Proof
ST ATEMENTS REASONS
I. Draw BM
..l A.2. Extend BM and AR until they intersect at S. Draw TS.
3. La == Lf3.
4. La == Ly.
5. L? == L?
6. RM=RM.
7. LRMB
==LRMS.
8. RB = RS.
9. BM = SM.
10. ~=~.
II. LIMB
==LTMS.
12. TB = TS.
13. AS < AT+ TS.
14. AR+RS = AS.
15. AR+RS < AT+TS.
16. AR + RB < AT+ TB.
I. Why?
2. Why?
3. Why?
4. Why?
5. Why?
6. Why?
7. Why?
8. Why?
9. Why?
10. Why?
II. Why?
12. Why?
13. Why?
14. Why?
15. Why?
16. Why?
Theorem 9.6
9.13. In a circle or in congruent circles, if two central angles have unequal measures, the greater central angle has the greater minor arc.
Theorem 9.6.
Given: 00 == OQ with mLO > mLQ.
Conclusion: mAR> mCD.
Proof
Q
I
INEQUALITIES 295
STATEMENTS REASONS
1. 00 ==
OQ.
2. mLO > mLQ.
3. mLO
= mAR, mLQ = mCD.
4. :.mAR > mCD.
1. Given.
2. Given.
3. §7.9.
4. 0-7.
Theorem 9.7
9.14. In a circle or in congruent circles, if two minor arcs are not congruent, the greater arc has the greater central angle.
(The proof of this theorem is left to the student.)
Theorem 9.89.15. In a circle or in congruent circles, the greater of two noncongruent chords has the greater minor arc.
0
Theorem 9.8.
Given: 00 == OQ with chord AB > chord CD.
Conclusion: mAR> mCD.
Proof
STATEMENTS REASONS
1. 00 ==
OQ.
2. Draw radii OA, OB, QC, QD.
3. OA = QC; OB = QD.
4. Chord AB > chord CD.
5. mLO > mLQ.
6. mAR> mCD.
I. Given.
2. Postulate 2.
3. Definition of
== @.4. Given.
5. Theorem 9.5.
6. Theorem 9.6.
296
FUNDAMENTALS OF COLLEGE GEOMETRYTheorem 9.9
9.16. In a circle or in congruent circles the greater of two noncongruent minor arcs has the greater chord.
(The proof of this theorem is left to the student.) Theorem 9.10
9.17. In a circle or in congruent circles, if two chords are not con- gruent, they are unequally distant from the center, the greater chord being nearer the center.
Given: 00 with chord AB > chord CD;
OE 1- AB; OF 1- CD.
Conclusion: OE
< OF.
Proof
ST A TEMENTS
c
Theorem 9.10.
'
1
-'-
i
;
1
"
.':
.
I'c .
i
REASONS
1. Draw chord AH == chord CD.
2. Draw OG 1- AH.
3. Draw GE.
4. OE 1- AB. OF 1- CD.
7 -
6. AB> CD.
7. AB > AH.
8. G is midpoint of AH; E is mid- point ofAB.
9. AE > AG.
10. mLOI > mL{3.
11. mLAGO = mLAEO.
12. mL y < mLcf>.
13. OE < OG.
14. :.OE < OF.
1. Postulate 11.
2. Theorem 5.4.
3. Postulate 2.
4. Given.
---
6. Given.
7. 0-7. £-1.
8. Theorem 7.7.
9. 0-5.
10. Theorem 9.1.
11. §1.20; Theorem 3.7.
12. 0-3.
13. Theorem 9.2.
14. 0-7.
Theorem 9.11
9.18. In a circle or in congruent circles, if two chords are unequally distant ~ from the center, they are not congruent, the chord nearer the center being
1the greater.
(The proof is left to the student.)
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..
INEQUALITIES 297 Exercises
In each of the following circles, 0 is assumed to be the center of a circle.
1. In ,0.ABC, mAB = 4 inches, mBC = 5 inches, mAC = 6 inches. Name the angles of the triangle in order of size.
2. ,0.RST is inscribed in a circle. mRS = 80 and mST = 120. Name the angles of the triangle in order of size.
3. In ,0.MNT, mLN= 60 and mLM < mLT. Which is the longest side of the triangle?
4. In quadrilateral LMNT, LM = MN, and mLL > mLN. Which is the longer, NT or LT? Prove your answer.
5. In quadrilateral QRST, QR> RS and LQ == LS. Which is the longer, QT or ST ? Prove your answer.
c
6. Given: AC > AB.
Prove: AC> AD.
Ex. 6.
A
7. Prove that, in a circle, if the measure of one minor arc is twice the measure of a second minor arc, the measure of the chord of the first arc is less than twice the measure of the chord of the second arc.
8. Prove that, if a square and an equilateral triangle are inscribed in a circle, the distance from the center of the circle to the side of the square is greater than that to the side of the triangle.
9. Given: P a point of diameter CD;
chord AB 1- CD; EF any other chord containing P.
Prove: AB < EF.
F
Ex. 9. D
--- ---
298
10. Given: 00 with OA
..l PR;OB ..l SR;
OB > OA.
Prove: mLPOR > mLROS.
11. Given: 00 with OA 1- PR;
OB..l SR;
mLPOR> mLROS.
Prove: OB > OA.
12. Given: 00 with PT = ST;
mLSTR> mLRTP.
Prove: mRS > mFR.
13. Given: 00 with PT = ST;
mRS > mPR.
Prove: mLSTR > mLRTP.
Exs. 10, 11.
R
Exs.12,13.
s
14. Prove that the measure of the hypotenuse of a right triangle is greater than the measure of either leg.
15. Prove that the shortest chord through a point inside a circle is perpendi- cular to the radius through the point.
16. Given: CM is a median of ,6.ABC;
CM is not 1- AB.
Prove: AC #- BC.
17. Given: CM is a median of ,6.ABC;
AC #- Be.
Prove: CM cannot be ..l AB.
18. Prove that the shortest dis- tance from a point within a circle to the circle is along a radius. (Hint: Prove PB <
PA, any A #- B.)
Exs. 16,17.
A B
Ex. lB.
19. Prove that the shortest dis- tance from a point outside a circle to the circle is along a radius produced. (Hint:
PS
+ SO > . . . ; SO = RO;
PS>... .)
20. Given: Chord AC > chord BD.
Prove: ChordAB > chord CD.
p
Ex. 19.
Ex. 20.
---
Summary Tests
Test 1
COMPLETION STATEMENTS
I
I. The wm of the meaw'
" of any 'wo ,id" of a "iangle i, - than the;:
measure of the third side.
.1
'2. Angle T is the largest angle in triangle RST. The largest side is -' ...
,
:
.:.. .,
'
3. Ifk> h, thenk+ l_h+ 1.
cm <.: 17and n <:
p,
LltCH rnp.
...
5. If I> w, then a-l_a-w.
6. If d < e, e < f,f= h, then d_h.
7. In 6.HJK, HJ > JK, mLJ = 80. Then mLH _50.
8. In quadrilateral LMNP, LM = MN and mLMLP > mLMNP.
mLNLP - mLLNP.
9. In 6.ABC, mLA = 50, mLB = 60, mLC = 70. ThenAB _AG.
10. In a circle or in congruent circles, if two central angles are not congruent, '.:
the greater central angle has the
- arc. \
11. The measure of an exterior angle of a triangle is equal to the _of the':
measures of the two nonadjacent interior angles.
12. Ifx+y = k, theny_k-x.
13. If a < band c > d, then a+d_b+c.
14. Ifx < y, thenx-a_y-a.
15. Ifx < y and z > y, then z_x.
16. Ifxy < Oandx > 0, theny_O. .
17. In quadrilateral PQRS, if PQ = QR, and mLP > mLR, then PS _RS 18. In quadrilateral PQRS, if PQ > QR, and mLP = mLR, then PS _RS,
300
19. In 6.RST, if mLR = 60 and mLS > mL T, then
- is the longest side of the triangle.20. The shortest chord through a point inside a circle is- to the radius through the point.
Test 2
TRUE-FALSE STATEMENTS
I. The shortest distance from a point to a circle is along the line joining that point and the center of the circle.
2. The measure of the perpendicular segment from a point to a line is the shortest distance from the point to the line.
3. Either leg of a right triangle is shorter than the hypotenuse.
4. If two triangles have two sides of one equal to two sides of the other, and the third side of the first less than the third side of the second, the measure of the angle included by the two sides of the first triangle is greater than the measure of the angle included by the two sides of the second.
5. :-../0two angles of a scalene triangle can have the same measure.
6. The measure of an exterior angle of a triangle is greater than the measure of any of the interior angles.
7. If two sides of a triangle are unequal, the measure of the angle opposite the greater side is less than the measure of the angle opposite the smaller side.
13- I[:L\,'.'(\ chnrds in the same circle are unequal, the smaller chord is nearer the center.
9. If John is older than Mary, and Alice is younger than Mary, John is older than Alice.
10. Bill has twice as much money as Tom, and Tom has one-third as much as Harry. Then Bill has more money than Harry.
II. Angle Q is the largest angle in 6.PQR. Then the largest side is PQ.
12. Ifk> mandm< t,thenk> t.
13. lfx> 0 andy > 0, then xy < O.
14. Ifx < y and z < 0, then xz < yz.
15. In a circle or in congruent circles, if two central angles are not congruent, the greater central angle has the greater major arc.
16. x < Y
~Y > x.
17. The difference between the lengths of two sides of a triangle is less than the length of the third side.
IS. The perimeter of a quadrilateral is less than the sum of its diagonals.
19. If a triangle is not isosceles, then a median to any side is greater than the altitude to that side.
301
STATEMENTS REASONS
20. The diagonals of a rhombus that is not a square are unequal.
Test 3
EXERCISES
I. Supply the reasons for the statements in the following proof:
Given: LABC with CD bisecting LACB.
Prove: AC> AD.
Proof
1. mLACD
= mLBCD.
2. mLADC > mLBCD;
mLBDC > mLACD.
3. mLADC > mLACD.
4. AC > AD.
c
/T\
A D B
Ex. I.
1.
2.
3.
4.
2. Given: PR = PT.
Prove: mLPRS > mLS.
3. Prove that the shortest chord through a point within a circle is perpendicular to the radius drawn through that point.
302
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l
p
R
Ex. 2.
c
Ex. J.