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{ }, m∠e = \boxed{ }, m∠f = \boxed{ }°

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: Solve the equation

Combine like terms: \( 7x - 9 = 180 \). Add 9 to both sides: \( 7x = 189 \). Divide by 7: \( x = 27 \).

Step4: Find each angle

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

Answer:

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