Question

fox is working to prove the base angles theorem. his proof is shown below. critique his reasoning. which statement or reason in his proof has a mistake? how can he fix his mistake? given: isosceles △abc with $overline{ab}congoverline{ac}$ prove: $angle bcongangle c$ identify mistake in steps to prove congruency between angles statements reasons 1. isosceles △abc with $overline{ab}congoverline{ac}$ 1. given 2. d, the mid - point of $overline{bc}$ $overline{ad}$ through points a and d 2. construction 3. $overline{bd}congoverline{dc}$ 3. definition of midpoint 4. $overline{ad}congoverline{ad}$ 4. reflexive property of congruence 5. △abd≅△acd 5. sas congruence theorem 6. $angle bcongangle c$ 6. cpctc theorem

Explanation:

Step1: Check each step

Examine statements and reasons one - by - one.

Step2: Analyze step 5

In step 5, for SAS congruence, we need to check the included angles.
Since we haven't shown that $\angle BAD\cong\angle CAD$, we can't use SAS.

Step3: Fix the proof

We can construct $\overline{AD}$ as the angle - bisector of $\angle BAC$ instead of just the line through $A$ and the mid - point of $\overline{BC}$. Then we have $\angle BAD\cong\angle CAD$ by the definition of angle - bisector, and with $\overline{AB}\cong\overline{AC}$ (given) and $\overline{AD}\cong\overline{AD}$ (reflexive property), we can use SAS.

Answer:

The mistake is in step 5. The reason "SAS Congruence Theorem" is incorrect as we haven't shown the included angles are congruent. To fix it, construct $\overline{AD}$ as the angle - bisector of $\angle BAC$.