Question

how can similarity be used to prove the pythagorean theorem? move the reasons to the table to complete the proof.

statement	reason
$\angle acb \cong \angle adc \cong \angle cdb$
$\triangle abc \sim \triangle acd$ and $\triangle abc \sim \triangle cbd$
$\frac{x}{b} = \frac{b}{c}$ and $\frac{y}{a} = \frac{a}{c}$
$b^2 = cx$ and $a^2 = cy$	multiplication
$a^2 + b^2 = cx + cy$
$a^2 + b^2 = c^2$	factoring and substitution

addition property of equality angle - angle similarity side - side - side similarity transitive property
right angles are congruent. similar triangles have proportional sides.

Explanation:

Step1: Match right angles congruence

$\angle ACB \cong \angle ADC \cong \angle CDB$: All are right angles, so this uses Right angles are congruent.

Step2: Match AA similarity

$\triangle ABC \sim \triangle ACD$ and $\triangle ABC \sim \triangle CBD$: Uses two pairs of congruent angles, so Angle-Angle Similarity.

Step3: Match proportional sides

$\frac{x}{b} = \frac{b}{c}$ and $\frac{y}{a} = \frac{a}{c}$: From similar triangles, sides are proportional, so Similar triangles have proportional sides.

Step4: Match addition property

$a^2 + b^2 = cx + cy$: Adding the two prior equations, so Addition Property of Equality.

Step5: Match factoring/substitution

$a^2 + b^2 = c^2$: Factor $cx+cy=c(x+y)$, and $x+y=c$, so Factoring and substitution (given).

Completed table (matching each statement to reason):

Statement	Reason
$\angle ACB \cong \angle ADC \cong \angle CDB$	Right angles are congruent.
$\triangle ABC \sim \triangle ACD$ and $\triangle ABC \sim \triangle CBD$	Angle-Angle Similarity
$\frac{x}{b} = \frac{b}{c}$ and $\frac{y}{a} = \frac{a}{c}$	Similar triangles have proportional sides.
$b^2 = cx$ and $a^2 = cy$	Multiplication
$a^2 + b^2 = cx + cy$	Addition Property of Equality
$a^2 + b^2 = c^2$	Factoring and substitution

Answer:

The completed proof matches reasons to statements as follows:

1. $\angle ACB \cong \angle ADC \cong \angle CDB$ → Right angles are congruent.
2. $\triangle ABC \sim \triangle ACD$ and $\triangle ABC \sim \triangle CBD$ → Angle-Angle Similarity
3. $\frac{x}{b} = \frac{b}{c}$ and $\frac{y}{a} = \frac{a}{c}$ → Similar triangles have proportional sides.
4. $a^2 + b^2 = cx + cy$ → Addition Property of Equality
5. $a^2 + b^2 = c^2$ → Factoring and substitution (provided)