Preparing for the Regents Examination Geometry, AK

9-5 The Sum of the
Measures of the Angles
of a Triangle
(pages 193–195)
1 (4) scalene
2 (3) obtuse
3 (3) 120
4 (2) right
5 (3) 112
6 a base angles are 30, vertex angle is 120
b base angles are 60, vertex angle is 60
c base angles are 52, vertex angle is 76
d base angles are 36, vertex angle is 108
e base angles are 20, vertex angle is 140
7 a base angles are 50, exterior angle is 130
b base angles are 40, exterior angle is 140
c base angles are 54, exterior angle is 126
d base angles are 60, exterior angle is 120
e base angles are 80, exterior angle is 100
8 base angles are 74, vertex angle is 32
9 mP 27, mQ 45, mR 108
10 base angles are 28, vertex angle is 124.
11 exterior angle is 30, base angles are 15,
vertex angle is 150
12 18, 54, 108
13 99, 45, 36
14 mc 35
15 mB 78
16 mx 150
17 mx 30
18 m1 150
19 mD 20
20 md 125
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.)
21 1. A C
2. ___
BD bisects ABC.
3. ABD CBD
4. ___
BD ___
BD
5. ABD CBD (AAS AAS)
48 Chapter 9: Parallel Lines
6. ADB CDB (CPCTC)
7. ADB is supplementary to CDB.
8. ADB and CDB are right angles.
9. −−
BD −−
AC (Segments forming right
angles are perpendicular.)
22 1. A C
2. ___
BD ___
AC
3. BDA and BDC are right angles.
4. BDA BDC
5. ___
BD ___
BD
6. BDA DBC (AAS AAS)
7. ABD CBD (CPCTC)
8. ___
BD bisects ABC. (Definition of angle
bisector)
23 Both ABC and DEC share C. Since the
remaining angles are the same as well, 1
and B are congruent corresponding angles.
Therefore, ___
BA ___
ED .
24 Compare ABC and DEC. Both are right
triangles with one pair of congruent acute
angles. Therefore, the other pair of acute
angles are also congruent, B CED. But
CED 1, vertical pairs. Then B 1
by the transitive postulate of congruence.
9-6 Proving Triangles
Congruent by Angle,
Angle, Side
(pages 198–200)
1 b and f
2 a Not sufficient. If the third side is congruent,
SSS. If included angles are congruent,
SAS.
b Sufficient, SSS
c Sufficient, hypotenuse-angle
d Not sufficient. If either pair of corresponding
angles are congruent, AAS.
e Sufficient, AAS
f Not sufficient. If any pair of corresponding
sides are congruent, AAS.
g Sufficient, hypotenuse-leg
h Sufficient, SAS