Question

1. in △wxy, if m∠w is five less than twice m∠y and m∠x is 21 more than m∠y, find the measure of each angle.

m∠w = _____
m∠x = _____
m∠y = _____

1. in △def, if m∠e is one more than three times m∠f and m∠d is three more than seven times m∠f, find the measure of each angle.

m∠d = _____
m∠e = _____
m∠f = _____

Explanation:

Step1: Define Variables

Let \( m\angle F = x \). Then \( m\angle E = 3x + 1 \) (since \( m\angle E \) is one more than three times \( m\angle F \)) and \( m\angle D = 7x + 3 \) (since \( m\angle D \) is three more than seven times \( m\angle F \)).

Step2: Use Triangle Angle Sum

In \( \triangle DEF \), the sum of angles is \( 180^\circ \). So:

$$ m\angle D + m\angle E + m\angle F = 180 $$

Substitute the expressions:

$$ (7x + 3) + (3x + 1) + x = 180 $$

Step3: Simplify and Solve for \( x \)

Combine like terms:

$$ 7x + 3 + 3x + 1 + x = 180 \\ 11x + 4 = 180 $$

Subtract 4 from both sides:

$$ 11x = 176 $$

Divide by 11:

$$ x = 16 $$

Step4: Find Each Angle

• \( m\angle F = x = 16^\circ \)
• \( m\angle E = 3x + 1 = 3(16) + 1 = 49^\circ \)
• \( m\angle D = 7x + 3 = 7(16) + 3 = 115^\circ \)

Answer:

\( m\angle D = 115^\circ \), \( m\angle E = 49^\circ \), \( m\angle F = 16^\circ \)