Download 3.2 Proof and Perpendicular Lines

3.2 Proof and
Perpendicular Lines
WHY?
PROVE!
Different Kinds of
Mathematical Proofs
• Two-Column Proofs
(Section 2.6)
• Paragraph Proofs
• Flow Proofs
5
6
Given: <5 and <6 are a linear pair.
<6 and <7 are a linear pair.
Prove: <5 is congruent to <7
7
2-Column Proof
Statements
Reasons
5
6
Statements
1. <5 and <6 are
a linear pair.
<6 and <7 are
a linear pair.
2. <5 and <6 are
supplementar
y.
<6 and <7 are
supplementar
y.
3. <5 is
congruent to
<7
7
Given: <5 and <6 are a linear pair.
<6 and <7 are a linear pair.
Prove: <5 is congruent to <7
Reasons
1. Given.
2. Linear Pair
Postulate.
3. Congruent
Supplements
Theorem
Because <5 and <6 are a linear
pair, the linear Pair Postulate
says that <5 and <6 are
supplementary. The same
reasoning shows that <6 and
<7 are supplementary.
Because <5 and <7 are both
supplementary to <6, the
Congruent Supplements
Theorem says that <5 is
congruent to <7
5
6
Given: <5 and <6 are a linear pair.
<6 and <7 are a linear pair.
Prove: <5 is congruent to <7
7
Statements
Reasons
1. <5 and <6 are a linear pair.
<6 and <7 are a linear pair.
2. <5 and <6 are supplementary.
<6 and <7 are supplementary.
3. <5 is congruent to <7
1. Given.
2. Linear Pair Postulate.
3. Congruent Supplements Theorem
FLOW PROOF
<5 and <6 are a
linear pair.
Given.
<6 and <7 are a
linear pair.
Given.
<5 and <6 are
supplementary.
Linear Pair Postulate.
<6 and <7 are
supplementary.
Linear Pair Postulate.
<5 is congruent
to <7.
Congruent
Supplements
Theorem.
• If I had two lines that
intersect to form a linear pair
of congruent angles then
what do we know about
them?
Theorem 3.1 (pg. 135)
If two lines intersect to form a
linear pair of congruent
angles, then the lines are
perpendicular.
g
h
g
1
2
h
Given: <1 is congruent to <2.
<1 and <2 are a linear pair.
Prove: g is perpendicular to h
FLOW PROOF
<1 and <2 are a linear pair.
Given.
<1 and <2 are supplementary.
<1 is congruent to <2
Given.
Linear Pair Postulate
m<1 +m<2 = 180
Def. of Supplementary <s
m<1 +m<21= 180
Substitution prop of equality
m<1 = m<2
Def. of congruent angles.
2(m<1)=180
Distributive prop.
m<1 = 90
<1 is a right <
g is perp. to h
Div. prop. Of equality
Def. of right angle
Def of perp. lines
• If two sides of two adjacent
acute angles are
perpendicular, what does that
mean?
Theorem 3.2 (pg. 135)
If two sides of two adjacent
acute angles are
perpendicular, then the angles
are complementary.
Given: <A and <B are
adjacent and acute
Prove: <A and <B are
complementary
FLOW PROOF
• If two lines intersect and form
four right angles what can you
tell me bout the lines?
Theorem 3.3 (pg. 135)
If two lines are perpendicular,
then they intersect to form
four right angles.
End of Lesson Questions
1. Draw the outlines of a two column proof and
a flow proof. Make sure to label where the
reasons and statements go.
2. What three things were learned about
perpendicular lines today?
3. Daily Puzzler (digital clock):
“I’m constructed from more than 4 distinct
segments. I contain 4 different pairs of parallel
segments. I contain 6 different pairs of
perpendicular segments.”