Question

if vx = wz = 40 cm and m∠zvx = m∠xwz = 22°, can δvzx and δwxz 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. Given \(VX = WZ=40\space\text{cm}\) and \(m\angle ZVX=m\angle XWZ = 22^{\circ}\).
2. We also know that \(\overline{ZX}\cong\overline{ZX}\) by the reflexive property (a segment is congruent to itself).
3. Now, let's check the sides and included angles:

• In \(\triangle VZX\), we have side \(VX\), side \(ZX\), and the included angle \(\angle ZVX\).
• In \(\triangle WXZ\), we have side \(WZ\), side \(ZX\), and the included angle \(\angle XWZ\).
• Since \(VX = WZ\), \(\angle ZVX=\angle XWZ\), and \(ZX = ZX\) (by reflexive property), the two triangles satisfy the SAS congruence criterion.

The option "Yes, the triangles are both obtuse" is incorrect because the type of triangle (obtuse or not) is not relevant to the SAS congruence criterion. The option "No, the sides of the triangles intersect" is incorrect because the intersection of sides does not prevent congruence. The option "No, there is not enough information given" is incorrect because we have enough information (two sides and the included angle with the help of the reflexive property) to apply SAS.

Answer:

Yes, along with the given information, \(\overline{ZX}\cong\overline{ZX}\) by the reflexive property.