Question

prove $overline{ij} cong overline{wv}$ by filling out the flowchart below.
for each box, choose a statement format from the dropdown menu. you will then be able to change the letters to match the diagram for this problem.

Explanation:

Step1: Identify Given Information

From the diagram, we observe:

• $\angle V \cong \angle J$ (marked angles)
• $VX \cong JH$ and $XW \cong HI$? Wait, no, looking at the markings: in $\triangle V W X$, sides $VX$ and $XW$? Wait, no, the markings on the sides: in $\triangle V W X$, two sides adjacent to $\angle X$ are marked, and in $\triangle J H I$, two sides adjacent to $\angle H$ are marked. Wait, actually, let's re - examine. The triangles: $\triangle VWX$ and $\triangle JHI$? Wait, no, the second triangle is $\triangle JHI$? Wait, the vertices are $H$, $J$, $I$ for the second triangle. So, let's list the given congruences:
• $\angle X \cong \angle H$ (marked angles)
• $VX \cong JH$ (marked sides)
• $WX \cong HI$ (marked sides)

So, by the Side - Angle - Side (SAS) congruence criterion, $\triangle VWX \cong \triangle JHI$.

Step2: Use Congruent Triangles

If $\triangle VWX \cong \triangle JHI$, then their corresponding sides are congruent. The side $WV$ in $\triangle VWX$ corresponds to side $IJ$ in $\triangle JHI$. So, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, $\overline{IJ} \cong \overline{WV}$.

Now, filling the flowchart:

First Box (Statement and Reason)

• Statement: $\angle X \cong \angle H$, $VX \cong JH$, $WX \cong HI$
• Reason: Given (from the diagram markings)

Second Box (Statement and Reason)

• Statement: $\triangle VWX \cong \triangle JHI$
• Reason: SAS (Side - Angle - Side) Congruence Criterion (because we have two sides and the included angle congruent: $VX \cong JH$, $\angle X \cong \angle H$, $WX \cong HI$)

Third Box (Statement and Reason)

• Statement: $\overline{IJ} \cong \overline{WV}$
• Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

To prove $\overline{IJ} \cong \overline{WV}$, we follow these steps:

1. Statement: $\angle X \cong \angle H$, $VX \cong JH$, $WX \cong HI$; Reason: Given (from diagram markings)
2. Statement: $\triangle VWX \cong \triangle JHI$; Reason: SAS Congruence Criterion
3. Statement: $\overline{IJ} \cong \overline{WV}$; Reason: CPCTC

So, the flowchart should be filled with these statements and reasons. The final conclusion is that $\overline{IJ} \cong \overline{WV}$ is proven by showing the triangles are congruent (SAS) and then using CPCTC.