Questions: With the information shown above, we can also determine that BC is congruent to BC. Therefore, which of the following theorems affirms that triangle GCB is congruent to triangle HCB? Choose two. A Side-Side-Side Congruence. B Angle - Side - Angle Congruence. C Side - Side - Angle Congruence. D Side - Angle - Side Congruence.

With the information shown above, we can also determine that BC is congruent to BC. Therefore, which of the following theorems affirms that triangle GCB is congruent to triangle HCB? Choose two. A Side-Side-Side Congruence. B Angle - Side - Angle Congruence. C Side - Side - Angle Congruence. D Side - Angle - Side Congruence.
Transcript text: With the information shown above, we can also determine that $\overline{B C} \cong \overline{B C}$. Therefore, which of the following theorems affirms that $\triangle G C B \cong \triangle H C B$ ? Choose two. A Side-Side-Side Congruence. B Angle - Side - Angle Congruence. C Side - Side - Angle Congruence. D Side - Angle - Side Congruence.
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Solution

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Solution Steps

To determine which theorems affirm that $\triangle G C B \cong \triangle H C B$, we need to identify the criteria for triangle congruence. The given information suggests that $\overline{B C}$ is congruent to itself, and we need to consider other sides and angles to apply the appropriate congruence theorems.

Step 1: Identify Given Information

We are given that \(\overline{BC} \cong \overline{BC}\). This means that side \(BC\) is congruent to itself.

Step 2: Determine Applicable Congruence Theorems

To determine which theorems affirm that \(\triangle GCB \cong \triangle HCB\), we need to consider the criteria for triangle congruence. The possible congruence theorems are:

  • Side-Side-Side (SSS) Congruence
  • Angle-Side-Angle (ASA) Congruence
  • Side-Side-Angle (SSA) Congruence
  • Side-Angle-Side (SAS) Congruence
Step 3: Apply the Given Information

Given that \(\overline{BC}\) is congruent to itself, we need to consider other sides and angles to apply the appropriate congruence theorems. The most likely applicable theorems based on the given information are:

  • Side-Side-Side (SSS) Congruence
  • Side-Angle-Side (SAS) Congruence

Final Answer

The theorems that affirm \(\triangle GCB \cong \triangle HCB\) are: \[ \boxed{\text{A: Side-Side-Side Congruence}} \] \[ \boxed{\text{D: Side-Angle-Side Congruence}} \]

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