Question

the lengths of the sides of triangle xyz are written in terms of the variable m, where m ≥ 6. which is correct regarding the angles of the triangle? m∠x < m∠z < m∠y m∠y < m∠z < m∠x m∠y < m∠x < m∠z m∠z < m∠y < m∠x

Explanation:

Step1: Compare side lengths

First, we analyze the side lengths of triangle \( XYZ \). The sides are \( XZ = m - 3 \), \( XY = m + 8 \), and \( YZ = 2m + 3 \) with \( m\geq6 \).

To compare the lengths, we can analyze the expressions:

• For \( XZ = m - 3 \) and \( XY = m + 8 \): Since \( - 3<8 \), we have \( m - 3

• For \( XY = m + 8 \) and \( YZ = 2m + 3 \): Subtract \( XY \) from \( YZ \): \( (2m + 3)-(m + 8)=m - 5 \). Since \( m\geq6 \), \( m - 5\geq1>0 \), so \( YZ>XY \).

• For \( XZ = m - 3 \) and \( YZ = 2m + 3 \): Subtract \( XZ \) from \( YZ \): \( (2m + 3)-(m - 3)=m + 6 \). Since \( m\geq6 \), \( m + 6\geq12>0 \), so \( YZ>XZ \).

So the order of the sides from shortest to longest is \( XZ

Step2: Relate side lengths to angle measures

In a triangle, the larger the side length, the larger the angle opposite to it.

• The angle opposite \( XZ \) is \( \angle Y \).

• The angle opposite \( XY \) is \( \angle Z \).

• The angle opposite \( YZ \) is \( \angle X \).

Since \( XZ

Answer:

\( m\angle Y < m\angle Z < m\angle X \) (the second option: \( \boldsymbol{m\angle Y < m\angle Z < m\angle X} \))