Question

make connections 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 = , m∠e = , m∠f =

Explanation:

Step1: Define variables for angles

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 property

The sum of angles in a triangle is \( 180^\circ \), so \( x + 3x + (3x - 9) = 180 \).
Simplify the left side: \( 7x - 9 = 180 \).

Step3: Solve for x

Add 9 to both sides: \( 7x = 180 + 9 = 189 \).
Divide both sides by 7: \( x = \frac{189}{7} = 27 \).

Step4: Find each angle

• \( m\angle D = x = 27^\circ \)
• \( m\angle E = 3x = 3\times27 = 81^\circ \)
• \( m\angle F = 3x - 9 = 81 - 9 = 72^\circ \)

Answer:

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