Question

given: 3ab = 7ad and 3ac = 7ae
prove: δabc ~ δade

complete the steps of the proof.

(image shows a diagram with triangles and a two - column proof table. the table has statements and reasons columns. statement 1: 3ab = 7ad, reason 1: given. statement 2: \\(\frac{ab}{ad}=\frac{7}{3}\\), reason 2: division property. statement 3: 3ac = 7ae, reason 3: given. statement 4: \\(\frac{ac}{ae}=\frac{7}{3}\\), reason 4: (dropdown with options: division property, reflexive property, substitution property). statement 5: \\(\frac{ab}{ad}=\frac{ac}{ae}\\), reason 5: substitution property. statement 6: ∠bac and ∠dae are vertical angles, reason 6: def. of vertical angles. statement 7: ∠bac ≅ ∠dae, reason 7: vertical angles theorem. statement 8: δabc ~ δade, reason 8: (dropdown). the diagram has points b, a, d on one line and c, a, e on another line, forming triangles abc and ade with a common vertex at a.)

Explanation:

Step1: Analyze Step 4 Reason

We have \(3AC = 7AE\). To get \(\frac{AC}{AE}=\frac{7}{3}\), we divide both sides of the equation \(3AC = 7AE\) by \(3AE\) (or use the division property of equality, similar to how we got step 2 from step 1). So the reason for step 4 is "division property".

Step2: Analyze Step 8 Reason

We have two sides in proportion (\(\frac{AB}{AD}=\frac{AC}{AE}\)) and the included angle \(\angle BAC\cong\angle DAE\) (vertical angles). By the Side - Angle - Side (SAS) Similarity Criterion, if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. So the reason for step 8 is "SAS similarity criterion". But looking at the dropdown for the first blank (step 4), the options are division property, reflexive property, substitution property. For step 4, since we derived \(\frac{AC}{AE}=\frac{7}{3}\) from \(3AC = 7AE\) by dividing both sides (similar to step 2 which used division property), the first blank (step 4 reason) is "division property". For the second blank (step 8 reason), but in the given dropdown for the second blank (the one for step 8), wait, no, the first dropdown is for step 4. Wait, the problem has two blanks: one for step 4 (the \(\boldsymbol{\star}\) in step 4's reason) and one for step 8 (the \(\boldsymbol{\star}\) in step 8's reason). But the first dropdown is for step 4. Let's focus on step 4 first.

From \(3AC = 7AE\), to get \(\frac{AC}{AE}=\frac{7}{3}\), we divide both sides of the equation \(3AC = 7AE\) by \(3AE\) (or divide both sides by \(3\) and then by \(AE\), which is the division property of equality, just like in step 2 where from \(3AB = 7AD\) we got \(\frac{AB}{AD}=\frac{7}{3}\) by division property. So step 4's reason is "division property".

For step 8, we have \(\frac{AB}{AD}=\frac{AC}{AE}\) (from step 5) and \(\angle BAC\cong\angle DAE\) (from step 7). By the SAS Similarity Theorem, \(\triangle ABC\sim\triangle ADE\). But in the given options for the second blank (if there was a dropdown, but in the problem, the first dropdown is for step 4). Wait, the problem's first dropdown is for step 4, with options division property, reflexive property, substitution property. So step 4's reason is division property.

Answer:

For the first blank (step 4 reason): division property

For the second blank (step 8 reason, if we consider the similarity criterion): SAS similarity criterion (but based on the given dropdown for the first blank, the answer for the first blank is division property)

(If we are to answer the first blank only, as per the dropdown in the problem, the answer is "division property")