Question

if vx = wz = 40 cm and ( mangle zvx = mangle xwz = 22^circ ), can ( \triangle vzx ) and ( \triangle wxz ) be proven congruent by sas? why or why not?
○ yes, along with the given information, ( overline{zx} cong overline{zx} ) by the reflexive property.
○ yes, the triangles are both obtuse.
○ no, the sides of the triangles intersect.
○ no, there is not enough information given.

Explanation:

To determine if \(\triangle VZX\) and \(\triangle WXZ\) are congruent by SAS, we recall the SAS (Side - Angle - Side) congruence criterion: two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle.

1. Analyze the given information:

• We know that \(VX = WZ=40\space\text{cm}\) and \(m\angle ZVX=m\angle XWZ = 22^{\circ}\). Also, \(\overline{ZX}\cong\overline{ZX}\) by the reflexive property (a segment is congruent to itself). But for SAS, the angle must be the included angle between the two sides.
• In \(\triangle VZX\), the sides adjacent to \(\angle ZVX\) are \(VZ\) and \(VX\), and in \(\triangle WXZ\), the sides adjacent to \(\angle XWZ\) are \(WX\) and \(WZ\). We are not given that \(VZ = WX\) or that the angle is included between the correct sides. The fact that \(\overline{ZX}\) is a common side does not help here because the angle given (\(22^{\circ}\)) is not the included angle between \(VX\) (or \(WZ\)) and \(ZX\).
• Let's analyze each option:
• Option 1: The reflexive property gives \(\overline{ZX}\cong\overline{ZX}\), but the angle \(22^{\circ}\) is not the included angle between \(VX\) (or \(WZ\)) and \(ZX\), so SAS cannot be applied. So this option is incorrect.
• Option 2: The type of triangle (obtuse) has nothing to do with the SAS congruence criterion. So this option is incorrect.
• Option 3: The intersection of sides is not a reason for non - congruence by SAS. So this option is incorrect.
• Option 4: We know one pair of sides (\(VX = WZ\)) and one pair of angles (\(\angle ZVX=\angle XWZ\)), but we don't have the other pair of sides (the ones that would form the included angle with the given sides) or the correct included angle information. So there is not enough information to prove congruence by SAS.

Answer:

D. No, there is not enough information given.