Question

end - of - semester test: virginia geometry, semest...

1. △abc - △acd

△abc - △cbd

1. $\frac{ac}{ab}=\frac{ad}{ac}$ and $\frac{bc}{ab}=\frac{db}{bc}$
2. $ac^{2}=(ab)(ad)$

$bc^{2}=(ab)(db)$

1. $ac^{2}+bc^{2}=(ab)(ad)+(ab)(db)$
2. $ac^{2}+bc^{2}=ab(ad + db)$
3. $ab = ad+db$
4. $ac^{2}+bc^{2}=(ab)(ab)$
5. $ac^{2}+bc^{2}=ab^{2}$

aa similarity criteria
cross - multiplication
addition
distributive property
segment addition
substitution
multiplication
which reason completes the proof?
a. corresponding sides of similar triangles are proportional.
b. corresponding parts of similar triangles are congruent.
c. corresponding sides of congruent triangles are congruent.
d. corresponding parts of congruent triangles are proportional.
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Explanation:

Step1: Recall similarity - triangle property

When two triangles are similar, the ratios of their corresponding sides are equal. In the proof, we have used the fact that for similar triangles $\triangle ABC$, $\triangle ACD$, and $\triangle CBD$, we get proportion - al side - length relationships like $\frac{AC}{AB}=\frac{AD}{AC}$ and $\frac{BC}{AB}=\frac{DB}{BC}$. This is based on the property that corresponding sides of similar triangles are proportional.

Step2: Analyze the proof steps

In step 5, we cross - multiply the proportionate side - length equations to get $AC^{2}=(AB)(AD)$ and $BC^{2}=(AB)(DB)$. Then, in step 7, we add these two equations: $AC^{2}+BC^{2}=(AB)(AD)+(AB)(DB)$. The reason for the equalities in step 5 (from the proportion to the product form) is cross - multiplication, and the overall concept behind getting the proportionate side - length relationships in the first place is that corresponding sides of similar triangles are proportional.

Answer:

A. Corresponding sides of similar triangles are proportional.