Question

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

complete the steps of the proof.

statements\treasons

1. 3ab = 7ad\t1. given
2. \\(\frac{ab}{ad} = \frac{7}{3}\\)\t2. division property
3. 3ac = 7ae\t3. given
4. \\(\frac{ac}{ae} = \frac{7}{3}\\)\t4. ♣
5. \\(\frac{ab}{ad} = \frac{ac}{ae}\\)\t5. substitution property
6. ∠bac and ∠dae are vertical angles\t6. def. of vertical angles
7. ∠bac ≅ ∠dae\t7. vertical angles theorem
8. δabc ~ δade\t8. ♦

options for ♦: sas similarity theorem, ssa similarity theorem, sss similarity theorem

Explanation:

Step 4 Reason:

To get \(\frac{AC}{AE}=\frac{7}{3}\) from \(3AC = 7AE\), we use the division property (dividing both sides of the equation \(3AC=7AE\) by \(3AE\) gives \(\frac{AC}{AE}=\frac{7}{3}\)). So the reason for step 4 is "division property".

Step 8 Reason:

We have \(\frac{AB}{AD}=\frac{AC}{AE}\) (from step 5) and \(\angle BAC\cong\angle DAE\) (from step 7). The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angle is congruent, then the two triangles are similar. Here, the sides \(AB, AC\) of \(\triangle ABC\) are proportional to sides \(AD, AE\) of \(\triangle ADE\) and the included angle \(\angle BAC\) (between \(AB\) and \(AC\)) is congruent to \(\angle DAE\) (between \(AD\) and \(AE\)). So the reason for step 8 is "SAS similarity theorem".

For step 4:

Step 1: Identify the operation

To derive \(\frac{AC}{AE}=\frac{7}{3}\) from \(3AC = 7AE\), we divide both sides by \(3AE\).

Step 2: State the property

This operation is justified by the division property of equality.

Step 1: Recall triangle similarity criteria

We know two sides of \(\triangle ABC\) (\(AB, AC\)) are proportional to two sides of \(\triangle ADE\) (\(AD, AE\)) and the included angle (\(\angle BAC\) and \(\angle DAE\)) is congruent.

Step 2: Apply the SAS similarity theorem

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angle is congruent, the triangles are similar.

Answer:

division property

For step 8: