Preparing for the Regents Examination Geometry, AK

i Sufficient, SSS
j Sufficient, AAS
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.)
3 1. ___
AB ___
EF
2. ABC FED
3. ___
BC ___
DE
4. 1 2
5. I II (ASA ASA)
4 1. 1 4
2. B D
3. ___
AC ___
AC
4. I II (AAS AAS)
5 1. ___
AB ___
DE
2. DEC BAC (Alternate interior
angles are congruent.)
3. ABC EDC (Alternate interior
angles are congruent.)
4. C is the midpoint of ___
BD .
5. ___
BC ___
CD (Definition of
midpoint)
6. ABC EDC (AAS AAS)
6 1. B D
2. BEC DEA
3. ___
BC ___
AD
4. AED CEB (AAS AAS)
5. ___
AE ___
CE (CPCTC)
7 1. A E
2. ___
BC ___
DC
3. ___
AC ___
CE
4. ___
AC ___
BC (Subtraction
___
CE ___
DC postulate)
or ___
AB ___
DE
5. ___
BG ___
AE
6. BGA is a right (Definition of
angle. right angle)
7. ___
DF ___
AE
8. ____
DFE is a right angle.
9. BGA DFE (Right angles are
congruent.)
10. BGA DFE (AAS AAS)
BG ___
DF (CPCTC)
11. ___
8 1. ___
AB ___
EF
2. ABC FED
3. A F
4. ___
AC ___
DF
5. I II (AAS AAS)
9 1. ___
AB ___
CD
2. ___
CD ___
AB
3. CDA is a right angle.
4. ___
AE ___
CB
5. AEC is a right angle.
6. BAC BCA (Isosceles triangle
theorem)
7. ___
AC ___
AC
8. CDA AEC (AAS AAS)
9. ___
CD ___
AE (CPCTC)
10 1. E is the midpoint of ___
AC .
2. ___
AE ___
CE
3. ___
AF ___
BD
4. AFE is a right angle.
5. ___
CD ___
BD
6. CDE is a right angle.
7. AFE CDE
8. CED AEF (Vertical angles are
congruent.)
9. AFE CDE (AAS AAS)
10. ___
AF ___
CD (CPCTC)
11 1. ___
QR ___
SR
2. RQS RSQ (Isosceles triangle
theorem)
3. 1 2
4. ___
QS ___
QS
5. QTS SMQ (AAS AAS)
6. ____
QM ___
ST (CPCTC)
12 1. ___
BA ___
CD
2. ___
BA ___
AD
3. BAQ is a right angle.
4. ___
CD ___
AD (Two lines perpendicular
to the same line
are parallel.)
5. CDP is a right angle.
6. BAQ CDP
7. B C
8. ___
AP ____
QD
9. ___
PQ ___
PQ
10. ___
AD ___
PD (Addition postulate)
11. BAQ CDP (AAS AAS)
BA ___
CD (CPCTC)
12. ___
9-6 Proving Triangles Congruent by Angle, Angle, Side 49