Studdy AI Logo
Math  /  Geometry

QuestionProve △ABC≅△EDC\triangle A B C \cong \triangle E D C \begin{tabular}{|l|l|} \hline STATEMENTS & REASONS \\ \hline C is the midpoint of AE‾\overline{A E} and BD‾\overline{B D}. & Given \\ \hline & \\ \hline & \\ \hline & Definition of midpoint \\ \hline & \\ \hline & \\ \hline \end{tabular} AC‾≅CD‾,BC‾≅CE‾::AC‾≅EC‾,BC‾≅DC‾\overline{A C} \cong \overline{C D}, \overline{B C} \cong \overline{C E} \quad:: \overline{A C} \cong \overline{E C}, \overline{B C} \cong \overline{D C} AC‾≅BC‾,DC‾≅EC‾\overline{A C} \cong \overline{B C}, \overline{D C} \cong \overline{E C} ∠ACB≅∠ECD\angle A C B \cong \angle E C D ∠ABC≅∠CDE\angle A B C \cong \angle C D E ∠BAC≅∠CED\angle B A C \cong \angle C E D △ABC≅△EDC\triangle A B C \cong \triangle E D C △ABC≅△CED\triangle A B C \cong \triangle C E D △BAC≅△DCE\triangle B A C \cong \triangle D C E Vertical Angles Congruence Theorem Alternate Interior Angles Theorem SAS Congruence Theorem

Studdy Solution

STEP 1

1. C C is the midpoint of AE‾ \overline{A E} and BD‾ \overline{B D} .
2. We need to prove △ABC≅△EDC \triangle A B C \cong \triangle E D C .

STEP 2

1. Use the definition of midpoint.
2. Establish congruent segments.
3. Use the Vertical Angles Congruence Theorem.
4. Apply the SAS Congruence Theorem.

STEP 3

Since C C is the midpoint of AE‾ \overline{A E} and BD‾ \overline{B D} , by definition of midpoint, we have:
AC‾≅CE‾andBC‾≅CD‾ \overline{A C} \cong \overline{C E} \quad \text{and} \quad \overline{B C} \cong \overline{C D}

STEP 4

Using the congruent segments from the midpoint definition, we can establish:
AC‾≅CE‾andBC‾≅CD‾ \overline{A C} \cong \overline{C E} \quad \text{and} \quad \overline{B C} \cong \overline{C D}

STEP 5

By the Vertical Angles Congruence Theorem, the angles ∠ACB \angle A C B and ∠ECD \angle E C D are congruent because they are vertical angles:
∠ACB≅∠ECD \angle A C B \cong \angle E C D

STEP 6

Now, we apply the SAS Congruence Theorem. We have two sides and the included angle congruent:
1. AC‾≅CE‾ \overline{A C} \cong \overline{C E}
2. ∠ACB≅∠ECD \angle A C B \cong \angle E C D
3. BC‾≅CD‾ \overline{B C} \cong \overline{C D}
Thus, by SAS, △ABC≅△EDC \triangle A B C \cong \triangle E D C .
The triangles are congruent:
△ABC≅△EDC \triangle A B C \cong \triangle E D C

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord