Question

for the following problems, hexagon orange @ hexagon fruity.

1. ( mangle e=square )
2. ( ge=square ) in
3. ( angle ngecong angle square )
4. ( overline{or}cong square )

Explanation:

Step1: Sum of interior angles of a hexagon

The sum of interior angles of a hexagon is \((6 - 2)\times180^{\circ}=720^{\circ}\).

Step2: Calculate \(m\angle E\)

\(m\angle E=720-(160 + 85+145+105+135)\)
\(=720 - 630\)
\(=90^{\circ}\)

Step3: Corresponding sides of similar hexagons

Since hexagon \(ORANGE\sim\) hexagon \(FRUITY\), and by observing the side - angle correspondence (assuming proper labeling based on similarity notation), if we consider the side - length ratios. But for \(GE\), if we assume the correspondence of sides (by looking at the order of the letters in the similarity statement \(ORANGE\sim FRUITY\)), \(GE\) corresponds to \(IT\). So \(GE = 30\) in.

Step4: Corresponding angles of similar hexagons

For \(\angle NGE\), by the property of similar polygons (corresponding angles are equal), \(\angle NGE\cong\angle UTY\)

Step5: Corresponding sides of similar hexagons

For \(\overline{OR}\), by the property of similar polygons (corresponding sides are equal), \(\overline{OR}\cong\overline{FR}\)

Answer:

1. \(90\)
2. \(30\)
3. \(UTY\)
4. \(\overline{FR}\)