Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK Preparing for the Regents Examination Geometry, AK
to be used as a guide—other possible solutions may vary.) 10 1. Rectangle ABCD 2. ABCD is a parallelogram. 3. −− BC −−− AD 4. −− AB −−− CD (Opposite sides of a parallelogram are 5. congruent.) −− BD −− CA (Diagonals of a rectangle are congruent.) 6. CDA BAD (SSS SSS) 7. CAD BDA 11 1. Rectangle PQRS (CPCTC) 2. PQRS is a parallelogram. 3. −− PQ −− PQ 4. −−− QR −− PS 5. SQP PQR (SSS SSS) 6. 1 2 (CPCTC) 12 1. Rectangle ABCD 2. ABP and NCD are right angles. 3. ABP and NCD are right triangles. 4. −− AP −−− DN 5. −−− DC −− AB (Opposite sides of a parallelogram are congruent.) 6. ABP DCN (HL HL) 7. DNC APB (CPCTC) 8. PAD APB 9. NDA APB (Alternate interior angles are congruent.) 10. PAD NDA (Transitive 11. postulate) −− AE −− DE (Definition of an isosceles triangle) 13 By the addition postulate of inequality, −− AY is not congruent to −− TX , and AMY is not congruent to THX. Therefore, −−− MY is not congruent to −−− HX . 14 1. Rectangle ABCD 2. N is the midpoint of −−− CD . 3. CN DN 4. −−− CN −−− DN 5. BCN NDA (A rectangle is 6. equiangular.) −− BC −−− AD (Opposite sides of a parallelogram are 7. congruent.) −− BN −−− AN (CPCTC) 15 1. −− AB −−− CD 2. BJH and AJH are linear angles. 3. mBJH mAJH 180 4. 1 _ mBJH 2 1 _ mAJH 90 2 (Division postulate) 5. −− JG bisects BJH. 6. mHJG 1 _ mBJH 2 7. −− EJ bisects AJH. 8. mEJH 1 _ mAJH 2 9. mHJG mEJH 90 ( Substitution) 10. mEJG 90 11. EJG is a right angle. 12. CHJ BJH 13. EHJ GJH 14. AJH DHJ 15. HJE JHG 16. −− HJ −− HJ 17. EJH GHJ 18. −− EJ −−− GH 19. −− EH −− JG 20. EJGH is a parallelogram. 21. EJGH is a rectangle. (Definition of a rectangle) 16 1. Rectangle PQRS 2. −− PA −− CS 3. −− PA −− AC −− CS −− AC 4. −− PC −− AS 5. −− QP −− RS 6. QPS RSP 7. QPC RSA (SAS SAS) 8. a 1 2 (CPCTC) 9. PQR and SRQ are right angles. 10. 3 is complementary to 1. 11. 4 is complementary to 2. 12. b 3 4 (Complements of congruent angles are congruent.) 13. c −− QB −− RB (Definition of an isosceles triangle) 10-5 Rhombuses (pages 229–231) 1 mB mD 105; mC 75. AB CD 7 2 mADB 66 10-5 Rhombuses 61
3 4x 2 3x 3 x 5 RS 18 4 Perimeter of MABC 8 5 mADC 110 6 2x 2 x 8 x 10 CD 18 7 mADC 94 8 a Midpoint of −− AC midpoint of −− BD (6, 5) b Slope of −− AC 1, slope of −− BD 1. Slopes are negative reciprocals, diagonals are perpendicular. 9 a (9, 5) b PQ PS 5 √ 2 c Slope of −− PR 1 _ , slope of 3 −− QS 3. Slopes are negative reciprocals, diagonals are perpendicular. 10 a Slope of −−− AD slope of −− BC k _ x . AD BC √ x 2 k 2 . ABCD is a parallelogram. (One pair of opposite sides have the same length and are parallel.) b Slope of −− CA k _ −− , slope of BD x k k _ x k , not negative reciprocals. The diagonals are not perpendicular. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 11 −−− QR −− AC , −− PS −− AC , −−− QR −− PS . Likewise, −− QP −− BD , −− RS −− BD , and −− QP −− RS . Thus, PQRS is a parallelogram. Since −− AC −− BD , −−− QR and −− PS are perpendicular to −− QP and −− RS . Thus, PQRS has at least one right angle, and PQRS is a rectangle. 12 1. Rhombus ABCD 2. −− ED 3. −− BF −− AC 4. −− EF −− AC 5. F bisects −− AC . 6. EA EC 62 Chapter 10: Quadrilaterals 7. −− EA −− EC 8. ACE is isosceles. (Definition of an isosceles triangle) 13 Prove that −− DF −− AB and −− DE −− BF by using the midpoints and alternate interior angles. Thus, EBFD is a parallelogram. By the division postulate, −− EB −− FB . EBFD is a rhombus because a rhombus is a parallelogram with two congruent consecutive sides. 14 1. Parallelogram ABCD 2. −− AB −−− CD 3. BAD 2 4. mBAD m2 (Corresponding angles are congruent.) 5. BAD 1 CAD 6. mBAD m1 (Whole is greater than a part.) 7. m2 m1 (Substitution postulate) 15 1. Rhombus PQRS 2. −− QS −− PR 3. −−− QC −− CS 4. B is the midpoint of −−− QC . 5. D is the midpoint of −− CS . 6. −− BC −−− CD 7. ___ AC ___ AC 8. ABC ADC 9. (SAS SAS) −− AB −−− AD (CPCTC) 10. BAD is isosceles. (Definition of an isosceles triangle) 16 If a quadrilateral is equilateral, it is a parallelogram with a pair of consecutive congruent sides. 17 The diagonal creates two congruent triangles (ASA ASA) and corresponding congruent sides are consecutive. 18 1. −−− CD BE 2. −−− CD −− BA 3. −−− AD BF 4. −−− AD −− BC 5. ABCD is a ( Definition of a parallelogram. parallelogram) 6. BAD BCD (Opposite angles 7. BG bisects FBE. are congruent.) 8. CBD ABD 9. ABCD is a rhombus. (Diagonal −− BD bisects opposite angles.)
- Page 13 and 14: 6. BAC DAE 6. Transitive property.
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- Page 25 and 26: 12. −−− HG −−− DC 13.
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- Page 41 and 42: 6. mDAB mCAD mDCB mACD 7. mCAB
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- Page 61 and 62: 7. MAD RCB 8. MAD RCB (SAS SAS)
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- Page 73 and 74: 11-6 Volume of a Prism (pages 269-2
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to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
10 1. Rectangle ABCD<br />
2. ABCD is a parallelogram.<br />
3. −−<br />
BC −−−<br />
AD<br />
4. −−<br />
AB −−−<br />
CD (Opposite sides of<br />
a parallelogram are<br />
5.<br />
congruent.)<br />
−−<br />
BD −−<br />
CA (Diagonals of a<br />
rectangle are<br />
congruent.)<br />
6. CDA BAD (SSS SSS)<br />
7. CAD BDA<br />
11 1. Rectangle PQRS<br />
(CPCTC)<br />
2. PQRS is a parallelogram.<br />
3. −−<br />
PQ −−<br />
PQ<br />
4. −−−<br />
QR −−<br />
PS<br />
5. SQP PQR (SSS SSS)<br />
6. 1 2 (CPCTC)<br />
12 1. Rectangle ABCD<br />
2. ABP and NCD are right angles.<br />
3. ABP and NCD are right triangles.<br />
4. −−<br />
AP −−−<br />
DN<br />
5. −−−<br />
DC −−<br />
AB (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
6. ABP DCN (HL HL)<br />
7. DNC APB (CPCTC)<br />
8. PAD APB<br />
9. NDA APB<br />
(Alternate interior<br />
angles are<br />
congruent.)<br />
10. PAD NDA (Transitive<br />
11.<br />
postulate)<br />
−−<br />
AE −−<br />
DE (Definition of an<br />
isosceles triangle)<br />
13 By <strong>the</strong> addition postulate of inequality, −−<br />
AY<br />
is not congruent to −−<br />
TX , and AMY is not<br />
congruent to THX. There<strong>for</strong>e, −−−<br />
MY is not<br />
congruent to −−−<br />
HX .<br />
14 1. Rectangle ABCD<br />
2. N is <strong>the</strong> midpoint of −−−<br />
CD .<br />
3. CN DN<br />
4. −−−<br />
CN −−−<br />
DN<br />
5. BCN NDA (A rectangle is<br />
6.<br />
equiangular.)<br />
−−<br />
BC −−−<br />
AD (Opposite sides of<br />
a parallelogram are<br />
7.<br />
congruent.)<br />
−−<br />
BN −−−<br />
AN (CPCTC)<br />
15 1. −−<br />
AB −−−<br />
CD<br />
2. BJH and AJH are linear angles.<br />
3. mBJH mAJH 180<br />
4. 1 _ mBJH <br />
2 1 _ mAJH 90<br />
2<br />
(Division postulate)<br />
5. −−<br />
JG bisects BJH.<br />
6. mHJG 1 _ mBJH<br />
2<br />
7. −−<br />
EJ bisects AJH.<br />
8. mEJH 1 _ mAJH<br />
2<br />
9. mHJG mEJH 90 ( Substitution)<br />
10. mEJG 90<br />
11. EJG is a right angle.<br />
12. CHJ BJH<br />
13. EHJ GJH<br />
14. AJH DHJ<br />
15. HJE JHG<br />
16. −−<br />
HJ −−<br />
HJ<br />
17. EJH GHJ<br />
18. −−<br />
EJ −−−<br />
GH<br />
19. −−<br />
EH −−<br />
JG<br />
20. EJGH is a parallelogram.<br />
21. EJGH is a rectangle. (Definition of a<br />
rectangle)<br />
16 1. Rectangle PQRS<br />
2. −−<br />
PA −−<br />
CS<br />
3. −−<br />
PA −−<br />
AC −−<br />
CS −−<br />
AC<br />
4. −−<br />
PC −−<br />
AS<br />
5. −−<br />
QP −−<br />
RS<br />
6. QPS RSP<br />
7. QPC RSA (SAS SAS)<br />
8. a 1 2 (CPCTC)<br />
9. PQR and SRQ are right angles.<br />
10. 3 is complementary to 1.<br />
11. 4 is complementary to 2.<br />
12. b 3 4 (Complements of<br />
congruent angles<br />
are congruent.)<br />
13. c −−<br />
QB −−<br />
RB (Definition of an<br />
isosceles triangle)<br />
10-5 Rhombuses<br />
(pages 229–231)<br />
1 mB mD 105; mC 75.<br />
AB CD 7<br />
2 mADB 66<br />
10-5 Rhombuses 61
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