Question

in △xyz provided, w is a point on yz such that xw is an angle - bisector of ∠zxy. what is m∠z? a 80° b 90°

Explanation:

Step1: Use angle - bisector property

Since \(XW\) is the angle - bisector of \(\angle ZXY\), \(\angle ZXW=\angle WXY = a^{\circ}\).

Step2: Apply exterior - angle property

In \(\triangle XYW\), the exterior angle \(\angle XWY = 65^{\circ}\). By the exterior - angle property of a triangle (\(\text{Exterior angle}=\text{Sum of non - adjacent interior angles}\)), we have \(65^{\circ}=a^{\circ}+(a + 20)^{\circ}\).
\[65=a+(a + 20)\]
\[65=2a+20\]
\[2a=65 - 20\]
\[2a = 45\]
\[a = 22.5\]

Step3: Use angle - sum property of a triangle

In \(\triangle XYZ\), the sum of interior angles of a triangle is \(180^{\circ}\). Let \(\angle Z=x\), \(\angle X = 2a^{\circ}\), and \(\angle Y=(a + 20)^{\circ}\). Substituting \(a = 22.5\), \(\angle X=45^{\circ}\) and \(\angle Y=42.5^{\circ}\).
\[x+45+(42.5)=180\]
\[x=180-(45 + 42.5)\]
\[x = 92.5^{\circ}\]

It seems there is a mistake in the problem - setup or options provided as the correct value of \(\angle Z\) is \(92.5^{\circ}\) and it's not in the given options. But if we assume some approximation or error in our understanding and go by the closest value logic in a multiple - choice scenario, we note that the closest value to our calculated result among the given options:

Answer:

B. \(90^{\circ}\)