Download 5.1 Paragraph Proofs

Paragraph Proofs
Writing paragraph proofs can be A LOT easier than two-column proofs. You can leave out some
steps as long as they are implied. You do not have to write reflexive steps (although you still can
if you want). If a conclusion is based on a postulate or theorem you must either state which
theorem or state what the theorem says (which I kind of did with the Angle Bisector Theorem in
Example #2 on page 3).
You can NEVER leave out reasons such as SAS, CPCTC, “If ||, then alt-int !'s " ”, and others
like them!
You still need to state the givens before you use them.
Things that no longer need justification and can be assumed:
angle addition postulate
segment addition postulate
definition of supplementary angles
definition of complementary angles
(but the reasoning still needs to be clear).
Some shortcuts are allowed, like:
1
3
2
4
If you know that ∠2 = ∠4, then you can say ∠1 = ∠3 because they are supplements of
congruent angles.
C
X
D
E
Y
H
If you know that CX = EY and XD = YH, then you can say that CD = EH because of the
addition property of equality.
You can state or clearly imply reasons, such as:
Because AB is perpendicular to CD, then ∠1 = 90 degrees.
Because l // m, then ∠2 + ∠1 = 180° because they are same-side interior angles
(diagram below)
1
2
Geometry
l
m
Paragraph Proof Explanation
Page 1
Here are two proof examples done in each style that will give you a comparison…
EXAMPLE #1:
Given: AC = XZ ; AB = XY
Prove: BC = YZ
A
X
B
C
Y
Z
Two-Column Proof...
Statements
1. AC = XZ ; AB = XY
2. AB + BC = AC!
XY + YZ = XZ
3. AB + BC = XY + YZ
4.
BC = YZ
VS.
Reasons
1. Given
2. SAP
3. Substitution
4. Subtraction
(Step 1 from 4)
Paragraph Proof...
Since it is given that AC = XZ and AB = XY, it follows
that BC = YZ by the subtraction property of equality.
OR
Since AC = XZ and AB = XY (Given), then BC = YZ
(Subtr. Prop. of =).
Note #1: SAP is implied from the diagram! You don’t have to state it.
Note #2: If you are going to use subtraction, be sure to state what you will be
subtracting from what.
Geometry
Paragraph Proof Explanation
Page 2
EXAMPLE #2:
A
Given: m!ABC = m!WXY
BD bisects !ABC
XZ bisects !WXY
Prove: m!1 = m!2
B
Two-Column Proof...
Statements
1. m!ABC = m!WXY
BD bisects !ABC
XZ bisects !WXY
1
2. m!1= m!ABC
2
1
m!2= m!WXY
2
1
3. m!2= m!ABC
2
4. m!1 = m!2
VS.
W
D
Z
1
2
C
X
Y
Reasons
1. Given
2. Angle Bisector Thm
3. Substitution
4. Substitution
(Steps 2 and 3)
Paragraph Proof...
Since !ABC " !WXY, BD bisects !ABC and XZ bisects
!WXY (all given), then !1"!2 because they are both
one half of congruent angles.
Geometry
Paragraph Proof Explanation
Page 3