Question

Write the following paragraph proof as a two-column proof.

Given:
A
⁢
B
=
C
⁢
D
and
B
⁢
C
=
D
⁢
E

Prove:
A
⁢
C
=
C
⁢
E



We're given that
A
⁢
B
=
C
⁢
D
. By the addition property of equality, we add
B
⁢
C
to both sides of the equation to get
A
⁢
B
+
B
⁢
C
=
C
⁢
D
+
B
⁢
C
. Since we're also given that
B
⁢
C
=
D
⁢
E
, we use the substitution property of equality to replace
B
⁢
C
with
D
⁢
E
on the right side of the equation. So,
A
⁢
B
+
B
⁢
C
=
C
⁢
D
+
D
⁢
E
. Next, by segment addition, we get that
A
⁢
B
+
B
⁢
C
is equal to
A
⁢
C
and that
C
⁢
D
+
D
⁢
E
is equal to
C
⁢
E
. Finally, we use the substitution property of equality on the equation
A
⁢
B
+
B
⁢
C
=
C
⁢
D
+
D
⁢
E
to replace
A
⁢
B
+
B
⁢
C
with
A
⁢
C
and
C
⁢
D
+
D
⁢
E
with
C
⁢
E
to get that
A
⁢
C
=
C
⁢
E
.

Type the correct answer in the box.

Answer 1

The addition property of equality tells us that adding the same number to each ... quantity b, and b equals the quantity, c, then a and c are equal as well.

Step-by-step explanation:

Answer 2

To convert the paragraph proof to a two-column proof, the given equations AB = CD and BC = DE are used along with the Addition and Substitution Properties of Equality to prove AC = CE, leveraging the Segment Addition Postulate.

Step-by-step explanation:

We are given two segments AB = CD and BC = DE. The goal is to prove that AC = CE using a two-column proof.

Two-Column Proof

StatementReason1. AB = CDGiven2. BC = DEGiven3. AB + BC = CD + BCAddition Property of Equality4. AB + BC = CD + DESubstitution Property of Equality (Step 2)5. AC = CESegment Addition Postulate6. AC = CESubstitution Property of Equality (Step 4)

By adding BC to both sides in step 3 and substituting DE for BC on the right side in step 4, we apply the segment addition postulate in step 5 to simplify both sides to AC and CE respectively, concluding the proof in step 6.