Question

below are statements that can be used to prove that the triangles are similar.

1. \\(\frac{ab}{zy} = \frac{bc}{yx} = 2\\)
2. \\(\angle b\\) and \\(\angle y\\) are right angles.

3.?
4.?
which two statements are missing in steps 3 and 4?
\\(\angle x \cong \angle c\\)
\\(\triangle abc \sim \triangle zyx\\) by the sas similarity theorem.
\\(\angle b \cong \angle y\\)

Explanation:

Step1: Recall 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 angles are congruent, then the two triangles are similar.

Step2: Analyze Given Information

We know that \(\frac{AB}{ZY}=\frac{BC}{YX} = 2\) (proportional sides) and \(\angle B\) and \(\angle Y\) are right angles (so \(\angle B\cong\angle Y\), the included angle).

Step3: Determine Step 3 and Step 4

For the SAS similarity theorem, after showing the proportional sides and the congruent included angle, we need to state the congruent included angle (\(\angle B\cong\angle Y\)) and then conclude the similarity of the triangles (\(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem). Wait, but looking at the options, the first part of the option: \(\angle X\cong\angle C\) is not correct for SAS. Wait, maybe I misread. Wait, the two statements in the option: first \(\angle X\cong\angle C\) (no, for SAS we need included angle), then \(\triangle ABC\sim\triangle ZYX\) by SAS. Wait, no, the correct included angle is \(\angle B\cong\angle Y\) (since \(\angle B\) is between \(AB\) and \(BC\), \(\angle Y\) is between \(ZY\) and \(YX\)). So step 3 should be \(\angle B\cong\angle Y\) (but in the options, the first option has \(\angle X\cong\angle C\) which is for AA, not SAS. Wait, maybe there is a mistake. Wait, the given ratios: \(\frac{AB}{ZY}=\frac{BC}{YX}\), and \(\angle B\) and \(\angle Y\) are right angles. So the included angle is \(\angle B\) and \(\angle Y\), so \(\angle B\cong\angle Y\). Then, by SAS similarity, \(\triangle ABC\sim\triangle ZYX\). So the two missing statements are \(\angle B\cong\angle Y\) (wait, no, \(\angle B\) and \(\angle Y\) are already stated as right angles, so maybe step 3 is \(\angle B\cong\angle Y\) (re - stating the congruence of the included angle) and step 4 is \(\triangle ABC\sim\triangle ZYX\) by SAS. But looking at the options, the option with \(\angle B\cong\angle Y\) (wait, the option is: \(\angle X\cong\angle C\) \(\triangle ABC\sim\triangle ZYX\) by SAS? No, that doesn't fit. Wait, maybe I made a mistake. Wait, the triangles: \(\triangle ABC\) with right angle at \(B\), \(\triangle ZYX\) with right angle at \(Y\). \(AB\) corresponds to \(ZY\), \(BC\) corresponds to \(YX\). So the included angle for \(AB\) and \(BC\) is \(\angle B\), for \(ZY\) and \(YX\) is \(\angle Y\). So \(\angle B\cong\angle Y\) (both right angles). Then, by SAS, \(\triangle ABC\sim\triangle ZYX\). So the two statements are: \(\angle B\cong\angle Y\) (but in the options, the first part of the option is \(\angle X\cong\angle C\) which is wrong for SAS. Wait, maybe the correct option is the one with \(\angle B\cong\angle Y\) (but in the options, the option is: \(\angle B\cong\angle Y\) (wait, the option is written as: \(\angle X\cong\angle C\) \(\triangle ABC\sim\triangle ZYX\) by SAS? No, that's not right. Wait, maybe the answer is the option with \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by SAS, but in the given options, maybe the first part is a typo, but according to the SAS theorem, the correct two statements are the congruence of the included angle (\(\angle B\cong\angle Y\)) and the conclusion of similarity by SAS. But looking at the options, the option that has \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem and \(\angle B\cong\angle Y\) (but in the options, the first option has \(\angle X\cong\angle C\) which is incorrect, the second option (the one with \(\angle B\cong\angle Y\) and the similarity conclusion) – wait, the user's option display is a bit messed up, but assuming the option is: \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem. Wait, no, the original problem's options:

Wait, the user's options:

- \(\angle X\cong\angle C\) \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem.

- \(\angle B\cong\angle Y\) (maybe the other option is cut off).

Wait, no, let's re - evaluate. The SAS similarity requires two sides in proportion and the included angle congruent. We have \(\frac{AB}{ZY}=\frac{BC}{YX}\) (two sides in proportion) and \(\angle B\) and \(\angle Y\) are right angles (so \(\angle B\cong\angle Y\), the included angle). So step 3 should be \(\angle B\cong\angle Y\) (stating the congruent included angle) and step 4 should be \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem. But in the options, if one of the options has \(\angle B\cong\angle Y\) and the similarity conclusion, that's the correct one. But in the given options (as per the user's input, maybe the correct option is the one with \(\angle B\cong\angle Y\) (wait, no, the user's input shows:

" \(\angle X \cong \angle C\)
\(\triangle ABC \sim \triangle ZYX\) by the SAS similarity theorem.
\(\angle B \cong \angle Y\) "

Wait, maybe the correct two statements are \(\angle B\cong\angle Y\) (but it's already stated as right angles, but maybe re - stating for the theorem) and \(\triangle ABC\sim\triangle ZYX\) by SAS. But the option with \(\angle X\cong\angle C\) is for AA, not SAS. So there must be a mistake. Wait, maybe the ratios are \(\frac{AB}{YX}=\frac{BC}{ZY}\)? No, the problem says \(\frac{AB}{ZY}=\frac{BC}{YX}\). So the sides around \(\angle B\) are \(AB\) and \(BC\), around \(\angle Y\) are \(ZY\) and \(YX\). So \(\angle B\) and \(\angle Y\) are the included angles. So \(\angle B\cong\angle Y\) (since they are right angles) and then \(\triangle ABC\sim\triangle ZYX\) by SAS. So the two missing statements are \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by SAS. But in the options, if the option is: \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by SAS, that's the correct one. But in the user's input, the first option is \(\angle X\cong\angle C\) \(\triangle ABC\sim\triangle ZYX\) by SAS, which is wrong for SAS (since \(\angle X\cong\angle C\) is not the included angle). The second option (if it's \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by SAS) is correct.

Answer:

The correct option (assuming the option is structured as: \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem) – but based on the given options in the user's input (maybe a formatting error), the correct two statements are \(\angle B\cong\angle Y\) (the included angle congruence) and \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem. So the answer is the option with \(\angle B\cong\angle Y\) (if that's part of the option) and \(\triangle ABC\sim\triangle ZYX\) by SAS. Wait, but in the user's input, the options are:

• \(\angle X \cong \angle C\) \(\triangle ABC \sim \triangle ZYX\) by the SAS similarity theorem.

• \(\angle B \cong \angle Y\) (maybe the second option is this pair).

So the correct answer is the option with \(\angle B\cong\angle Y\) and \(\triangle ABC\sim\triangle ZYX\) by the SAS similarity theorem.