Question

reasoning solve the problem. in triangle def, m∠e is three times m∠d, and m∠f is 9° less than m∠e. what is the measure of each angle? m∠d = \boxed{\space}°, m∠e = \boxed{\space}°, m∠f = \boxed{\space}°

Explanation:

Step1: Define Variables

Let \( m\angle D = x \). Then \( m\angle E = 3x \) (since \( m\angle E \) is three times \( m\angle D \)), and \( m\angle F = 3x - 9 \) (since \( m\angle F \) is \( 9^\circ \) less than \( m\angle E \)).

Step2: Use Triangle Angle Sum

The sum of angles in a triangle is \( 180^\circ \), so:

$$ x + 3x + (3x - 9) = 180 $$

Step3: Simplify and Solve

Combine like terms:

$$ 7x - 9 = 180 $$

Add 9 to both sides:

$$ 7x = 189 $$

Divide by 7:

$$ x = 27 $$

Step4: Find Each Angle

• \( m\angle D = x = 27^\circ \)
• \( m\angle E = 3x = 3(27) = 81^\circ \)
• \( m\angle F = 3x - 9 = 81 - 9 = 72^\circ \)

Answer:

\( m\angle D = 27^\circ \), \( m\angle E = 81^\circ \), \( m\angle F = 72^\circ \)