify at least three inscribed angles in the figure shown below.

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4) Theorem 23-A: If an angle is inscribed in a circle, then the measure of the angle is __________ of the measure of the intercepted arc.

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5) In the unit link to “Inscribed Angles”, there is a proof for Case 1 of Theorem 23-A. Study the proof and apply its concepts to prove case 2. Parts of the proof are shown below. Fill in the missing parts of the proof for steps 1 through 6.

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6) What is the measure of angle RST?

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7) Refer to the figure below to answer the following questions about circle V: (a) What is the measure of angle XWY? (b) What is the measure of angle VXW? (c) What is the measure of angle XVW?

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8) Theorem 23-B: If two inscribed angles intercept the same arc, then the angles are __________.

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9) What is the measure of (a) angle SWT? (b) angle VWU? (c) arc VU?

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10) What is the measure of (a) angle SVT? (b) angle SUT? (c) angle VSU? (d) angle VTU?

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11) What is the measure of (a) angle SWV? (b) angle TWU? (c) arc SV (d) arc TU?

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12) Since angles VSU and SUT are congruent, what is true about chord SV and chord TU?

a.) Chord SV and chord TU are NOT congruent.

b.) Chord SV and chord TU are intersecting chords.

c.) Chord SV and chord TU are diameters of circle W.

d.) Chord SV and chord TU are parallel.

13) Theorem 23-C: An angle that is inscribed in a circle is a right angle if and only if its intercepted arc is a __________.

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14) What is the measure of angle LMN?

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15) What is the measure of (a) angle CAB? (b) angle DAC? (c) angle DBA?

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16) Theorem 23-D: If a quadrilateral is inscribed in a circle, then its opposite angles are __________.

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17) What is the value (a) of “x”? (b) of “y”?

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18) What is the measure (a) of angle J? (b) of angle L? (c) of angle K? and (d) of angle M?

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19) Fill in the blanks: A tangent line to a circle is a line that intersects the circle at _________ _________ __________.

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20) Describe a “point of tangency”.

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21) Theorem 23-E: If a line is tangent to a circle, then it is __________ to the radius drawn to the point of tangency.

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22) In triangle ABC shown below, (a) explain why angle ABC is a right angle, (b) state the length of AC, and (c) determine the value of “x”?

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23) Theorem 23-F: If a radius is perpendicular to a line at the point at which the line intersects the circle, then the line is a __________.

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24) Fill in the blanks: A common tangent is a line or line segment that is tangent to ________ ________ in the same plane.

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25) Common external tangents (do, do not) intersect a line segment that has its endpoints on the centers of the two circles.

a.) do

b.) do not

26) Common internal tangents (do, do not) intersect a line segment that has its endpoints on the centers of the two circles.

a.) do

b.) do not

27) For the figure and the information shown below, explain how the Angle Angle Side Theorem (AAS) can be developed to show that segment AD is congruent to segment DB?

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28) Fill in the blank: A polygon with sides that are all tangents to a circle within the polygon is said to be __________________ about the circle.

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29) Complete the following statement about the activity above: The triangle is circumscribed about the circle and each side of the triangle is a __________ to the circle.

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30) Theorem 23-G: If two segments from the same exterior point are tangent to a circle, then the two segments are __________.

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31) Examine the figure and the information given below. Answer the following questions about the figure: (a) Why can it be stated that segments VU and VW are congruent? (b) What property of equality justifies the statement that segment TV is congruent to itself? (c) What theorem in this unit justifies the statement that segments TU and TW are congruent? (d) What postulate can be used to deduce that triangles TUV and TWV are congruent?

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32) What is the value of “x”? If necessary, refer to the unit links that provide a review of Algebra I topics. (Exclude negative values when dealing with lengths.)

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33) Describe a secant.

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34) Select the answer that completes Theorem 23-H: If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is _________ the measure of its intercepted arc.

a.) twice

b.) one-half

c.) one-third

d.) three times

e.) one-fourth

35) Select the answer that completes the Theorem 23-I: If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the __________ and its _________ __________.

a.) angle, inscribed angle

b.) angle, central angle

c.) angle, supplementary angle

d.) angle, vertical angle

e.) angle, right angle

36) In circle P, arcs JK and LM each measure 30 degrees. What is the measure of angle LNM?

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37) For the figure and the information provided below, answer the following questions: (a) What is the value of “x”? (b) What is the measure of arc AB? (c) What is the measure of arc CD? (d) What is the measure of angle DFC?

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38) Select the answer that completes the Theorem 23-J: The measure of an angle formed by two secants, a secant and a tangent, or two tangents intersecting in the exterior of a circle is equal to one-half the __________ __________ of the measures of the intercepted arcs.

a.) positive difference

b.) positive sum

c.) positive quotient

d.) positive product

39) Which illustration below does not represent Theorem 23-J? State the letter of the correct answer.

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40) Find the measure of angle S if arc RT measures 75 degrees and arc RU measures 148 degrees.

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41) For circle L and the information given below, answer the following questions: (a) What is the value of “x”? (b) What is the value of “y”? (c) What is the value of “z”?

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42) For circle A, find the value of “x”. Also, write an equation for the solution.

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43) For circle T, find the value of “x”. (Exclude negative values when dealing with lengths.) Also, write an equation in standard form that could be used to solve the problem.

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44) In circle R, what is the value of “x “?

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45) Activity: Step 1: Draw a circle and construct an inscribed hexagon within the circle. Step 2: Construct a perpendicular bisector for each side of the hexagon. Make sure that each bisector intersects the circle. Step 3: Connect all the points of intersection consecutively around the circle (in other words, connect the point of intersection formed by an arc, then the next point of intersection formed by a bisector, and so on). What regular polygon have you constructed?

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46) Activity: Modify the steps for constructing a hexagon to make the inscribed six-pointed star below. Explain how you created the star.

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