Question

∠s ≅ ∠u, \overline{uv} ⊥ \overline{vw}, and \overline{sw} ⊥ \overline{vw}. complete the proof that δuvw ≅ δswv

1 ∠s ≅ ∠u
2 \overline{uv} ⊥ \overline{vw} given
3 \overline{sw} ⊥ \overline{vw} given
4 ∠swv ≅ ∠uvw given
5 \overline{vw} ≅ \overline{vw}
6 δuvw ≅ δswv additive property of angle measure additive property of length all right angles are congruent angles forming a linear pair sum to 180° definition of angle bisector

Explanation:

Step1: Analyze Statement 5

The statement is $\overline{VW} \cong \overline{VW}$. This is a side that is common to both triangles $\triangle UVW$ and $\triangle SWV$. The reason for a side being congruent to itself is the Reflexive Property of Congruence. But looking at the given options, we can also think about what's being asked. Wait, actually, for the congruence of the triangles, we need to check the reasons. Wait, no, the question here is about the reason for $\overline{VW} \cong \overline{VW}$. The reflexive property states that any segment is congruent to itself. But in the dropdown, maybe we need to see. Wait, no, maybe I misread. Wait, the problem is to complete the proof. Let's re-examine.

Wait, the triangles are $\triangle UVW$ and $\triangle SWV$ (I think the original problem had a typo, maybe $\triangle SWV$ instead of $\triangle SW$). Let's list the given: $\angle S \cong \angle U$ (given), $\overline{UV} \perp \overline{VW}$ and $\overline{SW} \perp \overline{VW}$, so $\angle UVW$ and $\angle SWV$ are right angles? Wait, no, $\overline{UV} \perp \overline{VW}$ means $\angle UVW = 90^\circ$, and $\overline{SW} \perp \overline{VW}$ means $\angle SWV = 90^\circ$, so $\angle UVW \cong \angle SWV$ (all right angles are congruent). Then we have $\angle S \cong \angle U$ (given), $\overline{VW} \cong \overline{VW}$ (reflexive property), but wait, the statement 5 is $\overline{VW} \cong \overline{VW}$, so the reason should be the Reflexive Property, but in the dropdown, maybe the option is "Reflexive Property of Congruence" but looking at the given options, one of them is "All right angles are congruent" no, wait the options given in the dropdown (from the image) include "Reflexive Property" (maybe not shown, but actually, the correct reason for a segment being congruent to itself is the Reflexive Property. But wait, maybe the problem is about the congruence of the triangles. Wait, let's check the steps:

1. $\angle S \cong \angle U$ (given)
2. $\overline{UV} \perp \overline{VW}$ (given) $\implies \angle UVW = 90^\circ$
3. $\overline{SW} \perp \overline{VW}$ (given) $\implies \angle SWV = 90^\circ$
4. So $\angle UVW \cong \angle SWV$ (all right angles are congruent)
5. $\overline{VW} \cong \overline{VW}$ (reflexive property)
6. Then by AAS (Angle - Angle - Side) or ASA? Wait, $\angle S \cong \angle U$, $\angle SWV \cong \angle UVW$, and $\overline{VW} \cong \overline{VW}$. Wait, but the reason for statement 5: $\overline{VW} \cong \overline{VW}$ is the Reflexive Property of Congruence. But in the dropdown options, maybe "Reflexive Property" is not there, but wait the user's image shows options like "Additive Property of Angle Measure", "All right angles are congruent", etc. Wait, maybe I made a mistake. Wait, no, the segment $VW$ is common to both triangles, so it's congruent to itself by the Reflexive Property. But if we look at the options, maybe the correct reason for $\overline{VW} \cong \overline{VW}$ is "Reflexive Property of Congruence", but since that's not in the visible options, maybe the problem is different. Wait, no, maybe the question is about the congruence of the triangles. Wait, let's start over.

Wait, the goal is to prove $\triangle UVW \cong \triangle SWV$. Let's list the parts:

- $\angle S \cong \angle U$ (given)
- $\angle UVW \cong \angle SWV$: because $\overline{UV} \perp \overline{VW}$ and $\overline{SW} \perp \overline{VW}$, so both are right angles, and all right angles are congruent.
- $\overline{VW} \cong \overline{VW}$ (reflexive property)

So for statement 5, $\overline{VW} \cong \overline{VW}$, the reason is the Reflexive Property of Congruence. But if we look at the dropdown options, maybe "Reflexive Property" is not there, but maybe the user made a typo. Wait, no, the options given in the image (from the user's screenshot) include "All right angles are congruent", "Additive Property of Angle Measure", etc. Wait, maybe I misread the statement. Wait, statement 5 is $\overline{VW} \cong \overline{VW}$, so the reason is that a segment is congruent to itself, which is the Reflexive Property. But if that's not an option, maybe the problem is about the congruence of the triangles. Wait, no, the question is to complete the proof, so for statement 5, the reason is "Reflexive Property of Congruence" or "A segment is congruent to itself". But looking at the options, maybe the correct answer is that $\overline{VW}$ is a common side, so the reason is the Reflexive Property. But since the user's options include "All right angles are congruent" (no, that's for angles), "Additive Property..." (no), "Angles forming a linear pair..." (no), "Definition of angle bisector" (no), "Definition of equilateral triangle" (no). Wait, maybe I made a mistake. Wait, the triangles: $\angle S \cong \angle U$ (given), $\angle UVW \cong \angle SWV$ (because both are right angles, so all right angles are congruent), and $\overline{VW} \cong \overline{VW}$ (reflexive). Then by AAS, the triangles are congruent. But the question is about the reason for statement 5: $\overline{VW} \cong \overline{VW}$. The correct reason is the Reflexive Property of Congruence. But if that's not an option, maybe the user's options have "Reflexive Property" or maybe I misread the problem. Wait, maybe the original problem's triangle is $\triangle UVW$ and $\triangle SWV$, and the reason for $\overline{VW} \cong \overline{VW}$ is "Reflexive Property of Congruence". But since the user's options don't show that, maybe there's a mistake. Wait, no, maybe the problem is that $\overline{VW}$ is a side, and the reason is that it's a common side, so the reflexive property. So the answer for the reason of statement 5 is "Reflexive Property of Congruence" or "A segment is congruent to itself". But since the user's options include "All right angles are congruent" (no), maybe I made a mistake. Wait, no, let's check again.

Wait, the problem says "Complete the proof that $\triangle UVW \cong \triangle SWV$" (assuming the typo). The given:

1. $\angle S \cong \angle U$ (given)
2. $\overline{UV} \perp \overline{VW}$ (given) $\implies \angle UVW = 90^\circ$
3. $\overline{SW} \perp \overline{VW}$ (given) $\implies \angle SWV = 90^\circ$
4. So $\angle UVW \cong \angle SWV$ (all right angles are congruent)
5. $\overline{VW} \cong \overline{VW}$ (reflexive property)
6. $\triangle UVW \cong \triangle SWV$ (AAS: $\angle S \cong \angle U$, $\angle SWV \cong \angle UVW$, $\overline{VW} \cong \overline{VW}$)

So for statement 5, the reason is "Reflexive Property of Congruence" or "A segment is congruent to itself". But since the user's options don't have that, maybe the correct option from the dropdown (even if not visible) is the reflexive property. But maybe the user's options include "Reflexive Property" or maybe I misread. Wait, the user's image shows the reason dropdown with options like "Additive Property of Angle Measure", "All right angles are congruent", "Additive Property of Length", "Angles forming a linear pair sum to 180°", "Definition of angle bisector", "Definition of equilateral triangle". Wait, maybe the problem is not about statement 5, but about the congruence. Wait, no, the question is to complete the proof, so for statement 5, the reason is that $\overline{VW}$ is congruent to itself, which is the Reflexive Property. But since that's not an option, maybe the user made a mistake. Alternatively, maybe the triangles are proven by ASA: $\angle S \cong \angle U$, $\overline{VW} \cong \overline{VW}$, $\angle SWV \cong \angle UVW$. But no, ASA is angle - side - angle. Wait, $\angle S \cong \angle U$, $\overline{VW} \cong \overline{VW}$, $\angle SWV \cong \angle UVW$: that's AAS (angle - angle - side). But the reason for $\overline{VW} \cong \overline{VW}$ is reflexive.

But since the user's options don't have reflexive, maybe the correct answer is that the reason for $\overline{VW} \cong \overline{VW}$ is "A segment is congruent to itself" or "Reflexive Property of Congruence". But since the user's options include "All right angles are congruent" (no), maybe I'm missing something. Wait, no, the key here is that $\overline{VW}$ is a common side, so the reason is the Reflexive Property. So the answer for the reason of statement 5 is the Reflexive Property of Congruence.

Step2: Confirm the Reason

The reflexive property of congruence states that any geometric figure (segment, angle, triangle, etc.) is congruent to itself. So for the segment $\overline{VW}$, $\overline{VW} \cong \overline{VW}$ by the reflexive property.

Answer:

The reason for $\overline{VW} \cong \overline{VW}$ is the Reflexive Property of Congruence (or "A segment is congruent to itself"). However, if we consider the given options (even if not fully visible), the most appropriate reason is that a segment is congruent to itself, which is the Reflexive Property. But based on the options provided (like "All right angles are congruent" is for angles, not sides), the correct reason for $\overline{VW} \cong \overline{VW}$ is the Reflexive Property of Congruence.