Download Homework on Proofs

Geometry 1
Name: _______________________________
Homework #1 on Proofs
The following Definitions, Postulates and Theorems can be used in this assignment:
Definition of Midpoint: M is the midpoint of AB if and only if M is between A and B and
AM  BM .
Definition of Bisects: A line bisects a segment if it intersects the segment at its midpoint, and a
ray bisects an angle if it is interior to the angle, its vertex coincides with the vertex of the angle,
and it forms two congruent angles.
o
Definition of Complementary: Two angles are complementary if their measures add to 90 .
Definition of Supplementary: Two angles are supplementary if their measures add to 180o.
Definition of Adjacent Angles: Two angles are adjacent if they have a common side.
Definition of Linear Pair: Two angles form a linear pair if they are adjacent and their noncommon sides are opposite rays (form a line).
Definition of Parallelogram: A quadrilateral is a parallelogram if opposite sides are parallel.
Transitive: If two segments or angles are congruent to a third segment or angle then they are
congruent to each other.
Vertical Angle Theorem: Vertical angles are congruent.
Linear Pair Theorem: The angles in a linear pair are supplementary..
Parallel Postulate: Given a point and a line not containing that point, there is one and only one
line through the given point that is parallel to the given line.
Parallel Lines:
Two lines cut by a transversal are parallel if and only if corresponding angles are congruent.
Two lines cut by a transversal are parallel if and only if alternate interior angles are
congruent.
Two lines cut by a transversal are parallel if and only if alternate exterior angles are
congruent.
Two lines cut by a transversal are parallel if and only if same-side interior angles are
supplementary.
Two lines cut by a transversal are parallel if and only if same-side exterior angles are
supplementary.
Page 1 of 8
Problems:
1.
Mark the diagrams according to the given information, and fill in the missing
statements or reasons for each of the following proofs.
Q
Given:  P and Q are supplementary.
R
Prove:  R and S are supplementary.
Proof:
2.
P
S
Statement:
Reason:
1.
 P and Q are supplementary.
Given
2.
QR | | PS
3.
 R and S are supplementary.
B
Given: BAP  QBC and PAC  QBC
Prove: AP bisects BAC
P
A
Proof:
Statement:
Reason:
1.
BAP  QBC
Given
2.
PAC  QBC
3.
BAP  PAC
4.
AP bisects BAC
Page 2 of 8
Q
C
3
3.
Given: 1  2 and  1 ||  2
Prove: 1  3
4
2
1
1
2
3
Proof:
Statement:
Reason:
1.
 1 ||  2
Given
2.
2  3
3.
1  2
4.
1  3
Page 3 of 8
4.
3
Given:  1 ||  2 , 1  2 , and 3  4
Prove: 1  4
1
4
1
1
Proof:
Statement:
1.
 1 ||  2
2.
2  3
3.
1  3
4.
3  4
5.
1  4
2
3
2
Reason:
Page 4 of 8
5
4
5.
C
Given: ACE  CED and EAD  ADC
D
Prove: ACDE is a parallelogram.
A
Proof:
Statement:
1.
E
Reason:
ACE  CED
2.
Two lines cut by a transversal are parallel if
and only if alternate interior angles are
congruent.
3.
Given
4.
CD | | AE
5.
ACDE is a parallelogram.
Page 5 of 8
6.
Given: A  ACB and D  DCE
Prove:
A
Statement:
1.
A  ACB
2.
ACB  DCE
3.
A  DCE
4.
D  DCE
5.
A  D
D
C
AB | | DE
Proof:
6.
B
E
Reason:
Two lines cut by a transversal are parallel if
and only if alternate interior angles are
congruent.
Page 6 of 8
A
7.
Given: BD bisects EBC and CBD  AEB
Prove:
AE | | BD
E
Proof:
Statement:
1.
B
Reason:
BD bisects EBC
Definition of “bisects.”
2.
3.
CBD  AEB
4.
EBD  AEB
5.
AE | | BD
Transitive Property
Page 7 of 8
D
C
8.
Prove that the angles of a triangle add to 180o:
D
Given: ABC with a  mBAC ,
b  mABC , c  mACB
d
Prove: a + b + c = 180o
A
Proof:
Statement:
E
B
b
e
a
c
C
Reason:
1.
Let DE be the line containing point B
that is parallel to AC .
2.
Let d  mABD and e  mCBD
3.
d=a
Definition of d and e.
These are alternate interior angles for
parallel lines DE and AC with transversal
BC .
4.
5.
DBA and ABC are adjacent angles.
Definition of adjacent angles
6.
mDBC  d  b
Angle Addition Postulate
7.
DBC and CBE are a linear pair.
Definition of linear pair
8.
DBC and CBE are supplementary.
9.
mDBC  e  180
10. d + b + e= 180o
Substitution using steps 6 and 9.
11. a + b + c= 180o
Substitution using steps 3, 4 and 10.
Page 8 of 8