Question

5 in the figure, in △abc and △ade, m∠bac = m∠d. in order to make △abc and △ade similar, it is necessary to satisfy ______. a. $\frac{ac}{ad}=\frac{ab}{ae}$ b. $\frac{ac}{ad}=\frac{bc}{de}$ c. $\frac{ac}{ad}=\frac{ab}{de}$ d. $\frac{ac}{ad}=\frac{bc}{ae}$ 6 as shown in the figure, point d and point e are on $overline{ab}$ and $overline{ac}$ respectively. which of the following conditions can be used to deduce that △ade ~ △acb? ①m∠aed = m∠b; ②$\frac{ad}{ac}=\frac{ae}{ab}$; ③$\frac{de}{cb}=\frac{ad}{ac}$. a. ①② b. ①③ c. ②③ d. ③

Explanation:

Step1: Recall similarity - criterion

For two triangles \(\triangle ABC\) and \(\triangle ADE\), if \(\angle BAC=\angle D\), by the Side - Angle - Side (SAS) similarity criterion, we need the ratio of the sides including the equal angles to be equal. That is, \(\frac{AC}{AD}=\frac{AB}{DE}\).

Step2: Analyze the second problem

For \(\triangle ADE\) and \(\triangle ACB\),

• By the Angle - Angle (AA) similarity criterion, if \(\angle AED=\angle B\) and \(\angle A\) is common, then \(\triangle ADE\sim\triangle ACB\).
• By the Side - Angle - Side (SAS) similarity criterion, if \(\frac{AD}{AC}=\frac{AE}{AB}\), then \(\triangle ADE\sim\triangle ACB\).

Answer:

1. C
2. A