Question

m∠eqp = m∠hgs + 30 ratio of similarity
m∠eqp < m∠hgs + 30 triangle inequality
\frac{qe}{gh}=\frac{qp}{gs} definition of parallel lines
\frac{qe}{gh}=\frac{qp}{gs} ratio of similarity

Explanation:

• For the first option: If two triangles are similar, corresponding angles are equal. So \(m\angle EQP=m\angle HGS\) (not \(m\angle EQP = m\angle HGS+30\)), so the first option is wrong.
• For the second option: The triangle - inequality theorem is about the sum of the lengths of two sides of a triangle being greater than the length of the third side. It has nothing to do with the relationship \(m\angle EQP\lt m\angle HGS + 30\), so the second option is wrong.
• For the third option: The definition of parallel lines is about the relationship of slopes (in coordinate - geometry) or the property of non - intersection. The proportion \(\frac{QE}{GH}=\frac{QP}{GS}\) is related to similar triangles (ratio of similarity), not the definition of parallel lines.
• For the fourth option: If \(\triangle EQP\sim\triangle HGS\) (by AA similarity, for example, if \(EQ\parallel HG\)), then by the ratio of similarity (the ratio of corresponding sides of similar triangles), we have \(\frac{QE}{GH}=\frac{QP}{GS}\).

Answer:

\(\frac{QE}{GH}=\frac{QP}{GS}\) (Ratio of Similarity)