Complete the two column proof. Given: U is the midpoint of \(\overline{ST}\), \(\overline{SV} \cong \overline{TW}\), \(\overline{VU} \cong \overline{WU}\). Prove: \(\angle SVU \con... Complete the two column proof. Given: U is the midpoint of \(\overline{ST}\), \(\overline{SV} \cong \overline{TW}\), \(\overline{VU} \cong \overline{WU}\). Prove: \(\angle SVU \cong \angle TWU\).
Understand the Problem
The question is asking us to complete a two-column proof in geometry. Given certain conditions about line segments and a midpoint, we need to prove that two angles, ∠SVU and ∠TWU, are congruent. This involves using geometric theorems and definitions to justify each step of the proof.
Answer
See the answer section for the completed two-column proof.
Answer for screen readers
| Statements | Reasons |
|---|---|
| 1. U is the midpoint of $\overline{ST}$ | 1. Given |
| 2. $\overline{SV} \cong \overline{TW}$ | 2. Given |
| 3. $\overline{VU} \cong \overline{WU}$ | 3. Given |
| 4. $\overline{SU} \cong \overline{TU}$ | 4. Definition of Midpoint |
| 5. $\triangle SVU \cong \triangle TWU$ | 5. SSS Congruence Postulate |
| 6. $\angle SVU \cong \angle TWU$ | 6. CPCTC |
Steps to Solve
- State the second given
We are given that $\overline{SV} \cong \overline{TW}$.
- State the reason for the second statement
Since the statement $\overline{SV} \cong \overline{TW}$ is given to us, the reason is "Given".
- State that $\overline{SU} \cong \overline{TU}$
Because $U$ is the midpoint of $\overline{ST}$, $\overline{SU}$ must be congruent to $\overline{TU}$.
- State the reason why $\overline{SU} \cong \overline{TU}$
The reason is the definition of a midpoint which states that a midpoint divides a line segment into two congruent segments.
- Prove that the triangles are congruent
We know that $\overline{SV} \cong \overline{TW}$, $\overline{VU} \cong \overline{WU}$ and $\overline{SU} \cong \overline{TU}$. Therefore, $\triangle SVU \cong \triangle TWU$.
- State the reason why the triangles are congruent
The reason is the Side-Side-Side (SSS) Congruence Postulate.
- State that $\angle SVU \cong \angle TWU$
Since $\triangle SVU \cong \triangle TWU$, then $\angle SVU \cong \angle TWU$.
- State the reason why $\angle SVU \cong \angle TWU$
The reason is CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.
| Statements | Reasons |
|---|---|
| 1. U is the midpoint of $\overline{ST}$ | 1. Given |
| 2. $\overline{SV} \cong \overline{TW}$ | 2. Given |
| 3. $\overline{VU} \cong \overline{WU}$ | 3. Given |
| 4. $\overline{SU} \cong \overline{TU}$ | 4. Definition of Midpoint |
| 5. $\triangle SVU \cong \triangle TWU$ | 5. SSS Congruence Postulate |
| 6. $\angle SVU \cong \angle TWU$ | 6. CPCTC |
More Information
A two-column proof is a way to organize a mathematical argument. It includes statements and corresponding reasons to show why a statement is true. It makes it easy for others to follow your logic and verify its correctness.
Tips
A common mistake is mixing up the reasons for the statements. For example, using "Given" for a step that follows from the definition of a midpoint. Another mistake could be using the wrong congruence postulate, such as using SAS (Side-Angle-Side) when you don't have information about angles.
AI-generated content may contain errors. Please verify critical information
