Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK Preparing for the Regents Examination Geometry, AK
3. −− BX −− XC 3. Definition of a median. 4. −−− AX −−− AX 4. Reflexive property of congruence. 5. ABX ACX 5. SSS SSS. 6. ABX ACX 6. Corresponding parts of congruent triangles are congruent. 7. Construct a second median, −− BY , from vertex B. 7. A median of a triangle is a line segment with one endpoint at any vertex of the triangle, extending to the midpoint of the opposite side. 8. −− BA −− BC 8. Definition of an equilateral triangle. 9. −− AY −− YC 9. Definition of a median. 10. −− BY −− BY 10. Reflexive property of congruence. 11. BAY BCY 11. SSS SSS. 12. BAY BCY 12. Corresponding angles of congruent triangles are congruent. 13. ABC is equiangular. 13. BCY is the same as ACX, and all three angles of the triangle are equal. 5 mA mC 3x 5 mA mB mC 180 (3x 5) (4x 10) (3x 5) 180 10 x 180 x 18 mC 3(18) 5 59 6 The measure of each angle is 60. So, 3x 6 60; x 22. And 3y 6 60; y 18. 7 Show that RXZ TYZ SYX by SAS SAS, and −− XY −− YZ −− ZX because corresponding parts of congruent triangles are congruent. 8 Construct −− AC and −− BD , and label their intersection E. ABE CBE. −− AC and −− BD are perpendicular. AED and CDE are right angles; therefore, AED CDE. −− AE −− CE . −− ED −− ED . AED CED. CAD ACD. 9 mA mB mC 180 (6x 12) (8x 8) (3x 6) 180 17x 1 0 180 x 10 mA 72; mB 72; mC 36 The triangle is isosceles. 10 mB 180 2x 5-4 Working With Two Pairs of Congruent Triangles (pages 75–76) 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.) 1 1. ABC EFG 2. −− AC −− EG 3. −− BD is a median. 4. D is the midpoint (Definition of of −− AC . median) 5. DC 1 _ AC (Definition of mid- 2 point) 6. −− FH is a median. 7. H is the midpoint of −− EG . 8. HG 1 _ EG 2 9. AC EG 10. HG 1 _ AC 2 11. HG DC 5-4 Working With Two Pairs of Congruent Triangles 21
12. −−− HG −−− DC 13. −− FG −− BC 14. BCD FGH 15. BCD FGH (SAS SAS) 16. −− BD −− FH 2 a 1. −− PR and −− QS bisect each other at V. 2. −− PV −− RV 3. −−− QV −− SV 4. PVQ RVS (Vertical angles are congruent.) 5. PQV RSV (SAS SAS) b A continuation of part a. 6. UPV TRV (Corresponding parts of congruent triangles are congruent.) 7. PVU RVT (Vertical angles are congruent.) 8. PUV RTV 3 a 1. (ASA ASA) −− AC −−− AD 2. −− BC −− BD 3. −− AB −− AB 4. ABC ABD (SSS SSS) 5. BAC BAD (Corresponding parts of congruent triangles are 6. congruent.) −− AE −− AE 7. ACD ADE (SAS SAS) b A continuation of part a. 8. 1 2 (Corresponding parts of congruent triangles are 4 1. congruent.) −−− CD −− CB 2. 1 2 3. −− CF −− CF 4. CFB CFD (SAS SAS) 5. CBF CDF 6. EDF ABF 7. BFA EFD 8. (Corresponding parts of congruent triangles are congruent.) −− DF −− BF 9. BFA DFE (ASA ASA) 10. 3 4 (Corresponding parts of congruent triangles are congruent.) 22 Chapter 5: Congruence Based on Triangles 5 1. −− AC −− BC 2. −− AE −− BE 3. −− EC −− EC 4. ACE BCE (SSS SSS) 5. ACE BCE 6. −−− CD −−− CD 7. ACD BCD (SAS SAS) 8. −−− AD −− BD (Corresponding parts of congruent triangles are congruent.) 6 a 1. −− SP −−− QR 2. 1 2 3. −− AB bisects −− QS at C. 4. −− SC −−− QC (Definition of bisector) 5. −− SQ −− SQ 6. SQP QSR (SAS SAS) 7. SQP QSR (Corresponding parts of congruent triangles are congruent.) 8. ACQ BCS 9. ACQ BCS (ASA ASA) b Continuation of part a. 10. −− AC −− BC (Corresponding parts of congruent triangles are congruent.) 5-5 Proving Overlapping Triangles Congruent (page 77) 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.) 1 1. −− AE −− BE 2. EAB EBA (If two sides of a triangle are congruent, the angles opposite these sides are congruent.)
- Page 1 and 2: ANSWER KEY Preparing for the REGENT
- Page 3 and 4: Contents Chapter 1: Essentials of G
- Page 5 and 6: 1-4 Angles (pages 9-10) 1 (4) It is
- Page 7 and 8: 6 True 7 True 8 True 9 True 10 Answ
- Page 9 and 10: 3 1. e ∨ ~f 2. ~f → g 3. ~e 4.
- Page 11 and 12: CHAPTER 3 3-1 Inductive Reasoning (
- Page 13 and 14: 6. BAC DAE 6. Transitive property.
- Page 15 and 16: CHAPTER 4-1 Setting Up a Valid Proo
- Page 17 and 18: 6 Given: BA bisects CBE 1 2 2 4
- Page 19 and 20: 1 1. −−−− ABCD 2. 1 2 3. _
- Page 21 and 22: 18 1. I is the midpoint of −− E
- Page 23: 5-3 Isosceles and Equilateral Trian
- Page 27 and 28: 5 1. −− FG is the perpendicular
- Page 29 and 30: Chapter Review (pages 84-85) Note:
- Page 31 and 32: 20 Use constructing congruent angle
- Page 33 and 34: 6-3 Line Reflections and Symmetry (
- Page 35 and 36: 11 A(0, 8), B(2, 2), C(6, 4) (8) 10
- Page 37 and 38: 5 (3) (x, y) → (x, 2y) 6 (1) tran
- Page 39 and 40: 11. mABC mADC 12. 2mABD 2mADB 13.
- Page 41 and 42: 6. mDAB mCAD mDCB mACD 7. mCAB
- Page 43 and 44: g e 17 a _ f d b Undefined c a _
- Page 45 and 46: 5 BIG is isosceles because it has t
- Page 47 and 48: ___ 27 a M KA (5, 1), M ___ AT (4
- Page 49 and 50: 8 106 9 mA 75, mC 67 10 57 11 60
- Page 51 and 52: 9-5 The Sum of the Measures of the
- Page 53 and 54: 9-7 The Converse of the Isosceles T
- Page 55 and 56: 12 1. ___ CE ___ BA , ___ BD ___
- Page 57 and 58: 12 a mx 45, my 45 b mx 98, my 8
- Page 59 and 60: Quadrilaterals 10-2 The Parallelogr
- Page 61 and 62: 7. MAD RCB 8. MAD RCB (SAS SAS)
- Page 63 and 64: 5. RSQ TSV (Vertical angles) 6. QR
- Page 65 and 66: 3 4x 2 3x 3 x 5 RS 18 4 Perime
- Page 67 and 68: Note: Since there are many variatio
- Page 69 and 70: 28 Enclose PAT in a large rectangle
- Page 71 and 72: Geometry of Three Dimensions 11-1 P
- Page 73 and 74: 11-6 Volume of a Prism (pages 269-2
12. −−−<br />
HG −−−<br />
DC<br />
13. −−<br />
FG −−<br />
BC<br />
14. BCD FGH<br />
15. BCD FGH (SAS SAS)<br />
16. −−<br />
BD −−<br />
FH<br />
2 a 1. −−<br />
PR and −−<br />
QS bisect each o<strong>the</strong>r at V.<br />
2. −−<br />
PV −−<br />
RV<br />
3. −−−<br />
QV −−<br />
SV<br />
4. PVQ RVS (Vertical angles are<br />
congruent.)<br />
5. PQV RSV (SAS SAS)<br />
b A continuation of part a.<br />
6. UPV TRV (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
7. PVU RVT (Vertical angles are<br />
congruent.)<br />
8. PUV RTV<br />
3 a 1.<br />
(ASA ASA)<br />
−−<br />
AC −−−<br />
AD<br />
2. −−<br />
BC −−<br />
BD<br />
3. −−<br />
AB −−<br />
AB<br />
4. ABC ABD (SSS SSS)<br />
5. BAC BAD (Corresponding<br />
parts of congruent<br />
triangles are<br />
6.<br />
congruent.)<br />
−−<br />
AE −−<br />
AE<br />
7. ACD ADE (SAS SAS)<br />
b A continuation of part a.<br />
8. 1 2 (Corresponding<br />
parts of congruent<br />
triangles are<br />
4 1.<br />
congruent.)<br />
−−−<br />
CD −−<br />
CB<br />
2. 1 2<br />
3. −−<br />
CF −−<br />
CF<br />
4. CFB CFD (SAS SAS)<br />
5. CBF CDF<br />
6. EDF ABF<br />
7. BFA EFD<br />
8.<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
−−<br />
DF −−<br />
BF<br />
9. BFA DFE (ASA ASA)<br />
10. 3 4 (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
22 Chapter 5: Congruence Based on Triangles<br />
5 1. −−<br />
AC −−<br />
BC<br />
2. −−<br />
AE −−<br />
BE<br />
3. −−<br />
EC −−<br />
EC<br />
4. ACE BCE (SSS SSS)<br />
5. ACE BCE<br />
6. −−−<br />
CD −−−<br />
CD<br />
7. ACD BCD (SAS SAS)<br />
8. −−−<br />
AD −−<br />
BD<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
6 a 1. −−<br />
SP −−−<br />
QR<br />
2. 1 2<br />
3. −−<br />
AB bisects −−<br />
QS at C.<br />
4. −−<br />
SC −−−<br />
QC (Definition of<br />
bisector)<br />
5. −−<br />
SQ −−<br />
SQ<br />
6. SQP QSR (SAS SAS)<br />
7. SQP QSR (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
8. ACQ BCS<br />
9. ACQ BCS (ASA ASA)<br />
b Continuation of part a.<br />
10. −−<br />
AC −−<br />
BC (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
5-5 Proving Overlapping<br />
Triangles Congruent<br />
(page 77)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
1 1. −−<br />
AE −−<br />
BE<br />
2. EAB EBA (If two sides of<br />
a triangle are congruent,<br />
<strong>the</strong> angles<br />
opposite <strong>the</strong>se sides<br />
are congruent.)
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