Questions: Given: triangle EFG; with right angle angle F Prove: angle E and angle G are complementary. Statements: Reasons: 1. triangle EFG with right angle angle F 1. 2. m angle F=90 2. 3. m angle E+m angle F+m angle G=180 3. 4. m angle E+90+m angle G=180 4. 5. m angle E+m angle G=90 5. 6. angle E and angle G are complementary. 6.

Solution Steps

The problem states that we are given a right triangle \( \Delta EFG \) with a right angle at \( \angle F \). We need to prove that \( \angle E \) and \( \angle G \) are complementary.

Since \( \angle F \) is a right angle, its measure is \( 90^\circ \).

The sum of the interior angles in any triangle is \( 180^\circ \). Therefore, \( m\angle E + m\angle F + m\angle G = 180^\circ \).

Substitute \( m\angle F = 90^\circ \) into the equation from Step 3: \[ m\angle E + 90^\circ + m\angle G = 180^\circ \]

Subtract \( 90^\circ \) from both sides of the equation: \[ m\angle E + m\angle G = 90^\circ \]

By definition, if the sum of two angles is \( 90^\circ \), then the angles are complementary. Therefore, \( \angle E \) and \( \angle G \) are complementary.

Final Answer