Question

∠6 and ∠7 form a linear pair. twice the measure of ∠6 is twelve more than four times the measure of ∠7. find the measure of each angle.

m∠6 =
□°

m∠7 =
□°

Explanation:

Step1: Define variables and linear pair property

Let \( m\angle 6 = x \) and \( m\angle 7 = y \). Since \( \angle 6 \) and \( \angle 7 \) form a linear pair, they are supplementary, so \( x + y = 180 \). Also, from the problem, \( 2x = 4y + 12 \).

Step2: Solve the system of equations

From \( x + y = 180 \), we get \( x = 180 - y \). Substitute this into \( 2x = 4y + 12 \):
\( 2(180 - y) = 4y + 12 \)
\( 360 - 2y = 4y + 12 \)
Add \( 2y \) to both sides: \( 360 = 6y + 12 \)
Subtract 12: \( 348 = 6y \)
Divide by 6: \( y = 58 \)

Step3: Find \( x \)

Substitute \( y = 58 \) into \( x = 180 - y \): \( x = 180 - 58 = 122 \)

Answer:

\( m\angle 6 = 122^\circ \)
\( m\angle 7 = 58^\circ \)