Math / Geometry

QuestionGiven: $D E$ bisects $\angle B E C$ and $\overline{B E} \cong C E$ . Prove: $\triangle A D B \cong \triangle A D C$ . Step $\overline{D E}$ bisects $\angle B E C$ $\overline{B E} \cong \overline{C E}$
Given try Type of Statement

Studdy Solution

STEP 1

1. $\overline{DE}$ bisects $\angle BEC$ .
2. $\overline{BE} \cong \overline{CE}$ .
3. We need to prove $\triangle ADB \cong \triangle ADC$ .

STEP 2

1. Identify the given information and what needs to be proven.
2. Use the Angle Bisector Theorem.
3. Apply the Side-Angle-Side (SAS) Congruence Postulate.

STEP 3

Identify the given information and what needs to be proven:
- Given: $\overline{DE}$ bisects $\angle BEC$ , and $\overline{BE} \cong \overline{CE}$ . - To prove: $\triangle ADB \cong \triangle ADC$ .

STEP 4

Use the Angle Bisector Theorem:
Since $\overline{DE}$ bisects $\angle BEC$ , it implies:
$\angle BED \cong \angle CED$

STEP 5

Apply the Side-Angle-Side (SAS) Congruence Postulate:
- $\overline{BE} \cong \overline{CE}$ (Given) - $\angle BED \cong \angle CED$ (Angle Bisector) - $\overline{AD}$ is common to both $\triangle ADB$ and $\triangle ADC$ .
By the SAS Congruence Postulate, we have:
$\triangle ADB \cong \triangle ADC$
The triangles $\triangle ADB$ and $\triangle ADC$ are congruent.

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