Schaum’s outline Geometry Sixth Edition

Circles

6.1 The Circle; Circle Relationships

The following terms are associated with the circle. Although some have been defined previously, they are repeated here for ready reference.

A circle is the set of all points in a plane that are at the same distance from a fixed point called the center. The symbol for circle is ; for circles .

The circumference of a circle is the distance around the circle. It contains 360°.

A radius of a circle is a line segment joining the center to a point on the circle.

Note: Since all radii of a given circle have the same length, we may at times use the word radius to mean the number that is “the length of the radius.”

A central angle is an angle formed by two radii.

An arc is a continuous part of a circle. The symbol for arc is . A semicircle is an arc measuring one-half the circumference of a circle.

A minor arc is an arc that is less than a semicircle. A major arc is an arc that is greater than a semicircle.

Fig. 6-1

Fig. 6-2

Thus in Fig. 6-1, is a minor arc and is a major arc. Three letters are needed to indicate a major arc.

To intercept an arc is to cut off the arc.

Thus in Fig. 6-1, and intercept .

A chord of a circle is a line segment joining two points of the circumference.

Thus in Fig. 6-2, is a chord.

A diameter of a circle is a chord through the center. A secant of a circle is a line that intersects the circle at two points. A tangent of a circle is a line that touches the circle at one and only one point no matter how far produced.

Thus, is a diameter of circle O in Fig. 6-2, is a secant, and is a tangent to the circle at P. P is the point of contact or the point of tangency.

An inscribed polygon is a polygon all of whose sides are chords of a circle. A circumscribed circle is a circle passing through each vertex of a polygon.

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Fig. 6-3

Thus , , and quadrilateral ABCD are inscribed polygons of circle O in Fig. 6-3. Circle O is a circumscribed circle of quadrilateral ABCD.

A circumscribed polygon is a polygon all of whose sides are tangents to a circle. An inscribed circle is a circle to which all the sides of a polygon are tangents.

Thus, is a circumscribed polygon of circle O in Fig. 6-4. Circle O is an inscribed circle of .

Concentric circles are circles that have the same center.

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Fig. 6-4

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Fig. 6-5

Thus, the two circles shown in Fig. 6-5 are concentric circles. is a tangent of the inner circle and a chord of the outer one. is a secant of the inner circle and a chord of the outer one.

Two circles are equal if their radii are equal in length; two circles are congruent if their radii are congruent.

Two arcs are congruent if they have equal degree measure and length. We use the notation mAC to denote “measure of arc AC.”

6.1A Circle Principles

PRINCIPLE 1: A diameter divides a circle into two equal parts.

Thus, diameter divides circle O of Fig. 6-6 into two congruent semicircles, and .

PRINCIPLE 2: If a chord divides a circle into two equal parts, then it is a diameter. (This is the converse of Principle 1.)

Thus if in Fig. 6-6, then is a diameter.

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Fig. 6-6

PRINCIPLE 3: A point is outside, on, or inside a circle according to whether its distance from the center is greater than, equal to, or smaller than the radius.

F is outside circle O in Fig. 6-6, since is greater in length than a radius. E is inside circle O since is smaller in length than a radius. A is on circle O since is a radius.

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Fig. 6-7

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Fig. 6-8

PRINCIPLE 4: Radii of the same or congruent circles are congruent.

Thus in circle O of Fig. 6-7, .

PRINCIPLE 5: Diameters of the same or congruent circles are congruent.

Thus in circle O of Fig. 6-7, .

PRINCIPLE 6: In the same or congruent circles, congruent central angles have congruent arcs.

Thus in circle O of Fig. 6-8, if , then .

PRINCIPLE 7: In the same or congruent circles, congruent arcs have congruent central angles.

Thus in circle O of Fig. 6-8, if , then .

(Principles 6 and 7 are converses of each other.)

PRINCIPLE 8: In the same or congruent circles, congruent chords have congruent arcs.

Thus in circle O of Fig. 6-9, if , then .

PRINCIPLE 9: In the same or congruent circles, congruent arcs have congruent chords.

Thus in circle O of Fig. 6-9, if , then .

(Principles 8 and 9 are converses of each other.)

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Fig. 6-9

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Fig. 6-10

PRINCIPLE 10: A diameter perpendicular to a chord bisects the chord and its arcs.

Thus in circle O of Fig. 6-10, if , then bisects , , and .

A proof of this principle is given in Chapter 16.

PRINCIPLE 11: A perpendicular bisector of a chord passes through the center of the circle.

Thus in circle O of Fig. 6-11, if is the perpendicular bisector of , then passes through center O.

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Fig. 6-11

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Fig. 6-12

PRINCIPLE 12: In the same or congruent circles, congruent chords are equally distant from the center.

Thus in circle O of Fig. 6-12, if , if , and if , then .

PRINCIPLE 13: In the same or congruent circles, chords that are equally distant from the center are congruent.

Thus in circle O of Fig. 6-12, if , , and , then .

(Principles 12 and 13 are converses of each other.)

SOLVED PROBLEMS

6.1 Matching test of circle vocabulary

Match each part of Fig. 6-13 on the left with one of the names on the right:

Fig. 6-13

Solutions

(a) 1

(b) 6

(d) 9

(e) 8

(f) 4

(g) 3

(h) 5

(i) 2

(j) 13

(k) 12

(l) 10

(m) 11

6.2 Applying principles 4 and 5

In Fig. 6-14, (a) what kind of triangle is OCD; (b) what kind of quadrilateral is ABCD? (c) In Fig. 6-15 if circle O = circle Q, what kind of quadrilateral is OAQB?

Solutions

Radii or diameters of the same or equal circles have equal lengths.

(a) Since , is isosceles.

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Fig. 6-14

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Fig. 6-15

(b) Since diagonals and are equal in length and bisect each other, ABCD is a rectangle.

6.3 Proving a circle problem

PROOF:

6.4 Proving a circle problem stated in words

Prove that if a radius bisects a chord, then it is perpendicular to the chord.

Solution

PROOF:

6.2 Tangents

The length of a tangent from a point to a circle is the length of the segment of the tangent from the given point to the point of tangency. Thus, PA is the length of the tangent from P to circle O in Fig. 6-16.

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Fig. 6-16

6.2A Tangent Principles

PRINCIPLE 1: A tangent is perpendicular to the radius drawn to the point of contact.

Thus if is a tangent to circle O at P in Fig. 6-17, and is drawn, then .

PRINCIPLE 2: A line is tangent to a circle if it is perpendicular to a radius at its outer end.

Thus if radius at P of Fig. 6-17, then is tangent to circle O.

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Fig. 6-17

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Fig. 6-18

PRINCIPLE 3: A line passes through the center of a circle if it is perpendicular to a tangent at its point of contact.

Thus if is tangent to circle O at P in Fig. 6-18, and at P, then extended will pass through the center O.

PRINCIPLE 4: Tangents to a circle from an outside point are congruent.

Thus if and are tangent to circle O at P and Q (Fig. 6-19), then .

PRINCIPLE 5: The segment from the center of a circle to an outside point bisects the angle between the tangents from the point to the circle.

Thus bisects in Fig. 6-19 if and are tangents to circle O.

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Fig. 6-19

6.2B Two Circles in Varying Relative Positions

The line of centers of two circles is the line joining their centers. Thus, is the line of centers of circles O and O′ in Fig. 6-20.

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Fig. 6-20

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Fig. 6-21

Circles Tangent Externally

Circles O and O′ in Fig. 6-21 are tangent externally at P. is the common internal tangent of both circles. The line of centers passes through P, is perpendicular to , and is equal in length to the sum of the radii, R + r. Also bisects each of the common external tangents, and .

Circles Tangent Internally

Circles O and O′ in Fig. 6-22 are tangent internally at P. is the common external tangent of both circles. The line of centers if extended passes through P, is perpendicular to , and is equal in length to the difference of the radii, R – r.

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Fig. 6-22

Overlapping Circles

Circles O and O′ in Fig. 6-23 overlap. Their common chord is . If the circles are unequal, their (equal) common external tangents and meet at P. The line of centers is the perpendicular bisector of and, if extended, passes through P.

Circles Outside Each Other

Circles O and O′ in Fig. 6-24 are entirely outside each other. The common internal tangents, and meet at P. If the circles are unequal, their common external tangents, and if extended, meet at P′. The line of centers passes through P and P′. Also, AB = CD and EF = GH.

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Fig. 6-23

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Fig. 6-24

SOLVED PROBLEMS

6.5 Triangles and quadrilaterals having tangent sides

Points P, Q, and R in Fig. 6-25 are points of tangency.

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Fig. 6-25

(a) In Fig. 6-25(a), if AP = OP, what kind of triangle is OPA?

(b) In Fig. 6-25(b), if AP = PQ, what kind of triangle is APQ?

(d) In Fig. 6-25(c), if , what kind of quadrilateral is PABR?

Solutions

(a) is tangent to the circle at P; then by Principle 1, is a right angle. Also, AP = OP. Hence, is an isosceles right triangle.

(b) and are tangents from a point to the circle; hence by Principle 4, AP = AQ. Also, AP = PQ. Then is an equilateral triangle.

(c) By Principle 4, AP = AQ. Also, and are radii. And AP = OP. By Principle 1, is a rt. . Then AP = AQ = OP = OQ; hence, OPAQ is a rhombus with a right angle, or a square.

(d) By Principle 1, and . Then , since both are to . By Principle 1, ; also, (Given). Then , since both are to . Hence, PABR is a parallelogram with a right angle, or a rectangle.

6.6 Applying principle 1

(a) In Fig. 6-26(a), is a tangent. Find = 2:3.

(b) In Fig. 6-26(b), and are tangents. Find = 140°.

Solutions

(a) By Principle 1, = 90°. Then + = 90°. If = 2x and = 3x, then 5x = 90 and x = 18. Hence, = 36°.

Fig. 6-26

(b) By Principle 1, = = 90°. Since + + + = 360°, + = 180°. Since = 140°, = 40°. By Principle 5, = = 20°.

6.7 Applying principle 4

(a) , , and in Fig. 6-27(a) are tangents. Find y.

(b) in Fig. 6-27(b) is circumscribed. Find x.

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Fig. 6-27

Solutions

(a) By Principle 4, AR = 6, and RB = y. Then RB = AB – AR = 14 – 6 = 8. Hence, y = RB = 8.

(b) By Principle 4, PC = 8, QB = 4, and AP = AQ. Then AQ = AB – QB = 11. Hence, x = AP + PC = 11 + 8 = 19.

6.8 Finding the line of centers

Two circles have radii of 9 and 4, respectively. Find the length of their line of centers (a) if the circles are tangent externally, (b) if the circles are tangent internally, (c) if the circles are concentric, (d) if the circles are 5 units apart. (See Fig. 6-28.)

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Fig. 6-28

Solutions

Let R = radius of larger circle, r = radius of smaller circle.

(a) Since R = 9 and r = 4, OO′ = R + r = 9 + 4 = 13.

(b) Since R = 9 and r = 4, OO′ = R – r = 9 – 4 = 5.

(d) Since R = 9, r = 4, and d = 5, OO′ = R + d + r = 9 + 5 + 4 = 18.

6.9 Proving a tangent problem stated in words

PROOF:

6.3 Measurement of Angles and Arcs in a Circle

A central angle has the same number of degrees as the arc it intercepts. Thus, as shown in Fig. 6-29, a central angle which is a right angle intercepts a 90° arc; a 40° central angle intercepts a 40° arc, and a central angle which is a straight angle intercepts a semicircle of 180°.

Since the numerical measures in degrees of both the central angle and its intercepted arc are the same, we may restate the above principle as follows: A central angle is measured by its intercepted arc. The symbol may be used to mean “is measured by.’’ (Do not say that the central angle equals its intercepted arc. An angle cannot equal an arc.)

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Fig. 6-29

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords. An angle inscribed in an arc has its vertex on the arc and its sides passing through the ends of the arc. Thus, in Fig. 6-30 is an inscribed angle whose sides are the chords and . Note that intercepts and is inscribed in .

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Fig. 6-30

6.3A Angle-Measurement Principles

PRINCIPLE 1: A central angle is measured by its intercepted arc.

PRINCIPLE 2: An inscribed angle is measured by one-half its intercepted arc.

A proof of this principle is given in Chapter 16.

PRINCIPLE 3: In the same or congruent circles, congruent inscribed angles have congruent intercepted arcs.

Thus in Fig. 6-31, if , then .

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Fig. 6-31

PRINCIPLE 4: In the same or congruent circles, inscribed angles having congruent intercepted arcs are congruent. (This is the converse of Principle 3.)

Thus in Fig. 6-31, if , then .

PRINCIPLE 5: Angles inscribed in the same or congruent arcs are congruent.

Thus in Fig. 6-32, if and are inscribed in , then .

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Fig. 6-32

PRINCIPLE 6: An angle inscribed in a semicircle is a right angle.

Thus in Fig. 6-33, since is inscribed in semicircle , = 90°.

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Fig. 6-33

PRINCIPLE 7: Opposite angles of an inscribed quadrilateral are supplementary.

Thus in Fig. 6-34, if ABCD is an inscribed quadrilateral, is the supplement of .

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Fig. 6-34

PRINCIPLE 8: Parallel lines intercept congruent arcs on a circle.

Thus in Fig. 6-35, if , then . If tangent is parallel to , then .

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Fig. 6-35

PRINCIPLE 9: An angle formed by a tangent and a chord is measured by one-half its intercepted arc.

PRINCIPLE 10: An angle formed by two intersecting chords is measured by one-half the sum of the intercepted arcs.

PRINCIPLE 11: An angle formed by two secants intersecting outside a circle is measured by one-half the difference of the intercepted arcs.

PRINCIPLE 12: An angle formed by a tangent and a secant intersecting outside a circle is measured by one-half the difference of the intercepted arcs.

PRINCIPLE 13: An angle formed by two tangents intersecting outside a circle is measured by one-half the difference of the intercepted arcs.

Proofs of Principles 10 to 13 are given in Chapter 16.

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Fig. 6-36

6.3B Table of Angle-Measurement Principles

Note: To find the angle formed by a secant and a chord meeting on the circle, first find the measure of the inscribed angle adjacent to it and then subtract from 180°. Thus if secant meets chord atg C on the circle in Fig. 6-36, to find , first find the measure of inscribed . Obtain by subtracting from 180°.

SOLVED PROBLEMS

6.10 Applying principles 1 and 2

Fig. 6-37

Solutions

6.11 Applying principles 3 to 8

Find x and y in each part of Fig. 6-38.

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Fig. 6-38

Solutions

6.12 Applying principle 9

In each part of Fig. 6-39, CD is a tangent at P.

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Fig. 6-39

Solutions

6.13 Applying Principle 10

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Fig. 6-40

Solutions

6.14 Applying principles 11 to 13

Fig. 6-41

Solutions

6.15 Using equations in two unknowns to find arcs

In each part of Fig. 6-42, find x and y using equations in two unknowns.

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Fig. 6-42

Solutions

6.16 Measuring angles and arcs in general

Find x and y in each part of Fig. 6-43.

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Fig. 6-43

Solutions

6.17 Proving an angle measurement problem

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PROOF:

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6.18 Proving an angle measurement problem stated in words

Prove that parallel chords drawn at the ends of a diameter are equal in length.

Solution

PROOF:

SUPPLEMENTARY PROBLEMS

6.1. Provide the proofs requested in Fig. 6-44.

(6.3)

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Fig. 6-44

6.2. Provide the proofs requested in Fig. 6-45. Please refer to figure 6-45(a) for problems 6.2(a) and (b); to figure 6-45(b) for problems 6.2(c) and (d); and figure 6-45(c) for problems 6.2(e) and (f).

(6.3)

Fig. 6-45

6.3. Prove each of the following:

(6.4)

(a) If a radius bisects a chord, then it bisects its arcs.

(b) If a diameter bisects the major arc of a chord, then it is perpendicular to the chord.

6.4. Prove each of the following:

(6.4)

(a) A radius through the point of intersection of two congruent chords bisects an angle formed by them.

(b) If chords drawn from the ends of a diameter make congruent angles with the diameter, the chords are congruent.

(d) In a circle, chords that are equally distant from the center are congruent.

6.5. Determine each of the following, assuming t, t′, and t″ in Fig. 6-46 are tangents.

(6.5)

(a) If = 90° in Fig. 6-46(a), what kind of quadrilateral is PAQO?

(b) If BR = RC in Fig. 6-46(b), what kind of triangle is ABC?

(d) What kind of triangle is AOB in Fig. 6-46(c)?

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Fig. 6-46

6.6. In circle O, radii and are drawn to the points of tangency of and . Find if equals (a) 40°; (b) 120°; (c) 90°; (d) x°; (e) (180 – x)°; (f) (90 – x)°.

(6.6)

6.7. Find each of the following (t and t′ in Fig. 6-47 are tangents).

(6.6)

In Fig. 6-47(a)

(a) If = 80°, find .

(b) If = 25°, find and .

Fig. 6-47

In Fig. 6-47(b)

(d) If bisects , find .

(e) If = 35°, find .

(f) If PQ = QB, find .

6.8. In Fig. 6-48(a), is circumscribed. (a) If y = 9, find x. (b) If x = 25, find y.

(6.7)

In Fig. 6-48(b), quadrilateral ABCD is circumscribed. (c) Find AB + CD. (d) Find perimeter of ABCD.

In Fig. 6-48(c), quadrilateral ABCD is circumscribed. (e) If r = 10, find x. (f) If x = 25, find r.

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Fig. 6-48

6.9. If two circles have radii of 20 and 13, respectively, find their line of centers:

(6.8)

(a) If the circles are concentric

(b) If the circles are 7 units apart

(d) If the circles are tangent internally

6.10. If the line of centers of two circles measures 30, what is the relation between the two circles:

(6.8)

(a) If their radii are 25 and 5?

(b) If their radii are 35 and 5?

(d) If their radii are 25 and 10?

6.11. What is the relation between two circles if the length of their line of centers is (a) 0; (b) equal to the difference of their radii; (c) equal to the sum of their radii; (d) greater than the sum of their radii, (e) less than the difference of their radii and greater than 0; (f) greater than the difference and less than the sum of their radii?

(6.8)

6.12. Prove each of the following:

(6.9)

(a) The line from the center of a circle to an outside point bisects the angle between the tangents from the point to the circle.

(b) If two circles are tangent externally, their common internal tangent bisects a common external tangent.

(d) In a circumscribed quadrilateral, the sum of the lengths of the two opposite sides equals the sum of the lengths of the other two.

6.13. Find the number of degrees in a central angle which intercepts an arc of (a) 40°; (b) 90°; (c) 170°; (d) 180°; (e) 2x°; (f) (180 – x)°; (g) (2x – 2y)°.

(6.10)

6.14. Find the number of degrees in an inscribed angle which intercepts an arc of (a) 40°; (b) 90°; (c) 170°; (d) 180°; (e) 260°; (f) 348°; (g) 2x°; (h) (180 – x)°; (i) (2x – 2y)°.

(6.10)

6.15. Find the number of degrees in the arc intercepted by

(6.10)

(a) A central angle of 85°

(b) An inscribed angle of 85°

(d) An inscribed angle of i°

(e) The central angle of a triangle formed by two radii and a chord equal to a radius

(f) The smallest angle of an inscribed triangle whose angles intercept arcs in the ratio of 1:2:3

6.16. Find the number of degrees in each of the arcs intercepted by the angles of an inscribed triangle if the measures of these angles are in the ratio of (a) 1:2:3; (b) 2:3:4; (c) 5:6:7; (d) 1:4:5.

(6.10)

6.17.

Fig. 6-49

6.18. If quadrilateral ABCD is inscribed in a circle in Fig. 6-50, find

(a) if = 45°

(b) if = 90°

(d) if = (90 – x)°

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Fig. 6-50

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Fig. 6-51

6.19. If BC and AD are the parallel sides of inscribed trapezoid ABCD in Fig. 6-51, find

(6.11)

6.20. A diameter is parallel to a chord. Find the number of degrees in an arc between the diameter and chord if the chord intercepts (a) a minor arc of 80°; (b) a major arc of 300°.

(6.11)

6.21. Find x and y in each part of Fig. 6-52.

(6.11)

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Fig. 6-52

6.22. Find the number of degrees in the angle formed by a tangent and a chord drawn to the point of tangency if the intercepted arc has measure (a) 38°; (b) 90°; (c) 138°; (d) 180°; (e) 250°; (f) 334°; (g) x°; (h) (360 – x)°; (i) (2x + 2y)°.

(6.12)

6.23. Find the number of degrees in the arc intercepted by an angle formed by a tangent and a chord drawn to the point of tangency if the angle measures (a) 55°; (b) ; (c) 90°; (d) 135°; (e) (90 – x)°; (f) (180 – x)°; (g) (x – y)°; (h) .

(6.12)

6.24. Find the number of degrees in the acute angle formed by a tangent through one vertex and an adjacent side of an inscribed (a) square; (b) equilateral triangle; (c) regular hexagon; (d) regular decagon.

(6.12)

6.25. Find x and y in each part of Fig. 6-53 (t and t′ are tangents).

(6.12)

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