Question

which of the following statements is true if ( mangle e = mangle y ) and ( mangle f = mangle x )?
options:

• ( mangle e > mangle z )
• ( \triangle efg sim \triangle yxz )
• the measure of each segment in ( \triangle yxz ) is three times the size of those in ( \triangle efg ).
• there is a sequence of rigid motions that map ( \triangle efg ) onto ( \triangle yxz ).

Explanation:

Step1: Recall the properties of similar triangles

If two triangles have two pairs of equal angles (\(m\angle E = m\angle Y\) and \(m\angle F=m\angle X\)), then by the AA (Angle - Angle) similarity criterion, \(\triangle EFG\sim\triangle YXZ\).

Step2: Analyze other options

• For the statement \(m\angle E>m\angle Z\), we have no information about the relationship between \(\angle E\) and \(\angle Z\) based on \(m\angle E = m\angle Y\) and \(m\angle F = m\angle X\).
• If \(\triangle YXZ\) had segments three times the size of \(\triangle EFG\) segments, we would need information about side - length ratios. But from the angle - equality (\(m\angle E = m\angle Y\) and \(m\angle F = m\angle X\)), we cannot directly conclude this.
• Rigid motions (translations, rotations, reflections) preserve side - lengths and angles. Since the triangles are similar (not necessarily congruent, as we don't know about side - lengths) based on the given angle - equalities, there is no sequence of rigid motions that map \(\triangle EFG\) onto \(\triangle YXZ\) (unless they are congruent, which we can't confirm from the given angle - only information).

Answer:

\(\triangle EFG\sim\triangle YXZ\)