Math / Geometry

QuestionGiven: $\overline{D E}$ bisects $\angle B E C$ and $\overline{B E} \cong C E$ Prove: $\triangle A D B \cong \triangle A D C$ .
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Type of Statement $\angle B A D \cong \angle C A D$
Corresponding Parts of Congrueı are Congruent (CPCTC) An angle bisector divides an angl congruent angles
Reflexive Property SAS $\square$ $\overline{A D} \cong \overline{A D}$ Reflexive Property SAS

Studdy Solution

STEP 1

1. Line segment $\overline{D E}$ bisects $\angle B E C$ .
2. Line segments $\overline{B E}$ and $\overline{C E}$ are congruent.
3. We need to prove $\triangle A D B \cong \triangle A D C$ .

STEP 2

1. Identify the congruent angles created by the angle bisector.
2. Use the given congruence of line segments.
3. Apply the Reflexive Property.
4. Use the SAS (Side-Angle-Side) Congruence Postulate.
5. Conclude with CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

STEP 3

Identify the congruent angles created by the angle bisector:
Since $\overline{D E}$ bisects $\angle B E C$ , we have:
$\angle B E D \cong \angle C E D$

STEP 4

Use the given congruence of line segments:
$\overline{B E} \cong \overline{C E}$

STEP 5

Apply the Reflexive Property:
$\overline{A D} \cong \overline{A D}$

STEP 6

Apply the SAS (Side-Angle-Side) Congruence Postulate:
With $\overline{B E} \cong \overline{C E}$ , $\angle B E D \cong \angle C E D$ , and $\overline{A D} \cong \overline{A D}$ , we have:
$\triangle A D B \cong \triangle A D C$

STEP 7

Conclude with CPCTC (Corresponding Parts of Congruent Triangles are Congruent):
Since $\triangle A D B \cong \triangle A D C$ , it follows that:
$\angle B A D \cong \angle C A D$
The proof is complete, and we have shown that $\triangle A D B \cong \triangle A D C$ .

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Studdy Solution

STEP 1

1. Line segment $\overline{D E}$ bisects $\angle B E C$ .
2. Line segments $\overline{B E}$ and $\overline{C E}$ are congruent.
3. We need to prove $\triangle A D B \cong \triangle A D C$ .

STEP 2

1. Identify the congruent angles created by the angle bisector.
2. Use the given congruence of line segments.
3. Apply the Reflexive Property.
4. Use the SAS (Side-Angle-Side) Congruence Postulate.
5. Conclude with CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

STEP 3

Identify the congruent angles created by the angle bisector:
Since $\overline{D E}$ bisects $\angle B E C$ , we have:
$\angle B E D \cong \angle C E D$

STEP 4

Use the given congruence of line segments:
$\overline{B E} \cong \overline{C E}$

STEP 5

Apply the Reflexive Property:
$\overline{A D} \cong \overline{A D}$

STEP 6

Apply the SAS (Side-Angle-Side) Congruence Postulate:
With $\overline{B E} \cong \overline{C E}$ , $\angle B E D \cong \angle C E D$ , and $\overline{A D} \cong \overline{A D}$ , we have:
$\triangle A D B \cong \triangle A D C$

STEP 7

Conclude with CPCTC (Corresponding Parts of Congruent Triangles are Congruent):
Since $\triangle A D B \cong \triangle A D C$ , it follows that:
$\angle B A D \cong \angle C A D$
The proof is complete, and we have shown that $\triangle A D B \cong \triangle A D C$ .

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Studdy Solution

STEP 1

1. Line segment DE‾\overline{D E}DE bisects ∠BEC\angle B E C∠BEC.2. Line segments BE‾\overline{B E}BE and CE‾\overline{C E}CE are congruent.3. We need to prove △ADB≅△ADC\triangle A D B \cong \triangle A D C△ADB≅△ADC.

STEP 2

1. Identify the congruent angles created by the angle bisector.2. Use the given congruence of line segments.3. Apply the Reflexive Property.4. Use the SAS (Side-Angle-Side) Congruence Postulate.5. Conclude with CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

STEP 3

Identify the congruent angles created by the angle bisector:Since DE‾\overline{D E}DE bisects ∠BEC\angle B E C∠BEC, we have:∠BED≅∠CED\angle B E D \cong \angle C E D∠BED≅∠CED

STEP 4

Use the given congruence of line segments:BE‾≅CE‾\overline{B E} \cong \overline{C E}BE≅CE

STEP 5

Apply the Reflexive Property:AD‾≅AD‾\overline{A D} \cong \overline{A D}AD≅AD

STEP 6

STEP 7

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1. Line segment $\overline{D E}$ bisects $\angle B E C$ .
2. Line segments $\overline{B E}$ and $\overline{C E}$ are congruent.
3. We need to prove $\triangle A D B \cong \triangle A D C$ .

1. Identify the congruent angles created by the angle bisector.
2. Use the given congruence of line segments.
3. Apply the Reflexive Property.
4. Use the SAS (Side-Angle-Side) Congruence Postulate.
5. Conclude with CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Identify the congruent angles created by the angle bisector:
Since $\overline{D E}$ bisects $\angle B E C$ , we have:
$\angle B E D \cong \angle C E D$

Use the given congruence of line segments:
$\overline{B E} \cong \overline{C E}$

Apply the Reflexive Property:
$\overline{A D} \cong \overline{A D}$