Preparing for the Regents Examination Geometry, AK

9-7 The Converse of the
Isosceles Triangle Theorem
(pages 202–203)
1 2(3x 4) 2x 4 180
x 22
mA mC 70
mB 40
2 2(3x 3) 4x 16 180
x 19
mA mB mC 60
3 2x 180 82 82
x 8
4 (2x 14) (3x 2) (5x 8) 180
x 16
2x 14 2(16) 14 46
3x 2 3(16) 2 46
5x 8 5(16) 8 88
Since two angles are equal in measure,
ADC is isosceles.
5 mA mB 49
mx 180 49 49
my 49
mz 180 82 98
6 mQ 58
mx mz 64
my 116
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.)
7 1. A C
2. ___
BC ___
AB
3. ___
AD ___
EC
4. ABD CBE (SAS SAS)
8 1. ___
BD ___
AC
2. 1 ACB
3. 2 BAC
4. 1 2
5. ACB BAC (Substitution
postulate)
6. ABC is isosceles. (Definition of
isosceles triangle)
50 Chapter 9: Parallel Lines
9 1. 1 2
2. 3 4 (Supplements of
congruent angles are
congruent.)
3. ___
AB ___
BC
4. ABC is isosceles. (Definition of
isosceles triangle)
10 1. ___
BD ___
AE
2. ___
AC ___
CE
3. A E
4. A B (Two parallel lines are cut
by a transversal then the
corresponding angles are
congruent.)
5. E D (Two parallel lines are cut
by a transversal then the
corresponding angles are
congruent.)
6. B D (Transitive postulate of
congruence)
7. ___
BC ___
DC (Congruent angles imply
congruent sides.)
11 1. ___
BC ___
BD
2. BCD BDC
3. BDA BDE (Linear pairs of
congruent angles)
4. BAC BED (ASA ASA)
5. ___
AB ___
BE (CPCTC)
12 1. ___
AB ___
EB
2. BAE BEA
3. ___
BD ___
AE
4. BAE 1
5. BAE 2
6. 1 2 (Substitution
postulate)
13 1. ___
PQ ___
SR
2. ___
PQ ___
SR
3. QPR SRP (Alternate interior
angles are
congruent.)
4. ___
PR bisects QPS.
5. QRP RPS (Definition of
bisector)
6. SRP RPS (Transitive postulate
of congruence)
7. ___
PS ___
SR (Converse of the
isosceles triangle
theorem)