Question

1. in $\triangle ghi$, $gh = 12$, $hi = 16$, and $\angle g = 60^\circ$. in $\triangle jkl$, $jk = 6$, $kl = 8$, and $\angle j = 60^\circ$. are these triangles similar, and by which criterion?

a. yes, by the sas criterion
b. yes, by the aa criterion
c. no, the sides are not proportional
d. no, the angles are not equal

Explanation:

Step1: Check Side Ratios

Calculate the ratios of corresponding sides: $\frac{GH}{JK} = \frac{12}{6} = 2$, $\frac{HI}{KL} = \frac{16}{8} = 2$. So, the sides are proportional.

Step2: Check Included Angle

$\angle G = \angle J = 60^\circ$, which is the included angle between the proportional sides. By SAS (Side - Angle - Side) similarity criterion, if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.

Answer:

a. Yes, by the SAS Criterion