Math / Geometry

QuestionGiven: $\overline{D E} \cong \overline{C E}$ and $\overline{F E}$ bisects $\angle D E C$ . Prove: $\overline{F A} \cong \overline{F B}$ .
Step Statement Reason
1 $\overline{D E} \cong \overline{C E}$ $\overline{F E}$ bisects $\angle D E C$ Given try Type of Statement

Studdy Solution

STEP 1

1. $\overline{D E} \cong \overline{C E}$
2. $\overline{F E}$ bisects $\angle D E C$
3. The diagram accurately represents the given information
4. Points A and B are the intersections of $\overline{FD}$ and $\overline{FC}$ with $\overline{DC}$ respectively

STEP 2

1. Analyze the given information
2. Identify congruent triangles
3. Prove that $\overline{FA} \cong \overline{FB}$

STEP 3

Analyze the given information: $\overline{D E} \cong \overline{C E}$ implies that E is the midpoint of $\overline{DC}$ . $\overline{F E}$ bisects $\angle D E C$ , so $\angle DEF \cong \angle CEF$ .

STEP 4

Identify congruent triangles: We can prove that $\triangle DEF \cong \triangle CEF$ using the SAS (Side-Angle-Side) congruence criterion:
1. $\overline{D E} \cong \overline{C E}$ (given)
2. $\angle DEF \cong \angle CEF$ ( $\overline{F E}$ bisects $\angle D E C$ )
3. $\overline{EF}$ is common to both triangles
Therefore, $\triangle DEF \cong \triangle CEF$

STEP 5

Prove that $\overline{FA} \cong \overline{FB}$ :
3.1. Since $\triangle DEF \cong \triangle CEF$ , we know that $\overline{FD} \cong \overline{FC}$
3.2. E is the midpoint of $\overline{DC}$ , so $\overline{DE} \cong \overline{EC}$
3.3. $\triangle FDA \cong \triangle FCB$ by the SAS congruence criterion: - $\overline{FD} \cong \overline{FC}$ (from step 3.1) - $\angle FDA \cong \angle FCB$ (vertical angles) - $\overline{DA} \cong \overline{CB}$ (E is midpoint of $\overline{DC}$ , so $\overline{DA} = \overline{DE}$ and $\overline{CB} = \overline{CE}$ )
3.4. Since $\triangle FDA \cong \triangle FCB$ , we can conclude that $\overline{FA} \cong \overline{FB}$
Therefore, we have proved that $\overline{FA} \cong \overline{FB}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms