Preparing for the Regents Examination Geometry, AK

Chapter Review (pages 84–85)
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. −−−
MN bisects PNQ.
2. RND RNQ
3. RNP RQN
4. −−−
RN −−−
RN
5. RPN RQN (ASA ASA)
2 1. −−
PQ −−
PR
2. −−
QT is a median.
3. −−
PT −−
RT
4. −−
RS is a median.
5. −−
PS −−
QS
6. PQT PRS (SSS SSS)
3 1. 3 4
2. −−
DE −−
DF
3. −−−
DC −−−
DC
4. DEC DFC (SAS SAS)
5. DCE DCF (Corresponding
parts of congruent
triangles are
congruent.)
6. EDC FDC
7. mCAD m1 mEDC mDCE
180
8. mCBD m2 mFDC mDCF
180
9. mCAD mCBD
10. CAD CBD
11. ABC is isosceles. (Definition of isosceles
triangle)
4 1. ABC is an equilateral triangle.
2. ___
AB ___
AC (Definition of equilateral
triangle)
3. DCB DBC
4. DB DC
5. ___
AD ___
AD
6. ABD ACD (SSS SSS)
7. BAD CAD (Corresponding
parts of congruent
triangles are
congruent.)
8. −−−
AD bisects BAC. (Definition of an
angle bisector)
26 Chapter 5: Congruence Based on Triangles
5 1. ____
TM ___
TA
2. MTA is isosceles. (Definition of isosceles
triangle)
3. ___
TH bisects MTA.
4. ___
TH is an altitude (In an isosceles
of MTA. triangle, the
bisector is the
altitude.)
6 1. ___
AB ___
AD
2. ___
CB ___
CD
3. ___
AC ___
AC
4. ABC ADC (SSS SSS)
5. BAE DAE (Corresponding
parts of congruent
triangles are
congruent)
6. ___
AE ___
AE
7. ABE ADE (ASA ASA)
8. ___
BE ___
DE
9. DAB is isoceles. (Definition of
isosceles triangle)
10. ___
AE is the median (Definition of
of DAB. median)
11. ___
AE is the altitude (In an isosceles triof
DAB. angle, the median
is the altitude.)
7 1. ___
AB ___
BD
2. A D
3. DBA CBE
4. EBD EBD
5. DBA EBD CBE EBD
6. EBA CBD
7. ABE DBC (ASA ASA)
8 1. ADE BDC
2. EDB EDB
3. ADE EDB BDC EDB
4. ADB EDC
5. DAE DEC
6. ___
DA ___
DE
7. DAB DEC (ASA ASA)
9 1. 1 2
2. 3 4
3. ___
DE ___
DF
4. DEG DFH (ASA ASA)
5. ___
GD ____
HD (Corresponding
parts of congruent
triangles are
congruent.)
6. EDF EDF