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Math 611
Geometry Chapter 4 Review Problems
1. Fill in the blanks of the proof with the correct statements and reasons:
Z
Given: P is the midpoint of XZ .
1  2
Prove: XY  ZY
P
W
Y
X
Statements
Reasons
1. P is the midpt. of XZ .
1. Given
2. ZP = XP
2. If a point is the midpoint of a segment, then
it divides the segment into two congruent
segments
3. 1  2
3. Given
4. WX = WX
4. If angles, then sides.
5. WY  bis. YZ
5. If two points are each equidistant from the
endpoints of a segment, then they determine
the perpendicular bisector of the segment.
6. If a point is on the perpendicular bisector of
a segment, then it is equidistant from the
endpoints of that segment.
6. XY  ZY
2.
a) Identify a pair of corresponding angles
formed by BE and CD with transversal BC .
A
Angles ABE and ACD
b) Identify a pair of alternate interior angles
formed by BE and CD with transversal BD
B
E
Angles EBD and CDB
C
D
B
3.
Given: ADB  CDB
AD  DB
Prove: BD is an altitude.
A
Statements
1. Angle ADB = Angle CDB
2. Angle ADB and Angle CDB are right
angles
3. BD is an altitude
C
D
Reasons
1. Given
2. If two angles are both supplementary and
congruent, then they are right angles.
3. An altitude forms right angles with the side
to which is is drawn.
C
4.
Given: 1  4
FC bisects BFD
Prove: CF  AE
A
Statements
1. Angle 1 = Angle 4
2. FC bisects Angle BFD
3. Angle 2 = Angle 3
4. Angle CFA = Angle CFE
5. Angle CFA and Angle CFE are right angles
6. CF is perpendicular to AE
D
B
1
3
2
F
4
Reasons
1. Given
2. Given
3. If a ray bisects an angle then it divides the
angle into two congruent angles
4. Addition Property
5. If two angles are both supplementary and
congruent, then they are right angles.
6. If two lines intersect to form right angles,
then they are perpendicular.
E
5. Set up and complete a proof for the following:
If two isosceles triangles share the same base, then the line joining the vertex angles of
the triangles is the perpendicular bisector of the base.
A
Statements
1. Triangles ABC and
DBC are isosceles with
base BC
2. AB = AC
and DB = DC
A
OR:
D
B
C
B
D
C
3. AD is the
perpendicular bisector of
BC
Given: Triangles ABC and DBC are
isosceles with base BC
Prove: AD is the perpendicular bisector of BC
Reasons
1. Given
2. The legs of an
isosceles triangle are
congruent.
3. If two points are each
equidistant from the
endpoints of a segment,
then they determine the
perpendicular bisector of
the segment.
A
6.
Given: ABC is isosceles with base AC .
1  2
Prove: BD  AC
1
B
E
D
2
C
Statements
1. Triangle ABC is isosceles with base
AC
2. BA = BC
3. Angle 1 = Angle 2
4. DA = DC
5. BD is the perpendicular bisector of AC
Reasons
1. Given
2. The legs of an isosceles triangle are
congruent
3. Given
4. If angles, then sides
5. If two points are each equidistant from
the endpoints of a segment, then they
determine the perpendicular bisector of
the segment.
7.
Given:
O
M is the midpoint of AB .
Prove:
OM  AB
M
A
B
O
Statements
1. Circle O
2. Draw AO and BO
3. AO = BO
4. M is the midpoint of AB
5. AM = BM
Reasons
1. Given
2. Two points determine a segment
3. All radii of a circle are congruent
4. Given
5. If a point is the midpoint of a segment, then it
divides the segment into two congruent segments.
6. OM perpendicular bisector
6. If two points are each equidistant from the endpoints
of AB
of a segment, then they determine the perpendicular
bisector of the segment.
8. Set up a proof for the statement, “If two chords of a circle are congruent, then the segments
joining the midpoints of the chords to the center of the circle are congruent.” (A chord is
a segment whose endpoints are on the circle.)
A
M
O
B
C
D
Given: Circle O
AB = CD
M and N are midpoints
Prove: MO = NO
N
y
R (11 , 8)
8
C (15 , 7)
9.
a) If the median from A intersects BC at M,
what are the coordinates of M?
(9, 4)
b) Find the slope of BC .
1/2
6
A (2, 4 )
4
2
B (3 , 1)
x
5
c) Is AR parallel to BC ? Why or why not?
No, they have different slopes
d) Find the slope of the altitude from A to BC .
-2
10
15
X
1
10.
O
Given:
1  2
Prove: OY  WX
Z
O
Y
2
W
A
E
11.
Given: 1  2  3  4
BE  BF
Prove: ABE  CBF
1
2
B
D
3
4
F
C
Statements
1. Angle 1 = Angle 2 = Angle 3 = Angle 4
2. BE = BF
3. BD = BD
4. Triangle BED = Triangle BFD
5. Angle BED = Angle BFD
6. Angle AEB = Angle CFB
7. Triangle ABE = Triangle CBF
Reasons
1. Given
2. Given
3. Reflexive Property
4. SAS
5. CPCTC
6. If angles are supplementary to congruent
angles, then they are congruent
7. ASA
Y
12.
Given:
Prove:
O
DX  DY
DZ bisects XY .
O
D
Z
X
Statements
1. Circle O
2. Draw OY and OX
3. OY = OX
4. DX = DY
5. DO (which is also DZ) is the perpendicular
bisector of XY
13.
Reasons
1. Given
2. Two points determine a line
3. All radii of a circle are congruent
4. Given
5. If two points are each equidistant from the
endpoints of a segment, then they determine
the perpendicular bisector of the segment.
Given: WXY  ZYX
WX  ZY
Prove: WR  ZR
W
Z
R
Y
X
Statements
1. Angle XWY = Angle ZYX
2. WX = ZY
3. XY = YX
4. Triangle WXY = Triangle ZYX
5. Angle RXY = Angle RYX
6. WY = ZX
7. XR = YR
8. WR = ZR
Reasons
1. Given
2. Given
3. Reflexive Property
4. SAS
5. CPCTC
6. CPCTC
7. If angles, then sides
8. Subtraction Property
E
F
14.
Given: AB  AF
BD  FD
1  2
Prove: AD  CE
2
D
A
1
B
C
Statements
1. AB = AF
2. BD = FD
3. AD = AD
4. Triangle AFD = Triangle ABD
5. Angle FDA = Angle BDA
6. Angle 1 = Angle 2
7. Angle EDA = Angle CDA
8. Angle EDA and Angle CDA are right
angles
Reasons
1. Given
2. Given
3. Reflexive Property
4. SSS
5. CPCTC
6. Given
7. Addition Property
8. If two angles are both supplementary
and congruent, then they are right
angles.
9. If two segments intersect to form right
angles, then they are perpendicular.
9. AD perpendicular to CE
15.
C
Given: AD  bis. BC
Prove: 1  2
1
F
D
A
2
B
Statements
1. AD is the perpendicular bisector of BC
2. FC = FB and AC = AB
3. AF = AF
4. Triangle AFC = Triangle AFB
5. Angle 1 = Angle 2
Reasons
1. Given
2. If a point is on the perpendicular bisector of
a segment, then it is equidistant from the
endpoints of the segment.
3. Reflexive Property
4. SSS
5. CPCTC
y
16.
E (2 , 7)
F is a right angle. Explain why (9, 6)
could not be the coordinates of H.
H
F (4, 3 )
The slope of EF is -2.
If H was at (9, 6), then the slope of FH would be 3/5,
Which is not the opposite reciprocal of -2.
17. Prove: If the bisector of an angle whose vertex lies on a circle passes through the center of
the circle, then it is the perpendicular bisector of the segment that joins the points where
the sides of the angle intersect the circle.
x