Question

in the triangles, xy = mp and yz = pn. if m∠p = 85°, which is a possible measure for ∠y? m∠y = 80° m∠y = 100° m∠y = 85° m∠y = 180°

Explanation:

Step1: Apply the Hinge Theorem

In triangles \(XYZ\) and \(MPN\), \(XY = MP\), \(YZ=PN\), and \(XZ = 23\) in, \(MN = 19\) in. So \(XZ>MN\). By the Hinge Theorem, \(\angle Y>\angle P\).

Step2: Analyze the angle measure constraints

We know that the sum of angles in a triangle is \(180^{\circ}\), so \(\angle Y<180^{\circ}\). Given \(m\angle P = 85^{\circ}\), from \(\angle Y>\angle P\), we check the options:

• For \(m\angle Y = 80^{\circ}\), \(80^{\circ}<85^{\circ}\), not valid.
• For \(m\angle Y = 100^{\circ}\), \(100^{\circ}>85^{\circ}\) and \(100^{\circ}<180^{\circ}\), valid.
• For \(m\angle Y = 85^{\circ}\), \(85^{\circ}=85^{\circ}\), not valid as \(\angle Y>\angle P\).
• For \(m\angle Y = 180^{\circ}\), a triangle cannot have an angle of \(180^{\circ}\) (it would be a degenerate triangle).

Answer:

\(m\angle Y = 100^{\circ}\)