Question

1. look at the figure below.

name one additional piece of information that is sufficient to prove that the triangles are similar.

Explanation:

Step1: Recall the similarity criteria for triangles

The similarity criteria for triangles are AA (Angle - Angle), SAS (Side - Angle - Side), and SSS (Side - Side - Side).

Step2: Analyze the given information

We are given one pair of angles (\(\angle y\) and \(\angle s\)) and one pair of sides (\(xy = 3\mathrm{cm}\) and \(rs=9\mathrm{cm}\)).

Step3: Determine the additional information

If we have \(\frac{yz}{st}=\frac{3}{9}=\frac{1}{3}\) (using the SAS similarity criterion, where the ratio of sides adjacent to the given equal angles is equal), or \(\angle z=\angle t\) (using the AA similarity criterion as we already have one pair of equal angles \(\angle y = \angle s\)), or \(\frac{xz}{rt}=\frac{3}{9}=\frac{1}{3}\) (using SSS similarity if we can find the ratio of all three pairs of sides).

Answer:

One additional piece of information could be \(\frac{yz}{st}=\frac{1}{3}\) (or \(\angle z=\angle t\) or \(\frac{xz}{rt}=\frac{1}{3}\))