4.1 (To make this easier to type, I am not using line, segment or ray symbols, and
the = sign in proofs represents congruent.)
1.
Step 3:
Step 4:
Step 5:
Step 6:
Step 7:
WY = WY
SSS
CPCTC
AY = AY
SAS
2.
Statements
1. MN = NS
2. MP = PS
3. PN = PN
4. Triangle PMN = Triangle PSN
5. Angle MPN = Angle SPN
6. PQ = PQ
7. Triangle PMQ = Triangle PSQ
8. Angle MQP = Angle SQP
4.
Midpoint of AB = (1, 4)
Midpoint of BC = (6, 2)
Midpoint of AC = (1, 1)
5.
Midpoint of BC = (7, 2)
Reasons
1. Given
2. Given
3. Reflexive Property
4. SSS
5. CPCTC
6. Reflexive Property
7. SAS
8. CPCTC
12.
Statements
1. PR = PU
2. QR = QU
3. Draw PQ
4. PQ = PQ
5. Triangle PQR = Triangle PQU
6. Angle R = Angle U
7. RS = UT
8. Triangle PRS = Triangle PUT
9. Angle 1 = Angle 2
Reasons
1. Given
2. Given
3. 2 points determine a segment
4. Reflexive Property
5. SSS
6. CPCTC
7. Given
8. SAS
9. CPCTC
C
4.2
4. Given:
Prove:
CA = CB
CD bisects Angle ACB
CD perpendicular to AB
A
B
D
B
6. Given:
Prove:
AB = CB
BD is a median
Triangle ABD = Triangle CBD
A
Statements
1. AB = CB
2. BD is a median
3. AD = CD
4. BD = BD
5. Triangle ABD = Triangle CBD
C
D
Reasons
1. Given
2. Given
3. A median divides the side to
which it is drawn into two
congruent segments
4. Reflexive Property
5. SSS
A
7. Given: AB = AC
Prove: Angle 1 = Angle 2
1
B
Statements
1. AB = AC
2. Angle ABC = Angle ACB
3. Angle 1 = Angle 2
2
C
Reasons
1. Given
2. If sides, then angles
3. If angles are supplementary to
congruent angles, then they are
congruent.
B
8.
Given:
BD is a median and an altitude
Prove: Triangle ABC is isosceles.
C
D
A
Statements
1. BD is a median and an altitude
2. AD = CD
3. Angle BDA and Angle BDC are
right angles
4. BD = BD
5. Triangle BDA – Triangle BDC
6. BA = BC (or Angle A = Angle C)
7. Triangle ABC is isosceles
Reasons
1. Given
2. A median divides the side to which
it is drawn into two congruent
segments
3. An altitude forms right angles with
the side to which it is drawn.
4. Reflexive Property
5. SAS
6. CPCTC
7. If at least two sides (or angles) of a
triangle are congruent, then the
triangle is isosceles.
A
9. Given:
Prove:
AB = AC
D and E trisect BC
AD = AE
B
Statements
1. AB – AC
2. Angle B – Angle C
3. D and E trisect BC
4. BE = DE = CE
5. Triangle ABD – Triangle ACE
6. AD = AE
D
E
C
Reasons
1. Given
2. If sides, then angles
3. Given
4. If two points trisect a segment, then
they divide the segment into three
congruent segments.
5. SAS
6. CPCTC
X
Y
10. Given:
Prove:
WY bisects Angle XWZ
YW bisects Angle XYZ
Angle X = Angle Z
W
Z
Statements
1. WY bisects Angle XWZ
2. Angle XWY = Angle YWZ
3. YW bisects Angle XYZ
4. Angle XYW = Angle WYZ
5. WY = WY
6. Triangle WXY = Triangle WZY
7. Angle X = Angle Z
Reasons
1. Given
2. If a ray bisects an angle then it
divides the angle into two congruent
angles.
3. Given
4. If a ray bisects an angle then it
divides the angle into two congruent
angles.
5. Reflexive Property
6. ASA
7. CPCTC
C
4.3
2. Given:
Prove:
AC = BC
CD bisects angle ACB
CD perpendicular to AB
A
Statements
1. AC = BC
2. Angle A = Angle B
3. CD bisects Angle ACB
4. Angle ACD = Angle BCD
5. Triangle ACD = Triangle BCD
6. Angle CDA = Angle CDB
7. Angle CDA and Angle CDB are
right angles
8. CD perpendicular to AB
D
B
Reasons
1. Given
2. If sides then angles
3. Given
4. If a ray bisects an angle, then it
divides the angle into two
congruent angles.
5. ASA
6. CPCTC
7. If two angles are both
supplementary and congruent, then
they are right angles
8. If two segments intersect to form
right angles, then they are
perpendicular.
4.
Statements
1. XR bisects Angle YXZ
2. Angle YXR = Angle ZXR
3.
4.
5.
6.
7.
Angle Y = Angle Z
YX = ZX
Triangle YXR = Triangle ZXR
Angle XRY = Angle XRZ
Angle XRY and Angle XRZ are
right angles
8. XR is an altitude
Reasons
1. Given
2. If a ray bisects and angle, then it
divides the angle into two
congruent angles.
3. Given
4. If angles, then sides
5. ASA
6. CPCTC
7. If two angles are both
supplementary and congruent, then
they are right angles
8. An altitude forms right angles with
the side to which it is drawn.
5. (-1 8)
6. 8
7. 4 pi = 12.6
8. (1, 3) or (2, 1)
11.
Statements
Reasons
1. PR bisects QS
2. QT = ST
Statements
3. Angle RQT = Angle RST
1.
O
4. Circle
QR = SR
2.
OARTQ
and OB
5. Draw
Triangle
= Triangle RTS
3.
= OB
6. OA
Angle
RTQ = Angle RTS
4.
is the
midpoint
of AB
7. M
Angle
RTQ
and Angle
RTS are
5. AM
BM
right=angles
8. QS perpendicular to PR
6. OM perpendicular to AB
14. y = 37, x = 26. 5 YES
4.4
1.
2.
1. Given
2. If a line bisects a segment, then it
divides the segment into two
congruent segments.
Reasons
3. Given
1.
4. Given
If angles, then sides
2.
5. Two
SAS points determine a segment
3.
radii of a circle are congruent
6. All
CPCTC
4.
7. Given
If two angles are both
5. If
a point is the and
midpoint
of a then
supplementary
congruent,
segment,
then angles.
it divides the
they are right
into twointersect
congruent
8. segment
If two segments
to form
segments.
right angles, then they are
6. If
two points are each equidistant
perpendicular.
from the endpoints of a segment,
then they determine the
perpendicular bisector of the
segment.
Statements
1. Circles P and Q
2. Draw PR, PS, QR, and QS
3. PR = PS and QR = QS
4. PQ is the perpendicular bisector of
RS
Reasons
1. Given
2. Two points determine a segment
3. All radii of a circle are congruent
4. Two points each equidistant from
the endpoints of a segment
determine the perpendicular
bisector of the segment
3.
4.
Statements
1. AD is the perpendicular bisector of
BC
2. EB = EC and AB = AC
Statements
3. AE = AE
1. WZ is the perpendicular bisector of
4. Triangle ABE = Triangle ACE
XY
2. WX = WY
3. Triangle WXY is isosceles
Reasons
1. Given
2. If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of
that segment.
Reasons
3. Reflexive Property
1. Given
4. SSS
2. If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of
the segment.
3. If at least two sides of a triangle are
congruent, then the triangle is
isosceles.
5.
Statements
1. Circle O
2. Draw OB and OC
3. OB = OC
4. AB = AC
5. AD is the perpendicular bisector of
BC
Reasons
1. Given
2. Two points determine a segment
3. All radii of a circle are congruent
4. Given
5. Two points each equidistant from
the endpoints of a segment
determine the perpendicular
bisector of the segment.
6.
7.
Midpoint of OA = (6, 1)
Midpoint of AB = (10. 4)
4 greater.
8. (6. 82, 1)
9. (15, 3)
Statements
1. AB = BC
2. AE = EC
3. BE is the perpendicular bisector of
AC
Reasons
1. Given
2. Given
3. Two points each equidistant from
the endpoints of a segment
determine the perpendicular
bisector of that segment.
4. DA = DC
4. If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of
that segment.
10.
a) x = 49
YES AB is perpendicular to DC
b) measure of angle EBC is 90
AB and BE are the same line!
12.
Statements
1. AG is the perpendicular bisector of
BC
2. AB = AC
3. AG is the perpendicular bisector of
DE
4. AD = AE
5. BD = CE
Reasons
1. Given
2. If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of
that segment.
3. Given
4. If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of
that segment,
5. Subtraction Property
20.
Statements
1. AB = BC
2. AE = EC
3. Draw AC
4. BE is the perpendicular bisector of
AC
5. AD = DC
Reasons
1. Given
2. Given
3. Two points determine a segment.
4. Two points each equidistant from
the endpoints of a segment
determine the perpendicular
bisector of that segment.
5. If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of
that segment.