Question

given
ac = ab + bc
segment addition
assumption
bd = bc + cd
segment addition
ab + bc ≠ bc + cd
substitution property
of equality
subtraction property
of equation
contradiction
length cannot be both
equal and unequal
the negation is false.
ac ≠ bd
ac = bd
ab = cd
ab ≠ cd

Explanation:

Step1: Identify proof structure

This is a proof by contradiction, so we first assume the negation of the statement we want to prove.

Step2: Match given and assumption

The proof uses segment addition for $AC=AB+BC$ (given) and $BD=BC+CD$ (from assumption). We substitute into $AB+BC
eq BC+CD$, then use subtraction property to get a contradiction. The negation (assumption) must be the opposite of the conclusion. If the contradiction shows the negation is false, the conclusion is $AB=CD$, so the assumption (negation) is $AB
eq CD$, and the given is $AC=BD$.

1. First dashed box (given): $AC=BD$
2. Second dashed box (assumption): $AB

eq CD$

1. Third dashed box (subtraction step): Subtract $BC$ from both sides of $AB+BC

eq BC+CD$, so $AB
eq CD$ (this is the step leading to contradiction)

1. Final dashed box: Since negation ($AB

eq CD$) is false, the true statement is $AB=CD$.

Answer:

1. Given (first dashed box): $\boldsymbol{AC=BD}$
2. Assumption (second dashed box): $\boldsymbol{AB

eq CD}$

1. Subtraction property step (third dashed box): $\boldsymbol{AB

eq CD}$

1. Final conclusion (fourth dashed box leads to): $\boldsymbol{AB=CD}$

Filling the boxes in order of the flow:

• Given: $AC=BD$
• Assumption: $AB

eq CD$

• Subtraction property result: $AB

eq CD$

• Final verified statement: $AB=CD$