Question

which would prove that δabc ~ δxyz? select two options.
□ \\(\frac{ba}{yx} = \frac{bc}{yz} = \frac{ac}{xz}\\)
□ \\(\frac{ba}{yx} = \frac{bc}{yz}, \angle c \cong \angle z\\)
□ \\(\frac{ac}{xz} = \frac{ba}{yx}, \angle a \cong \angle x\\)
□ \\(\frac{ba}{yx} = \frac{ac}{yz} = \frac{bc}{xz}\\)
□ \\(\frac{bc}{xy} = \frac{ba}{zx}, \angle c \cong \angle x\\)

Explanation:

To determine which options prove \(\triangle ABC \sim \triangle XYZ\), we use the criteria for triangle similarity: SSS (Side - Side - Side) Similarity (if the ratios of all three corresponding sides are equal) and SAS (Side - Angle - Side) Similarity (if the ratio of two corresponding sides is equal and the included angle is congruent).

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

For SSS similarity, the ratios of all three pairs of corresponding sides must be equal. Let's check the correspondence of the sides:

• \(BA\) corresponds to \(YX\)
• \(BC\) corresponds to \(YZ\)
• \(AC\) corresponds to \(XZ\)

If \(\frac{BA}{YX}=\frac{BC}{YZ}=\frac{AC}{XZ}\), by the SSS (Side - Side - Side) similarity criterion, \(\triangle ABC \sim \triangle XYZ\). So this option is valid.

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

For SAS similarity, the angle must be the included angle between the two sides. Let's see the sides and the angle:

• The sides are \(BA\) (corresponding to \(YX\)) and \(BC\) (corresponding to \(YZ\)). The angle \(\angle C\) is not the included angle between \(BA\) and \(BC\) (the included angle between \(BA\) and \(BC\) is \(\angle B\)), and \(\angle Z\) is not the included angle between \(YX\) and \(YZ\) (the included angle between \(YX\) and \(YZ\) is \(\angle Y\)). So this does not satisfy SAS similarity. Thus, this option is invalid.

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

Let's check the correspondence:

• The sides are \(AC\) (corresponding to \(XZ\)) and \(BA\) (corresponding to \(YX\)). The angle \(\angle A\) is the included angle between \(BA\) and \(AC\) in \(\triangle ABC\), and \(\angle X\) is the included angle between \(YX\) and \(XZ\) in \(\triangle XYZ\).
• If \(\frac{AC}{XZ}=\frac{BA}{YX}\) (ratio of two corresponding sides) and \(\angle A\cong\angle X\) (the included angle), by the SAS (Side - Angle - Side) similarity criterion, \(\triangle ABC \sim \triangle XYZ\). So this option is valid.

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

Let's check the correspondence of the sides:

• \(BA\) corresponds to \(YX\)
• \(AC\) corresponds to \(YZ\) (this is an incorrect correspondence as \(AC\) should correspond to \(XZ\) and \(YZ\) should correspond to \(BC\) for the similarity of \(\triangle ABC\) and \(\triangle XYZ\))
• \(BC\) corresponds to \(XZ\) (incorrect correspondence)

Since the side correspondences are incorrect (the ratios are not of corresponding sides), this does not satisfy the SSS similarity criterion. So this option is invalid.

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

The side correspondences are incorrect (\(BC\) should correspond to \(YZ\), \(BA\) should correspond to \(YX\), \(XY\) is \(YX\) reversed, \(ZX\) is \(XZ\) reversed) and the angle correspondence is also incorrect (\(\angle C\) should correspond to \(\angle Z\) and \(\angle X\) should correspond to \(\angle A\)). So this does not satisfy any similarity criterion. Thus, this option is invalid.

Answer:

The two options that prove \(\triangle ABC \sim \triangle XYZ\) are:

1. \(\boldsymbol{\frac{BA}{YX}=\frac{BC}{YZ}=\frac{AC}{XZ}}\)
2. \(\boldsymbol{\frac{AC}{XZ}=\frac{BA}{YX},\angle A\cong\angle X}\)