Download 3 Styles of Proofs 1. Two

Geometry
Guided Notes
Styles of Proofs
Name: ___________________________
Date: _______________ Period: _____
3 Styles of Proofs
1. Two-Column Proof
a. __________________________________________________________________
b. __________________________________________________________________
2. Flow Proof
a. __________________________________________________________________
b. __________________________________________________________________
3. Paragraph Proof
a. __________________________________________________________________
b. __________________________________________________________________
New Theorems
Theorem: If two lines are perpendicular, then they intersect to form four right angles.
Example:
Theorem: All right angles are congruent.
Example:
Theorem: If two lines intersect to form a pair of adjacent congruent angles, then the lines are
perpendicular.
Example:
Geometry
Guided Notes
Styles of Proofs
Name: ___________________________
Date: _______________ Period: _____
l1
2-Column Proof
(1). Given: l1 l2
Prove: m∡1 = 90°, m∡2 = 90°, m∡3 = 90°, m∡4 = 90°
Statements
1. l1
l2
1 2
3 4
l2
Reasons
1.
2. ∡1 is a right angle
2.
3. m∡1 = 90°
3.
4. m∡1 = m∡4
4.
5. m∡4 = 90°
5.
6. m∡1 + m∡2 = 180°
m∡1 + m∡3 = 180°
6.
7. 90° + m∡2 = 180°
90° + m∡3 = 180°
7.
8. m∡2 = 90°
m∡3 = 90°
8.
Paragraph Proof:
Given: l1 l2
Prove: m∡1 = m∡2 = m∡3 = m∡4 = 90°
We are given that l1
l2. Therefore, we know that ∡1 is a right angle by the definition of
perpendicular lines. By the definition of a right angle, we know that m∡1 = 90°. Since ∡1 and ∡4 are
vertical angles, we know that m∡1 = m∡4 by the Vertical Angle Theorem. Therefore, by the
Substitution Property, m∡4 = 90°. By the Linear Pair Postulate, m∡1 + m∡2 = 180° and m∡1 + m∡3 =
180°. We can substitute 90° for m∡1 using the Substitution Property; 90° + m∡2 = 180° and 90° + m∡3
= 180°. Subtract 90° from both sides of the equation to yield m∡2 = 90° and m∡3 = 90° by the
Subtraction Property of Equality. We have now shown that m∡1 = m∡2 = m∡3 = m∡4 = 90°.
Geometry
Guided Notes
Styles of Proofs
Name: ___________________________
Date: _______________ Period: _____
Flow Proof:
l1 l2
Given
∡1 is a right angle
Def. Of Perpendicular Lines
m∡1 + m∡2 = 180°
m∡1 + m∡3 = 180°
Linear Pair Postulate
m∡1 = 90°
Def. Of Right Angle
m∡1 = m∡4
Vertical Angle Th.
m∡4 = 90°
Substitution
90° + m∡2 = 180°
90° + m∡3 = 180°
Substitution
m∡2 = 90°
m∡3 = 90°
Subtraction POE
Geometry
Guided Notes
Styles of Proofs
Name: ___________________________
Date: _______________ Period: _____
(2). Given: ∡1 and ∡2 are right angles
Prove: ∡1 ≅ ∡2
1
Statements
Reasons
1. ∡1 and ∡2 are right angles
1.
2. m∡1 = 90°
m∡2 = 90°
2.
3. m∡1 = m∡2
3.
4. ∡1 ≅ ∡2
4.
Flow Proof
2
Geometry
Guided Notes
Styles of Proofs
Name: ___________________________
Date: _______________ Period: _____
m
(3). Given: ∡1 ≅ ∡2;
∡1 and ∡2 are a linear pair
Prove: m n
1
Statements
Reasons
1. ∡1 ≅ ∡2
1.
2. ∡1 and ∡2 are linear pair
2.
3. m∡1 + m∡2 = 180°
3.
4. m∡1 + m∡1 = 180°
4.
5. 2(m∡1) = 180°
5.
6. m∡1 = 90°
6.
7. m
7.
n
Flow Proof
2
n