Preparing for the Regents Examination Geometry, AK

3. −−
AB −−
AB
4. −−
AC −−
BD
5. ABC BAD
2 1. 1 3
2. 2 2
3. 1 3 2 3
(SAS SAS)
4. ADB EDC
5. 4 5
6.
(Partition
postulate)
−−−
AD −−
ED
7. ADB EDC (ASA ASA)
3 1. −−
AE −−
BE
2. D is the midpoint of −−
AB .
3. −−−
AD −−
BD
4. −−
ED −−
ED
5. ADE BDE
4 1. −−
TP −−
TQ
2. TPQ TQP
3. SPQ RQP
4. SPT RQT
5. STP RTQ (Vertical angles are
congruent.)
6. STP RTQ
7.
(SAA SAA)
−−
SP −−
PQ
8. −−
PQ −−
PQ
9. SPQ RQP
5 1. −−
EF −−
GF
2. −−
DF −−
HF
3. DFH DFH
4. DFG EFH (SAS SAS)
5. 1 2 (Corresponding
parts of congruent
triangles are
congruent.)
6 1. −−−
AD and −−
FC bisect each other at G.
2. −−−
AG −−−
DG
3. −−
FG −−
CG
4. AGC DGF (Vertical angles are
congruent.)
5. AGC DGF (SAS SAS)
6. GDE GAC
7. AGB DGE
(Corresponding
parts of congruent
triangles are
congruent.)
8. AGB DGE (ASA ASA)
5-6 Perpendicular
Bisectors of a Line
Segment
(pages 79–80)
1 Given −−
AB and P and Q, such that PA PB
and QA QB, let −−
AB intersect −−
PQ at
point M. APQ BPQ. APM BPM.
APM BPM. −−−
AM −−−
BM . Therefore,
AMP BMP.
2 a) Given −−
AB and point P such that PA PB;
if two points are each equidistant from the
endpoints of a line segment, then the points
determine the perpendicular bisector of the
line segment. Select a second point that is
equidistant from both A and B, for example
the midpoint of −−
AB , M. Then −−−
PM is the perpendicular
bisector of −−
AB .
b) Given point P on the perpendicular bisector
of −−
AB , let point M be the midpoint of −−
AB .
PMA PMB.
3 The perpendicular bisector of −−
PQ
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.)
4 1. −−
JL −−
KL
2. −−
JM −−−
KM
3. JL KL (Congruent segments are
equal in length.)
4. JM KM
5. −−−
LM is the perpendicular bisector of −−
JK .
(Two points, each equidistant from the
endpoints of a line segment, determine
the perpendicular bisector of the line
segment.)
6. N is the midpoint of JK.
(A point on the perpendicular bisector
of a line segment is equidistant from the
endpoints of the line segment.)
7. −−
JN −−−
KN (Definition of a midpoint)
5-6 Perpendicular Bisectors of a Line Segment 23