Question 1170173: I have two right triangles that are connected at point x. On the right, there is triangle one which has point v at the top, w at the corner, and x which connects to triangle two which has point z at the bottom, connected to y which connects to x.
I am given that line WV is perpendicular to line WY. I am also given that line ZY is perpendicular to line WY, as well as line WV is congruent to ZY. I am supposed to prove that the triangles are congruent and X is the midpoint of line VZ.
I do have a 4th statement that I attempted to do. That is that angle W is congruent to angle Y because of the def. of perpendicularity. I also have about half of the fifth statement which is that angle 1 and angle 2 are congruent. I don't know how to prove this with reason. I need help.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down this geometry proof step-by-step.
**Given:**
* Triangle VWX and Triangle YXZ
* WV ⊥ WY
* ZY ⊥ WY
* WV ≅ ZY
**To Prove:**
1. Triangle VWX ≅ Triangle YXZ
2. X is the midpoint of VZ
**Proof:**
Here's the breakdown of the proof with reasons:
1. **Statement:** WV ⊥ WY
**Reason:** Given
2. **Statement:** ZY ⊥ WY
**Reason:** Given
3. **Statement:** ∠W and ∠Y are right angles.
**Reason:** Definition of perpendicular lines.
4. **Statement:** ∠W ≅ ∠Y
**Reason:** All right angles are congruent.
5. **Statement:** WV ≅ ZY
**Reason:** Given
6. **Statement:** ∠WX V ≅ ∠YX Z
**Reason:** Vertical angles are congruent.
7. **Statement:** Triangle VWX ≅ Triangle YXZ
**Reason:** Angle-Angle-Side (AAS) Congruence Theorem. (We have ∠W ≅ ∠Y, ∠WX V ≅ ∠YX Z, and WV ≅ ZY)
8. **Statement:** VX ≅ XZ
**Reason:** Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
9. **Statement:** X is the midpoint of VZ
**Reason:** Definition of a midpoint (a point that divides a segment into two congruent segments).
**Explanation of Step 6:**
* When two lines intersect, the angles opposite each other at the intersection point are called vertical angles.
* Vertical angles are always congruent.
* In this case, line VX intersects line ZY at point X, creating ∠WX V and ∠YX Z, which are vertical angles.
**In summary, here is the proof in a table format:**
| Statement | Reason |
|---|---|
| 1. WV ⊥ WY | Given |
| 2. ZY ⊥ WY | Given |
| 3. ∠W and ∠Y are right angles | Definition of perpendicular lines |
| 4. ∠W ≅ ∠Y | All right angles are congruent |
| 5. WV ≅ ZY | Given |
| 6. ∠WX V ≅ ∠YX Z | Vertical angles are congruent |
| 7. Triangle VWX ≅ Triangle YXZ | AAS Congruence Theorem |
| 8. VX ≅ XZ | CPCTC |
| 9. X is the midpoint of VZ | Definition of a midpoint |
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