Let's go through each question step-by-step to help you understand the solutions.

Problem 8

To prove that $\triangle BCD \cong \triangle EDC$, we can use the SSS (Side-Side-Side) congruence criterion since:

• $BC = ED = 4$ inches,
• $BD = EC$ (this is a shared diagonal in the square),
• $CD = DC = 3$ inches.

All corresponding sides are equal, so $\triangle BCD \cong \triangle EDC$ by SSS.

Problem 9

To prove that $\triangle GJK \cong \triangle GJL$:

• $GK = GL$ (marked with one line, indicating they are equal),
• $GJ$ is a common side between the triangles,
• $JK = JL$ (marked with two lines, indicating equality).

These triangles are congruent by SSS (Side-Side-Side) as all corresponding sides are equal.

Problem 10

In this problem, we need to prove that $\triangle ECB \cong \triangle DBC$ using SAS (Side-Angle-Side).

• Side 1: $EC \cong DB$ (given in the problem).
• Angle: $\angle EBC = \angle DBC$ because they are vertical angles (formed by the intersection of lines and thus are congruent).
• Side 2: $BC$ is a common side in both triangles.

Since we have two sides and the included angle congruent, we can conclude that $\triangle ECB \cong \triangle DBC$ by the SAS criterion.

Would you like further details on any of these proofs or more in-depth explanations?

Here are some related questions to deepen your understanding:

1. How does the SSS criterion differ from the SAS criterion in proving triangle congruence?
2. Why are vertical angles always congruent?
3. How would you prove congruence if only two angles and one side were given?
4. What other methods besides SSS and SAS can be used to prove triangle congruence?
5. Can you think of a real-life situation where congruent triangles are used in design or construction?

Tip: When proving triangle congruence, always check if you have enough information to use a specific criterion (like SSS, SAS, ASA, or AAS) before starting your proof.