Question

1. if $m\angle pqr = 141^{circ}$, find each measure. which equation will correctly solve for x? a. $13x + 4=10x - 1$ b. $13x + 4+10x - 1 = 141$ c. $141+10x - 1 = 13x + 4$ d. $10x - 1+13x + 4 = 180$ 7. if $m\angle def=(7x + 4)^{circ}$, $m\angle deg=(5x + 1)^{circ}$, and $m\angle gef = 27^{circ}$, find each measure. 8.

Explanation:

7.

Step1: Set up the angle - addition equation

Since \(m\angle DEF=m\angle DEG + m\angle GEF\), we have the equation \((7x + 4)=(5x + 1)+27\).

Step2: Simplify the right - hand side of the equation

\(7x+4 = 5x+1 + 27\) simplifies to \(7x+4=5x + 28\).

Step3: Subtract \(5x\) from both sides

\(7x-5x+4=5x-5x + 28\), which gives \(2x+4 = 28\).

Step4: Subtract 4 from both sides

\(2x+4-4=28 - 4\), so \(2x=24\).

Step5: Divide both sides by 2

\(\frac{2x}{2}=\frac{24}{2}\), and \(x = 12\).

Step6: Find \(m\angle DEG\)

Substitute \(x = 12\) into the expression for \(m\angle DEG\): \(m\angle DEG=(5x + 1)=(5\times12+1)=61^{\circ}\).

Step7: Find \(m\angle DEF\)

Substitute \(x = 12\) into the expression for \(m\angle DEF\): \(m\angle DEF=(7x + 4)=(7\times12+4)=88^{\circ}\).

Step1: Use the property of vertical angles

Since \(x\) and \(43^{\circ}\) are vertical angles, \(x = 43^{\circ}\).

Step2: Use the property of supplementary angles

\(x\) and \(y\) are supplementary (a straight - line pair), so \(x + y=180^{\circ}\). Substitute \(x = 43^{\circ}\) into the equation: \(43^{\circ}+y=180^{\circ}\), then \(y=180^{\circ}-43^{\circ}=137^{\circ}\).

Step3: Use the property of vertical angles again

Since \(y\) and \(z\) are vertical angles, \(z = y\), so \(z = 137^{\circ}\).

Answer:

\(x = 12\)
\(m\angle DEG=61^{\circ}\)
\(m\angle DEF=88^{\circ}\)

8.