Question

in triangle xyz, m∠z = (5m − 16)° and the exterior angle to ∠z measures (3m + 12)°. determine the value of m.
m = 3
m = 10.8
m = 18
m = 23

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In this case, the exterior angle to \(\angle Z\) is equal to \(\angle Z\) plus the other non - adjacent interior angle (but since we are only dealing with \(\angle Z\) and its exterior angle, we know that the measure of an exterior angle and its adjacent interior angle are supplementary. Also, more directly, for a triangle, the measure of an exterior angle is equal to the sum of the two non - adjacent interior angles. Here, if we consider the basic relationship between an interior angle and its exterior angle: \( (3m + 12)+(5m - 16)=180\) (because an interior angle and its exterior angle are supplementary, i.e., they add up to \(180^{\circ}\)).

$$3m + 12+5m-16 = 180$$

Step2: Combine like terms

Combine the \(m\) terms and the constant terms:

$$ (3m+5m)+(12 - 16)=180$$

$$8m-4 = 180$$

Step3: Isolate the term with \(m\)

Add \(4\) to both sides of the equation:

$$8m-4 + 4=180 + 4$$

$$8m=184$$

Step4: Solve for \(m\)

Divide both sides of the equation by \(8\):

$$m=\frac{184}{8}$$

$$m = 23$$

Answer:

\(m = 23\)