Deck 6: Circles

Question 1

Supply missing statements and missing reasons for the following proof.
Given: Chords and intersect at point N in Prove: )
S1. R1.
S2. Draw R2.
S3. R3. The measure of an ext. of a is the sum
of measures of the two nonadjacent int. . S4. and R4.
S5. R5. Substitution Property of Equality
S6. R6. Substitution Property of Equality

Answer 1

S1. Chords and intersect at point N in R1. Given
R2. Through 2 points, there is exactly one line.
R4. In a circle, the measure of an inscribed angle is one-half that of its intercepted arc.
S5. S6. )

Question 2

Supply missing statements and missing reasons for the proof of the following theorem.
"An angle inscribed in a semicircle is a right angle."
Given: with diameter and (as shown)
Prove: is a right angle.
S1. R1.
S2. R2.
S3. R3. The measure of a semicircle is 180.
S4. or R4.
S5. R5.

Answer 2

S1. with diameter and (as shown)
R1. Given
R2. The measure of an inscribed angle is on-half the degree measure of its intercepted arc.
S3. R4. Substitution Property of Equality
S5. is a right angle.
R5. Definition of a right angle.

Question 3

Use the drawing provided to explain why the following theorem is true.
"The tangent segments to a circle from an external point are congruent."
Given: and are tangent to Prove: [Hint: Use auxiliary line segment .]

Answer 3

Draw . Now and because the measure of an angle formed by a tangent and chord at the point of contact is one-half the measure of the intercepted arc.
Then by substitution, so . Then because these sides lie opposite the congruent angles of .

Question 4

Supply missing statements and missing reasons for the following proof.
Given: Chords , , , and as shown
Prove: S1. R1.
S2. R2.
S3. R3. If 2 inscribed intercept the same arc, these are .
S4. R4.

Question 5

Supply missing statements and missing reasons for the following proof.
Given: in Prove: is an isosceles triangle
S1. R1.
S2. R2.
S3. R3.
S4. ? and ? R4. The degree measure of an iscribed angle is equal to one-half
the degree measure of its intercepted arc.
S5. R5.
S6. R6. Definition of congruent angles
S7. R7. If two angles of a triangle are congruent, then the two sides
that lie opposite those angles are also congruent.
S8. R8.

Question 6

Supply missing reasons for the following proof.
Given: with diameter Prove: S1. with diameter R1.
S2. Draw radius R2.
S3. R3.
S4. R4.
S5. R5.
S6. R6.
S7. or R7. S8. R8.
S9. But R9.
S10. Then R10.

Question 7

Explain why the following must be true.
Given: Points A, B, and C lie on in such a way that ;
also, chords , , and (no drawing provided)
Prove: must be an isosceles triangle.

Question 8

Supply all statements and all reasons for the proof that follows.
Given: ; Prove:

Question 9

Supply missing statements and missing reasons for the following proof.
Given: In the circle, Prove: S1. R1.
S2. Draw R2.
S3. R3.
S4. R4. Congruent angles have equal measures.
S5. ? and ? R5. The measure of an inscribed angle equals one-half
the measure of its intercepted arc.
S6. R6.
S7. R7.
S8. R8.

Question 10

Supply missing statements and missing reasons for the following proof.
Given: ; chords and intersect at point V
Prove: S1. R1.
S2. Draw and . R2.
S3. R3. Vertical angles are congruent.
S4. R4.
S5. R5. AA
S6. R6.
S7. R7. Means-Extremes Property of a Proportion