Question

solve each problem.

1. 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?
2. in triangle rst, m∠t is 5° more than m∠r, and m∠s is 10° less than m∠t. what is the measure of each angle?

Explanation:

Problem 21:

Step1: Define variables

Let \( m\angle D = x \). Then \( m\angle E = 3x \) and \( m\angle F = 3x - 9 \).

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

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

Step1: Define variables

Let \( m\angle R = x \). Then \( m\angle T = x + 5 \) and \( m\angle S = (x + 5) - 10 = x - 5 \).

Step2: Use triangle angle sum

The sum of angles in a triangle is \( 180^\circ \), so \( x + (x + 5) + (x - 5) = 180 \).

Step3: Solve the equation

Combine like terms: \( 3x = 180 \). Divide by 3: \( x = 60 \).

Step4: Find each angle

\( m\angle R = 60^\circ \), \( m\angle T = 60 + 5 = 65^\circ \), \( m\angle S = 60 - 5 = 55^\circ \).

Answer:

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

Problem 22: