Question

according to the lengths given in the diagram, which of the following statements is true?
△hig and △kij are not similar.
△hig ~ △kij by aa
△hig ~ △kij by sas
△hig ~ △kij by sss

Explanation:

Step1: Identify Vertical Angles

$\angle HIG$ and $\angle KIJ$ are vertical angles, so $\angle HIG = \angle KIJ$.

Step2: Calculate Side Ratios

For sides around the vertical angles:

• $\frac{HI}{KI} = \frac{12}{4} = 3$
• $\frac{GI}{JI} = \frac{18}{6} = 3$

So, $\frac{HI}{KI} = \frac{GI}{JI}$ and included angles are equal.

Step3: Apply SAS Similarity

By SAS (Side - Angle - Side) similarity criterion, if two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar. So $\triangle HIG \sim \triangle KIJ$ by SAS.

Answer:

$\triangle HIG \sim \triangle KIJ$ by SAS