LAOB ~ LCOD

LMBE

Postulate 19. In a plane one, and only one, circle can be drawn with a given POZ as center and a given line segment as radius

I. LAOB ~ LCOD

2. IIILAOB

= mLCOD.

;t IIILAOB = mAR; mLCOD = mcn.

4. IIIAn = mCD.

0. AB ~ CD.

1. Given.

2. Definition of ~ .6.

3. Definition of measure of arc.

4. Substitution property.

5. Definition of

^~

arcs.

Theorem 7.2

7.12. If two arcs of a circle or congruent circles are congruent, then the central angles intercepted by these arcs are congruent.

C rhe proof of this theorem is left as an exercise for the student.)

Exercises (A)

Indicate which of the following statements are always true (mark T) and which are not always true (mark F).

I. All central angles of the same circle are congruent.

2. If the radii of two circles are not congruent, the circles are not congruent.

3. Two circles each with radii of 10 inches have congruent diameters.

4. A point is outside of a circle if its distance from the center of the circle ell uals the measure of the diameter of the circle.

5. The measure of a major arc is more than the measure of a minor arc.

6. The vertex of a central angle is on the circle.

7. Every circle has exactly two semicircles.

8. Every arc of a circle subtends a central angle.

9. All semicircles are congruent.

10. If two arcs of the same circle are congruent, their central angles must be congruent.

II. A chord is a diameter.

12. Some radii of a circle are chords of that circle.

13. Every diameter is a chord.

14. In a given circle it is possible for a chord to be congruent to a radius.

15. The intersection of a circle and one of its chords is a null set.

16. The intersection of a plane and a sphere can be a point.

17. The intersection of two diameters of a given circle is four points.

18. No chord of a circle can equal a diameter.

19. A sphere is a set of points.

20. Every s!-,here has only one great circle.

216

FUNDAMENTALS OF COLLEGE GEOMETRY

Exercises (B)

Find the number of degrees asked for in each of the following.

0 will indicate centers of circles.

21. Given: mBe = 70; AC is a diameter.

Find: mLAOB.

B

A

Ex. 21.

23. Given: mLOAB

= 30;

AC is a diameter.

Find: mBG'.

25. Given: mLAOB = 60;

AC is a diameter.

Find: (a) mLABC; (b) mBe.

Ex. 23.

A

Ex. 25.

22. Given: mLOAB = 36.

Find: mAE.

B

c

A

24. Given: mAR = 70; AC is a diameter.

Find: mLOBC.

26. Given: mAD = 140;

BD and AC are diameters.

Find: mLOBC.

C

c

A

B

D

CIRCLES 217 7.13. Inscribed angle. An angle is inscribed in an arc of a circle if the end- points of the arc are points on the sides of the angle and if the vertex of the angle is a point, but not an endpoint, of the arc. In Fig. 7.15a, LABC is inscribed in the minor arc ABC. In Fig. 7.15b, LDEF is inscribed in major lJEF. The angles are called inscribed angles.

A

B E F

C

(a) _{Fig. 7.15. (b)}

7.14. Intercepted arcs. An angle inter-

Cf'jJls an arc if the endpoints of the arc

lie on the sides of the angle, and if each side of the angle contains at least one end point of the arc, and if, except for the endpoints, the arc lies in the interior of the angle. Thus, in Fig. 7.15, LABC intercepts AC

^{and LDEF intercepts}

fSF.

In Fig. 7.1t},£AEB-int{,K~s--AEattdn A.B, and LEPD intercepts DE and Be.

A p

Fig. 7.16.

Theorem 7.3

7.15. The measure of an inscribed angle is equal to half the measure of its intercepted arc.

Given: LBAC inscribed in 00.

Conclusion: mLBAC = tmBC.

A A A

B B

B D D

Theorem 7.3.

218

FUNDAMENTALS OF COLLEGE GEOMETRY

CASEI: When one side of the angle is a diameter.

Proof

ST ATEMENTS REASONS

1. LBAC is inscribed in 00.

2. DrawOC.

3. OA ::=OC.

4. LA ::= LC.

5. mLBOC = mLA + mLC.

6. mLBOC=mLA+mLA.

1. Given.

2. Postulate 2.

3. Definition of a O.

4. Theorem 4.16.

5. Theorem 5.16.

6. Symmetric and substitution pro-!

perties of equality. (With State- ments 4 and 5).

7. Definition of measure of arc.

8. Theorem 3.5.

9. Division property of equality.

7. mLBOC = mBC.

8. mLA + mLA = mBC.

9. mLA = tmBC. ..

C.ME n, When f)" an'" 'I'k"i"k Ii" within th, in"""'"Itk, anK/£'

' I

^......

Plan: Draw diameter AD and apply CASE I to La and Lf3. Use the additive:

property of equality. .

CASE III: When the center of the circle lies in the exterior of the angle.

1

.:.

Piau: Draw diam~ter AD. Apply CASE I to La and Lf3. LIse the subtrac- lift

tive property of equaiIty.~--

... . m...~ ~ _me

--

The proofs of the following corollaries are left to the student.

7.16. Corollary:

An angle inscribed in a semicircle is a right angle.

7.17. Corollary: Angles inscribed in the same arc are congruent.

7.18. Corollary: Parallel lines cut off congruent arcs on a circle.

Plan of proof: LABC ::= LBCD.

mLABC

= tmAC. mLBCD = tmBD.

ThenAC::= BD.

Corollary 7.18.

Exercises

In each of the following problems, 0 is the center of a circle.

I. Given: 0 center of 0;

mAR

=

100;

mAD = 140;

mDC = 66.

Find: the measure of each of the four central L!;.

2. Given: Inscribed quadrilateral diagonals RT and WS.

Which angles have measures as

(a) L WRT?

(c) LRTS?

WRST with

(b) LWTR?

3. Given: Chord LM

_{IIchord UV;}

mLVLH = 25.

- - "'=" ,-.-c-

----~-~~7~=",",,"_-'-'-_-=-7-'--_=C-- -

Find: (llJlnVM; (b)mUL.

4. Given: mLA

= 68.

Find: mLC.

the same

B Ex. 1.

Ex. 2.

Ex. 3.

D

c

A Ex. 4.

V I'UNUAMENTALS OF COLLEGE GEOMETRY

5. Given: mLSOR

=

80.

Find: mLT.

6. Given: Chords AB and CD intersecting at E;

mAG

= 40; mBD = 70.

Find: mLAEC.

7. Given: Aii

^.1

to diameter TD;

TC is a chord;

mTC = 100.

Find: mLBTC.

8. Given: AB

^.1

to diameter RS;

OT is a radius; TS is a chord; mLROT = 1I0.

Find: mLTSA.

T

Ex. 5.

Ex. 6.

A

Ex. 7.

R

Ex. 8.

:A

s

,B

9. Given: mLAEC

= 80;

mAC = 100.

Find: mBD.

Find:

WS bisects LRST;

mRS = 120;

mWR = 62.

(a) mLTKS; (b) mTS.

10. Given:

11. Given: Chord AD .1 diameter ST;

mRS = 50.

Find: (a) mLRST; (b) mL,JIBR.

12. Given: 00 with diameter AB; chordBD.

mBD = mDE.

~

Prove: i5A

bisects LBAC.

CIRCLES 221

c

Ex. 9.

Ex. 10.

T

s

Ex.lJ.

A B

Ex. 12.