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Determine if the congruence statements are correct and explain why or why not

Let's go through each of the congruence statements to determine if they are true and why.

### Problem 5: \( \triangle RST \cong \triangle TQP \)
- **Analysis**: In both triangles \( \triangle RST \) and \( \triangle TQP \), we observe matching pairs of sides and angles. The marks on the sides indicate that \( \overline{RS} \cong \overline{TQ} \), \( \overline{ST} \cong \overline{QP} \), and \( \overline{RT} \cong \overline{TP} \).
- **Reasoning**: Since all three corresponding sides are congruent (SSS), we can conclude that \( \triangle RST \cong \triangle TQP \) by the Side-Side-Side (SSS) postulate.

### Problem 6: \( \triangle ABD \cong \triangle CDB \)
- **Analysis**: In this figure, we see that \( \overline{AB} \cong \overline{CD} \) and \( \overline{BD} \) is common to both triangles, which suggests two sides are congruent.
- **Reasoning**: The vertical angles at point \( B \) are also congruent. Therefore, by the Side-Angle-Side (SAS) postulate, \( \triangle ABD \cong \triangle CDB \) because they have two pairs of congruent sides and the included angle.

### Problem 7: \( \triangle DEF \cong \triangle DGF \)
- **Analysis**: In this case, we see that \( \overline{DE} \cong \overline{DG} \) and \( \overline{DF} \) is common to both triangles.
- **Reasoning**: The angles at \( E \) and \( G \) are also marked as congruent. Therefore, \( \triangle DEF \cong \triangle DGF \) by the Side-Angle-Side (SAS) postulate as well.

Each congruence statement given in these problems is true based on the properties observed in each triangle pair.

Would you like more details on any of the congruence criteria or a breakdown of another method?

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### Follow-up Questions
1. What other congruence postulates could apply in similar triangle problems?
2. How does the Angle-Side-Angle (ASA) postulate differ from Side-Angle-Side (SAS)?
3. Can two triangles be congruent if they only have two pairs of corresponding sides congruent?
4. What role do vertical angles play in determining triangle congruence?
5. How would you determine congruence if only the angles in two triangles were given?

### Tip
Remember, identifying shared or common sides and vertical angles is often helpful when determining triangle congruence.

Determine if the congruence statements are correct and explain why or why not.

Explore how to determine triangle congruence by verifying congruence statements using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. Understand why triangles are congruent in problems involving side and angle comparisons. Suitable for students in Grades 8-10.

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