Question

in the diagram, m∠f = 60°. to prove that the triangles are similar by the sas similarity theorem, it needs to be proven that ∠j measures 30°. ∠i measures 30°. ∠j measures 60°. ∠i measures 60°.

Explanation:

Step1: Recall SAS Similarity Theorem

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

Step2: Calculate Side Ratios

For triangle \(EFG\) and triangle \(HIJ\):

• In \(\triangle EFG\), \(FG = 20\), \(FE=40\)
• In \(\triangle HIJ\), \(IJ = 10\), \(IH = 20\)

Calculate the ratios of the corresponding sides:
\(\frac{FG}{IJ}=\frac{20}{10} = 2\) and \(\frac{FE}{IH}=\frac{40}{20}=2\)

So, the sides \(FG\) and \(FE\) of \(\triangle EFG\) are proportional to sides \(IJ\) and \(IH\) of \(\triangle HIJ\) with a ratio of \(2:1\).

Step3: Determine the Included Angle

The included angle for the sides \(FG\) and \(FE\) in \(\triangle EFG\) is \(\angle F\) (with \(m\angle F = 60^{\circ}\)). The included angle for the sides \(IJ\) and \(IH\) in \(\triangle HIJ\) should be \(\angle I\) (since \(IJ\) and \(IH\) meet at \(I\)). For the SAS similarity theorem to hold, \(\angle I\) should be congruent to \(\angle F\), so \(m\angle I=60^{\circ}\).

Answer:

\(\angle I\) measures \(60^{\circ}\)