Download Parallel Line Proof Puzzle #1

Linear Pair Theorem Puzzle Proof
If two angles form a linear pair,
Then the angles are supplementary.
Given: 1 and 2 form a linear pair.
Prove:

1 and 2 are supplementary.
Statements
1 2
T


A
M
Reasons
1 and 2 form a linear pair
Given
TAM is a straight angle.
Definition of Linear Pair
Two angles form a linear pair
iff their non-shared sides form a straight angle..
Definition of straight angle
m TAM = 180
An angle is a straight angle,
iff its measure is 180.
Angle Addition Postulate
If two angles are adjacent,
then the sum of their individual measures equals the
measure of the angle formed by their non-shared
sides.
m1 + m 2 = mTAM
Substitution Property of Equality
m1 + m 2 = 180
If m 1 + m  2 = m TAM and mTAM = 180
Then m1 + m 2 = 180
Definition of Supplementary
1 and 2 are supplementary
The measure of two angles sums to 180
iff the angles are supplementary.
QED
Vertical Angles Theorem Puzzle Proof
If two angles are vertical angles,
Then they have equal measures.
2
Given: 1 and 2 are vertical angles
1 3
Prove: m1 = m 2
Statements
Reasons
1 and 2 are vertical angles
Given
1 and 3 form a linear pair
2 and 3 form a linear pair
Definition of Linear Pair
1 and 3 are supplementary
2 and 3 are supplementary
Linear Pair Theorem
m1 + m 3 = 180
Definition of Supplementary
m 2 + m  3 = 180
The measure of two angles sums to 180
iff the angles are supplementary.
m1 = 180 – m 3
Subtraction Property of
Equality
Two angles form a linear pair
iff their non-shared sides form a straight angle.
If two angles form a linear pair,
then they are supplementary.
m2 = 180 – m  3
If the same number is subtracted from both sides of an
equation,
Then the new equation is equivalent to the original.
Substitution Property of
Equality
m 1 = m 2
Substitute (m 1) for (180 - m  3) in m2 = (180 - m  3)
QED
Alternate Interior Angles (AIA)
Given: Lines l and m are parallel lines cut by a transversal a.
a
l
m
1 2
3 4
5 6
7  8
Prove: Alternate Interior angles are congruent. (Prove:  6   3)
Statements
Reasons
Lines l and m are parallel lines cut by a Given
transversal a.
 6 &  2 are corresponding angles
Definition of Corresponding
Angles
62
If Corresponding Angles are
formed by parallel lines cut by
a transversal, those angles are
congruent.
Definition of Vertical Angles
 2 &  3 are vertical angles
23
If two angles are vertical, then
they are .
63
Transitive Property of
Congruence.
QED
Alternate Exterior Angles (AEA)
Given: Lines l and m are parallel lines cut by a transversal a.
a
l
m
Prove:
1 2
3 4
5 6
7  8
Alternate Exterior angles are congruent. (Prove:  2   7)
Statements
Reasons
Lines l and m are parallel lines cut by Given
a transversal a.
 6 &  2 are corresponding angles
Definition of Corresponding Angles
62
 7 &  6 are vertical angles
If Corresponding Angles are formed
by parallel lines cut by a transversal,
those angles are congruent.
Definition of Vertical Angles.
76
If vertical angles, then congruent.
72
Transitive Property of Congruence.
27
Symmetric Property of Congruence
QED
Same Side Interior Angles (SSI)
Given: Lines l and m are parallel lines cut by a transversal a.
a
l
1 2
3 4
m
7
Prove:
6
5
8
Same Side Interior angles are supplementary. (Prove:  6 and
 4 are supplementary)
Lines l and m are parallel lines cut Given
by a transversal a.
 6 &  2 are corresponding
angles
62
 2 &  4 form a linear pair of
angles
 2 &  4 are supplementary
Definition of Corresponding Angles
If parallel lines cut by a transversal
form Corresp. Angles, they are .
Definition of Linear Pair
If two angles form a linear pair,
then they are supplementary.
m 2 + m 4 =180
Definition of supplementary
m 6 = m 2
Definition of Congruent Angles
m 6 + m 4 =180
Substitution property of equality
( 6   2)
(m 6 = m 2 into m 2 + m 4 =180)
 6 &  4 are supplementary
Definition of supplementary.
QED
Same Side Exterior Angles (SSE)
Given: Lines l and m are parallel lines cut by a transversal a.
a
l
1 2
3 4
m
5
7
6
8
Prove: Same Side Exterior angles are supplementary.
(Prove:  1 and  7 are supplementary)
Lines l and m are parallel lines
cut by a transversal a.
 1 &  5 are corresponding
angles
15
 5 &  7 form a linear pair
Given
Definition of Corresponding Angles
If parallel lines cut by a transversal
form Corresp. Angles, angles are .
Definition of Linear Pair
m 5 + m 7 =180
If two angles form a linear pair, then
they are supplementary.
Definition of supplementary
m 1 = m 5
Definition of Congruent Angles(15)
m 1 + m 7 =180
Substitution property of equality
 1 &  7 are supplementary
Definition of supplementary.
 5 &  7 are supplementary
QED
Converse of AIA Puzzle Proof
If two lines are cut by a transversal so that alternate interior angles are congruent,
then the lines are parallel.
Given:  6   3
l
2
3
6
m
Prove: l // m
t
Statements
Reasons
63
Given
t is transversal of l and m
Given / Def. of transversal
 2 &  3 are vertical angles
Definition of Vertical Angles
32
If two angles are vertical angles,
then they are .
62
Transitive Property of Congruence.
 6 &  2 are corresponding
angles
Definition of Corresponding Angles
l // m
If two lines are cut by a transversal
such that corresponding angles are
congruent, then the lines are parallel.
QED
Converse of AEA Puzzle Proof
If two lines are cut by a transversal so that alternate exterior angles are congruent,
then the lines are parallel.
j
Given:  A   C
B
A
C
k
Prove: j // k
m
Statements
Reasons
A  C
Given
 B &  A are vertical angles
Definition of Vertical Angles
BA
If two angles are vertical angles,
then they are .
BC
Transitive Property of Congruence.
m is transversal of j and k
Given / Def. of transversal
 B &  C are corresponding
angles
Definition of Corresponding Angles
j // k
If two lines are cut by a transversal
such that corresponding angles are
congruent, then the lines are parallel.
QED
Converse of SSI Proof
If two lines are cut by a transversal so that same side interior angles are
supplementary, then the lines are parallel.
Given:  6 &  4 are supplementary
l
2
4
6
m
Prove: l // m
t
Statements
Reasons
 6 &  4 are supplementary
Given
t is transversal of l and m
Def. of transversal
 2 &  4 form a Linear Pair
Definition of Linear Pair
 2 &  4 are supplementary
If Linear Pair, then supplementary
m 2 + m 4 = 180
Definition of Supplementary
m 6 + m 4 = 180
Definition of Supplementary.
m 6 + m 4 = m2+ m 4
Substitution Prop of equality
m 4 = m 4
Reflexive Property of =
m 6 = m 2
Addition/Subtraction Property of =
62
Definition of  Angles
 6 &  2 are Corresponding
Angles
l // m
Definition of Corresponding Angles
If two lines are cut by a transversal
such that corresponding angles are
congruent, then the lines are parallel.
QED
Converse of SSE Proof
If two lines are cut by a transversal so that same side exterior angles are
supplementary, then the lines are parallel.
Given:  2 &  8 are supplementary
l
2
4
m
Prove: l // m
t
Statements
8
Reasons
 2 &  8 are supplementary
Given
t is transversal of l and m
Def. of transversal
 2 &  4 form a Linear Pair
Definition of Linear Pair
 2 &  4 are supplementary
If Linear Pair, then supplementary
m 2 + m 4 = 180
Definition of Supplementary
m 2 + m 8 = 180
Definition of Supplementary
m 2 + m 8 = m2+ m 4
Substitution Prop of equality
m 2 = m 2
Reflexive Property of =
m 8 = m 4
Addition/Subtraction Property of =
84
Definition of  Angles
 8 &  4 are Corresponding
Angles
l // m
(from given)
Definition of Corresponding Angles
If two lines are cut by a transversal
such that corresponding angles are
congruent, then the lines are parallel.
QED