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3.4 Warm Up
October 28, 2015 3.3 Proofs with Parallel Lines
1. Find the values of x and y.
Substitute the given values of m, x, and y into the
equation y = mx + b and solve for b.
2. m = 2, x = 3, and y = 0
3. m = -1, x = 5, and y = -4
Geometry
3.4 Proofs with Perpendicular Lines
October 28, 2015 3.4 Proofs with Perpendicular Lines
Essential Question
What conjectures can you make about
perpendicular lines?
Distance between a Point and a Line
The distance from a point to a line is the
length of the perpendicular segment from
the point to the line.
This perpendicular segment is the shortest
distance between the point and the line.
October 28, 2015 3.4 Proofs with Perpendicular Lines
Example 1
October 28, 2015 3.4 Proofs with Perpendicular Lines
2 22 1 2 1= ( - ) + ( - )d x x y y
Perpendicular Bisector
October 28, 2015 3.4 Proofs with Perpendicular Lines
October 28, 2015 3.4 Proofs with Perpendicular Lines
Theorem 3.10If two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular.
1 2
m
n
m
Linear Pair
Perpendicular
Theorem
October 28, 2015 Geometry 3.2 Proof and Perpendicular Lines 8
Theorem 3.10: The ProofIf two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular.
1 2n
m
Given: 1 & 2 form a linear pair.
1 2.
Prove: m n. Plan:
Show that 1 and 2 are
supplementary, and then
since the angles are
congruent, each is 90°. This
means the lines are
perpendicular.
October 28, 2015 Geometry 3.2 Proof and Perpendicular Lines 9
Theorem 3.10: 2-column proof
1 2
m
n
Given: 1 & 2 form a linear pair. 1 2.
Prove: m n.
Statements Reasons1. 1 & 2 are lin. pr. 1. Given2. 1 supp. 2 2. Lin. Pr. Post. 3. m1 + m2 =180 3. Def. Supp.4. 1 2 4. Given5. m1 = m2 5. Def. s6. m1 + m1 = 180 6. Substitution7. 2m1 = 180 7. Simplify8. m1 = 90 8. Division9. 1 is a right angle 9. Def Rt 10. m n 10. Def. lines
QED
October 28, 2015 Geometry 3.2 Proof and Perpendicular Lines 10
Theorem 3.10: Paragraph Proof
1 2
m
n
Given: 1 & 2 form a
linear pair. 1 2.
Prove: m n.
Since 1 & 2 form a linear pair, their sum
is 180 degrees. They are also congruent,
which means they have the same measure.
So, each angle must be 90 degrees. This
means the angles are right angles, and hence
the lines are perpendicular by definition.
October 29, 2015 Geometry 3.2 Proof and Perpendicular Lines 11
Theorem 3.10: Flow Proof
Given: 1 & 2 form a linear pair.
1 2.
Prove: m n.
1 & 2 are lin. pr.
1 2
m1 + m2 =180
m1 + m1 = 180 2m1 = 180
m1 = 90
m n
SimplifyGiven
Lin. Pr. Post
Substitution
Given
Division
Def. of
October 28, 2015 3.4 Proofs with Perpendicular Lines
Theorem 3.11
If a transversal is perpendicular to one of
two parallel lines, then it is perpendicular
to the other.
Perpendicular
Transversal
Theorem
October 28, 2015 3.4 Proofs with Perpendicular Lines
Proving the Perpendicular Transversal
Theorem (3.11)
Given: h || k, j h
Prove: j k.Plan:
Show that by knowing h
and k are parallel we know
something about
corresponding angles, then
show 2 and 6 are
congruent and measure 90°.
This means the lines are
perpendicular.
If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the other.
j
h
k5 67 8
13 4
2
October 28, 2015 3.4 Proofs with Perpendicular Lines
Theorem 3.11: 2-Column Proof
Statements Reasons
1. h || k, j h
2. m2 = 903. 2 64. m2 = m65. m6 = 906. j k
QED
Given: h || k, j h
Prove: j k.
j
h
k5 67 8
13 4
2
𝟏. 𝑮𝒊𝒗𝒆𝒏𝟐.𝑫𝒆𝒇. 𝒐𝒇
𝟑. 𝑪𝑨 ≅𝟒.𝑫𝒆𝒇 ≅ ∠𝟓. 𝑺𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒊𝒐𝒏𝟔.𝑫𝒆𝒇. 𝒐𝒇
October 28, 2015 3.4 Proofs with Perpendicular Lines
Theorem 3.12In a plane, if two lines are perpendicular
to the same line, then the lines are parallel.
t
m n
m t
n t
m || n
Lines Perpendicular to a Transversal Theorem
October 28, 2015 3.4 Proofs with Perpendicular Lines
Example 1True or False?
Line a is perpendicular to line b.
Line b is perpendicular to line c.
Therefore, line a is parallel to
line c.
Th. 3.12
a
b
c
October 28, 2015 3.4 Proofs with Perpendicular Lines
Example 2True or False?
Line a is perpendicular to line b.
Line b is perpendicular to line c.
Therefore, line a is
perpendicular to line c.
False
a
b
c
October 28, 2015 3.4 Proofs with Perpendicular Lines
Example 3: Building a Fence
The vertical slats are parallel.
The horizontal
slats are
perpendicular to
all of the
vertical slats.
Why are the horizontal slats also parallel?
Theorem 3.12
If two lines are perpendicular to the same
line, then the lines are parallel.
Example 4The photo shows the layout of a neighborhood.
Determine which lines, if any, must be parallel in
the diagram. Explain your reasoning.
October 28, 2015 3.4 Proofs with Perpendicular Lines
s t u
p
q
October 29, 2015 3.4 Proofs with Perpendicular Lines
To Summarize Linear Pair Perpendicular Theorem (3.10):
If two lines intersect to form a linear pair of
congruent angles, then the lines are
perpendicular.
Perpendicular Transversal Theorem (3.11):
If a transversal is perpendicular to one of two
parallel lines, it is perpendicular to the other.
Lines Perpendicular to a Transversal
Theorem (3.12): If two lines are perpendicular
to the same line, then those lines are also
parallel.
October 28, 2015 3.4 Proofs with Perpendicular Lines
Assignment
