Question

1. which of the following additional statements would you need to prove these two triangles are similar by sas similarity?

∠i = ∠k
∠i = ∠l
∠h = ∠k
∠g = ∠j

Explanation:

Step1: Recall SAS similarity criterion

SAS (Side - Angle - Side) similarity states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the two triangles are similar.
For \(\triangle GHI\) and \(\triangle JKL\), we have \(\frac{HI}{KL}=\frac{12}{18}=\frac{2}{3}\) and \(\frac{GI}{JL}=\frac{18}{27}=\frac{2}{3}\).

Step2: Identify the included angles

The included angle for sides \(HI\) and \(GI\) in \(\triangle GHI\) is \(\angle I\). The included angle for sides \(KL\) and \(JL\) in \(\triangle JKL\) is \(\angle K\).
If \(\angle I=\angle K\), then by SAS similarity \(\triangle GHI\sim\triangle JKL\)

Answer:

\(\angle I = \angle K\) (i.e., the first option)