Question

1. m∠1 = ____ m∠2 = __ m∠3 = __ 7) in △mnp, if m∠m=(4x - 3)°, m∠n=(9x - 6)°, and m∠p=(6x - 1)°, find the value of x and the measure of each angle. x = __ m∠m = __ m∠n = __ m∠p = ____

Explanation:

Step1: Find \(m\angle1\) (linear - pair)

Since the angle adjacent to \(\angle1\) is \(118^{\circ}\), and they form a linear - pair, \(m\angle1 = 180^{\circ}-118^{\circ}=62^{\circ}\)

Step2: Find \(m\angle2\) (angle - sum property of a triangle)

Let's consider the triangle with angles \(m\angle1 = 62^{\circ}\), \(73^{\circ}\), and \(\angle2\). By the angle - sum property of a triangle (\(m\angle1 + 73^{\circ}+m\angle2=180^{\circ}\)), we substitute \(m\angle1 = 62^{\circ}\) into the equation: \(62^{\circ}+73^{\circ}+m\angle2 = 180^{\circ}\), so \(m\angle2=180^{\circ}-(62^{\circ} + 73^{\circ})=45^{\circ}\)

Step3: Find \(m\angle3\) (exterior - angle property)

The exterior angle of the triangle with non - adjacent interior angles \(m\angle2\) and \(49^{\circ}\) is related to \(\angle3\). The exterior angle is \(73^{\circ}\). By the exterior - angle property (\(73^{\circ}=m\angle2 + 49^{\circ}\)), we know \(m\angle3 = 73^{\circ}-49^{\circ}=24^{\circ}\)

Step4: Solve for \(x\) in \(\triangle MNP\)

In \(\triangle MNP\), by the angle - sum property of a triangle (\(m\angle M+m\angle N+m\angle P = 180^{\circ}\)), we substitute \(m\angle M=(4x - 3)^{\circ}\), \(m\angle N=(9x - 6)^{\circ}\), and \(m\angle P=(6x - 1)^{\circ}\) into the equation: \((4x - 3)+(9x - 6)+(6x - 1)=180\). Combine like terms: \(4x+9x + 6x-3-6 - 1=180\), \(19x-10 = 180\), \(19x=190\), \(x = 10\)

Step5: Find \(m\angle M\)

Substitute \(x = 10\) into \(m\angle M=(4x - 3)^{\circ}\), \(m\angle M=(4\times10 - 3)^{\circ}=37^{\circ}\)

Step6: Find \(m\angle N\)

Substitute \(x = 10\) into \(m\angle N=(9x - 6)^{\circ}\), \(m\angle N=(9\times10 - 6)^{\circ}=84^{\circ}\)

Step7: Find \(m\angle P\)

Substitute \(x = 10\) into \(m\angle P=(6x - 1)^{\circ}\), \(m\angle P=(6\times10 - 1)^{\circ}=59^{\circ}\)

Answer:

\(m\angle1 = 62^{\circ}\)
\(m\angle2 = 45^{\circ}\)
\(m\angle3 = 24^{\circ}\)
\(x = 10\)
\(m\angle M = 37^{\circ}\)
\(m\angle N = 84^{\circ}\)
\(m\angle P = 59^{\circ}\)