Question

in △xyz, m∠y = a + 65, m∠z = 7a + 35, and m∠y = m∠z. what is the shortest side of this triangle?
xy
xz
cannot be determined from information given
yz

Explanation:

Step1: Solve for \(a\)

Since \(m\angle Y=m\angle Z\), we have \(a + 65=7a+35\).
Subtract \(a\) from both sides: \(65 = 6a+35\).
Subtract \(35\) from both sides: \(6a=65 - 35=30\).
Divide both sides by \(6\): \(a = 5\).

Step2: Find the measures of \(\angle Y\) and \(\angle Z\)

Substitute \(a = 5\) into \(m\angle Y=a + 65\), we get \(m\angle Y=5+65 = 70^{\circ}\).
Since \(m\angle Y=m\angle Z\), then \(m\angle Z = 70^{\circ}\).

Step3: Find the measure of \(\angle X\)

Using the triangle - angle sum theorem (\(m\angle X+m\angle Y+m\angle Z = 180^{\circ}\)), we have \(m\angle X=180-(70 + 70)=40^{\circ}\).

Step4: Determine the shortest side

In a triangle, the side opposite the smallest angle is the shortest side.
Since \(\angle X\) is the smallest angle (\(m\angle X = 40^{\circ}\)), and the side opposite \(\angle X\) is \(YZ\).

Answer:

\(YZ\)