Question

1. if hexagon abcdef is similar to hexagon klmnop, which of the following must be true?

$overline{de} \cong \overline{no}$
$overline{ef} \cong \overline{op}$
$\angle e \cong \angle o$
$\angle a \cong \angle p$

Explanation:

When two polygons are similar, their corresponding angles are congruent, and their corresponding sides are proportional (not necessarily congruent). For similar hexagons \(ABCDEF\) and \(KLMNOP\), we need to identify the corresponding angles. The order of the vertices matters in similar polygons, so angle \(E\) in hexagon \(ABCDEF\) corresponds to angle \(O\) in hexagon \(KLMNOP\) (since the order is \(A - B - C - D - E - F\) and \(K - L - M - N - O - P\), so the 5th angle of the first hexagon (\(E\)) corresponds to the 5th angle of the second hexagon (\(O\))).

• For the side options (\(\overline{DE} \cong \overline{NO}\) and \(\overline{EF} \cong \overline{OP}\)): Similarity implies sides are proportional, not necessarily congruent, so these are not necessarily true.
• For \(\angle A \cong \angle P\): Angle \(A\) (first angle) should correspond to angle \(K\) (first angle of the second hexagon), not angle \(P\) (last angle), so this is false.
• For \(\angle E \cong \angle O\): As per the correspondence of vertices in similar polygons, angle \(E\) and angle \(O\) are corresponding angles, so they must be congruent.

Answer:

\(\boldsymbol{\angle E \cong \angle O}\) (the option with \(\angle E \cong \angle O\))