Question

complete the proof showing that the triangles \\( \triangle ghs \\) and \\( \triangle qep \\) are similar. note that the diagram is not to scale.

step
statement
reason
1.
\\( m \angle e q p = 180 - \\)
\\( ( m \angle p e q + m \angle q p e ) \\)
\\( m \angle e q p = 58 \\)
angle sum theorem
2.
\\( \angle h g s \cong \angle e q p \\)
definition of congruent angles
3.
4.
\\( \triangle g h s \sim \triangle q e p \\)
side-angle-side (sas) similarity
theorem

Explanation:

Step1: Calculate ratio of sides

For \(\triangle GHS\) and \(\triangle QEP\), \(\frac{GH}{QE}=\frac{3}{6}=\frac{1}{2}\), \(\frac{GS}{QP}=\frac{2}{4}=\frac{1}{2}\)

Step2: Check angle equality

We know from step 2 that \(\angle HGS\cong\angle EQP\) (both equal to \(58^{\circ}\))

Answer:

Since \(\frac{GH}{QE}=\frac{GS}{QP}=\frac{1}{2}\) and \(\angle HGS=\angle EQP = 58^{\circ}\), by the Side - Angle - Side (SAS) Similarity Theorem, \(\triangle GHS\sim\triangle QEP\)