Question

which would prove that \\( \triangle abc \sim \triangle xyz \\)? select two options. \\( \square \frac{ba}{yx}=\frac{bc}{yz}=\frac{ac}{xz} \\) \\( \square \frac{ba}{yx}=\frac{bc}{yz}, \angle c \cong \angle z \\) \\( \square \frac{ac}{xz}=\frac{ba}{yx}, \angle a \cong \angle x \\) \\( \square \frac{ba}{yx}=\frac{ac}{yz}=\frac{bc}{xz} \\) \\( \square \frac{bc}{xy}=\frac{ba}{zx}, \angle c \cong \angle x \\)

Explanation:

Step1: Recall the similarity criteria for triangles

For two triangles \(\triangle ABC\) and \(\triangle XYZ\), the Side - Side - Side (SSS) similarity criterion states that if \(\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}\), then \(\triangle ABC\sim\triangle XYZ\). The Side - Angle - Side (SAS) similarity criterion states that if \(\frac{AB}{XY}=\frac{AC}{XZ}\) and \(\angle A\cong\angle X\) (or \(\frac{AB}{XY}=\frac{BC}{YZ}\) and \(\angle B\cong\angle Y\) or \(\frac{BC}{YZ}=\frac{AC}{XZ}\) and \(\angle C\cong\angle Z\)) then \(\triangle ABC\sim\triangle XYZ\)

Step2: Analyze each option

• Option 1: \(\frac{BA}{YX}=\frac{BC}{YZ}=\frac{AC}{XZ}\)

This satisfies the SSS similarity criterion.

• Option 2: \(\frac{BA}{YX}=\frac{BC}{YZ},\angle C\cong\angle Z\)

The sides \(\frac{BA}{YX}=\frac{BC}{YZ}\) and the included angle for \(\triangle ABC\) is \(\angle B\) and for \(\triangle XYZ\) is \(\angle Y\). Since \(\angle C\) and \(\angle Z\) are not the included angles for the given side - ratios, this does not satisfy the SAS similarity criterion.

• Option 3: \(\frac{AC}{XZ}=\frac{BA}{YX},\angle A\cong\angle X\)

Here, \(\frac{AC}{XZ}=\frac{BA}{YX}\) (which can be written as \(\frac{BA}{YX}=\frac{AC}{XZ}\)) and \(\angle A\) and \(\angle X\) are the included angles for the sides \(BA, AC\) in \(\triangle ABC\) and \(YX,XZ\) in \(\triangle XYZ\) respectively. This satisfies the SAS similarity criterion.

• Option 4: \(\frac{BA}{YX}=\frac{AC}{YZ}=\frac{BC}{XZ}\)

The ratios of the sides are not in the correct order for SSS similarity. For SSS similarity, the ratios should be of corresponding sides.

• Option 5: \(\frac{BC}{XY}=\frac{BA}{ZX},\angle C\cong\angle X\)

The side - ratios are not of corresponding sides and the angle is not the included angle for the given side - ratios

Answer:

\(\frac{BA}{YX}=\frac{BC}{YZ}=\frac{AC}{XZ}\) and \(\frac{AC}{XZ}=\frac{BA}{YX},\angle A\cong\angle X\)