Question

are ∠bad and ∠bcd congruent? explain why or why not. choose the correct answer below. a. ∠bad and ∠bcd are congruent because (overline{ab}congoverline{ad}) and (overline{bc}congoverline{cd}) b. ∠bad and ∠bcd are congruent because (\triangle abdcong\triangle cbd) by sss and corresponding parts of congruent triangles are congruent c. ∠bad and ∠bcd are not congruent because ∠dac is not congruent to ∠dca since (\triangle adc) is isosceles d. ∠bad and ∠bcd are not congruent because ∠cab is not congruent to ∠acb

Explanation:

Step1: Recall congruent - triangle criteria

If two triangles are congruent, their corresponding parts are congruent.

Step2: Analyze the given information

We need to check if we can prove $\triangle ABD\cong\triangle CBD$. If we can prove $\triangle ABD\cong\triangle CBD$ by SSS (Side - Side - Side) criterion (i.e., $AB = AD$, $BC = CD$, and $BD=BD$ (common side)), then $\angle BAD$ and $\angle BCD$ (corresponding angles of congruent triangles) are congruent.

Step3: Evaluate each option

• Option A: Just stating $AB\cong AD$ and $BC\cong CD$ is not enough to prove the congruence of $\angle BAD$ and $\angle BCD$. We need to prove the congruence of the triangles first.
• Option B: If $\triangle ABD\cong\triangle CBD$ by SSS (since $AB\cong AD$, $BC\cong CD$ and $BD$ is common), then by the property of corresponding parts of congruent triangles (CPCTC), $\angle BAD\cong\angle BCD$. This option is correct.
• Option C and D: Trying to use non - relevant angle non - congruence in other triangles to prove non - congruence of $\angle BAD$ and $\angle BCD$ is incorrect. We should focus on the congruence of $\triangle ABD$ and $\triangle CBD$.

Answer:

B. $\angle BAD$ and $\angle BCD$ are congruent because $\triangle ABD\cong\triangle CBD$ by SSS and corresponding parts of congruent triangles are congruent