Preparing for the Regents Examination Geometry, AK

6 Given: BA bisects CBE
1 2
2 4
Prove: 3 4
Narrative Proof: A bisector divides an angle
into two congruent angles, so 1 2.
Using the transitive property of congruence,
1 4. By substitution, 3 4.
7 4x 3 2x 21
x 12
mBCA 4x 3
4(12) 3
45
mBCE 2(45) 90
8 4x 4 2(3x 21)
x 23
AB 4x 4
4(23) 4
96
9 Since DE CE, then 2x y 5y, and
x 2y.
AC BD
6 5y (2x y) (x y)
6 5y 3x 2y
6 5y 3(2y) 2y
6 5 y 6y 2y
6 3y
2 y
x 2(2) 4
BD 2(4) 2 4 2 16
10 AB DE 8, so BD 4 and CD 2. Therefore,
CE 8 2 10.
4-2 Proving Theorems
About Angles
(pages 59–61)
1 If two angles are right angles, they both have
a measurement of 90. Angles with the same
measure are congruent.
2 If two angles are straight angles, they both
have a measurement of 180. Angles with the
same measure are congruent.
3 If 1 is a complement of A, then m1
90 mA. If 2 is a complement of A,
then m2 90 mA. Then m1 m2
by the symmetric property and transition,
and 1 2 by definition of congruence.
14 Chapter 4: Congruence of Lines, Angles, and Triangles
4 If 1 2, 3 is the complement of
1, and 4 is the complement of 2, then
m3 90 m1 and m4 90 m2.
Since m1 m2 by congruence, m3
m4 by symmetric property and transition,
and 3 4 by congruence.
5 If 1 is a supplement of A, then m1
180 mA. If 2 is a supplement of A,
then m2 180 mA. Then m1 m2
by the symmetric property and transition,
and 1 2 by definition of congruence.
6 If 1 2 and 3 is the supplement of
1 and 4 is the supplement of 2, then
m3 90 m1 and m4 90 m2.
Since m1 m2 by congruence, m3
m4 by symmetric property and transition,
and 3 4 by congruence.
7 By definition the sum of their measures is a
straight angle or 180.
8 The two angles are a linear pair so the sum
of their measures is 180. If they are congruent,
each measure is 90; they form right
angles and are therefore perpendicular.
9 Each angle forms a linear pair with the same
angle. They are supplementary to the same
angle and are therefore congruent.
10 m2 m3 because they are vertical angles.
By substitution, 1 is complementary
to 3. 1 4 because complements of the
same angle are congruent.
11 3 forms a linear pair with 1, 3 is supplementary
to 1. Since 1 2, 3 is
supplementary to 2. 4 forms a linear pair
with 2, 4 is supplementary to 2.
3 4 because supplements of the same
angle are congruent.
12 Since they form a linear pair, 2 is supplementary
to EDG. Since 1 and 2 have
the same measure (they are congruent), 1 is
supplementary to EGD.
13 Since 1 is complementary to 3,
m1 m3 90. Since ACB is a right
angle, m1 m2 90. By subtraction and
substitution, m2 m3, so they are congruent
angles.
14 By the transitive postulate, 1 3.
Because they are vertical angle pairs,
1 4, and 3 6. By substitution
and transition, 4 6.