Question

the lengths of the sides of triangle xyz are written in terms of the variable m, where m ≥ 6.
triangle xyz with xz labeled ( m - 3 ), xy labeled ( m + 8 ), yz labeled ( 2m + 3 )
which is correct regarding the angles of the triangle?
○ ( mangle x < mangle z < mangle y )
○ ( mangle y < mangle z < mangle x )
○ ( mangle y < mangle x < mangle z )
○ ( mangle z < mangle y < mangle 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 \( YZ = 2m + 3 \). Since \( m \geq 6 \), we can compare the lengths:

• \( XZ = m - 3 \)
• \( XY = m + 8 \)
• \( YZ = 2m + 3 \)

We know that \( m - 3 < m + 8 \) because subtracting 3 from \( m \) gives a smaller value than adding 8 to \( m \). Now compare \( m + 8 \) and \( 2m + 3 \):

\( m + 8 < 2m + 3 \)

Subtract \( m \) from both sides: \( 8 < m + 3 \)

Subtract 3 from both sides: \( 5 < m \). Since \( m \geq 6 \), this is true. So the order of side lengths from shortest to longest is \( XZ < XY < YZ \).

Step2: Relate side lengths to angles

In a triangle, the larger the side length, the larger the angle opposite it. Let's identify the angles opposite each side:

• Side \( XZ \) (length \( m - 3 \)) is opposite angle \( Y \) ( \( \angle Y \) )
• Side \( XY \) (length \( m + 8 \)) is opposite angle \( Z \) ( \( \angle Z \) )
• Side \( YZ \) (length \( 2m + 3 \)) is opposite angle \( X \) ( \( \angle X \) )

Since \( XZ < XY < YZ \), the angles opposite them follow the same order: \( \angle Y < \angle Z < \angle X \)

Answer:

\( m\angle Y < m\angle Z < m\angle X \) (corresponding to the option \( m\angle Y < m\angle Z < m\angle X \))