Question
Parallelogram V W Z X is shown. Point Y is at the bottom center of the shape. Lines are drawn from points V to X through point Y and from points W to Z through point Y. 4 triangles are formed by the lines.
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, ZX ≅ 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.
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, ZX ≅ 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.
Answers
Answered by
GPT-4o mini
To determine if triangles ΔVZX and ΔWXZ can be proven congruent by the SAS (Side-Angle-Side) criterion, we need to analyze the given information:
1. **Sides VX and WZ are given as equal**:
- \( VX = WZ = 40 \, \text{cm} \)
2. **Angles m∠ZVX and m∠XWZ are given as equal**:
- \( m∠ZVX = m∠XWZ = 22^\circ \)
3. **Common Side**:
- The side ZX is common to both triangles (reflexive property), meaning \( ZX \cong ZX \).
Now, applying the SAS criterion for congruence:
- We have one pair of sides that are equal (\( VX \cong WZ \)).
- We have the included angle that is equal (\( m∠ZVX \cong m∠XWZ \)).
- We have the other side ZX that is common to both triangles (\( ZX \cong ZX \)).
Thus, by SAS, we can conclude that ΔVZX and ΔWXZ are indeed congruent because we have two sides (VX and WZ) and the included angle (m∠ZVX and m∠XWZ) that are each congruent, as well as the shared side ZX.
So the correct response is:
**Yes, along with the given information, ZX ≅ ZX by the reflexive property.**
1. **Sides VX and WZ are given as equal**:
- \( VX = WZ = 40 \, \text{cm} \)
2. **Angles m∠ZVX and m∠XWZ are given as equal**:
- \( m∠ZVX = m∠XWZ = 22^\circ \)
3. **Common Side**:
- The side ZX is common to both triangles (reflexive property), meaning \( ZX \cong ZX \).
Now, applying the SAS criterion for congruence:
- We have one pair of sides that are equal (\( VX \cong WZ \)).
- We have the included angle that is equal (\( m∠ZVX \cong m∠XWZ \)).
- We have the other side ZX that is common to both triangles (\( ZX \cong ZX \)).
Thus, by SAS, we can conclude that ΔVZX and ΔWXZ are indeed congruent because we have two sides (VX and WZ) and the included angle (m∠ZVX and m∠XWZ) that are each congruent, as well as the shared side ZX.
So the correct response is:
**Yes, along with the given information, ZX ≅ ZX by the reflexive property.**