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Math  /  Geometry

QuestionProve: △AEC≅△DFB\triangle A E C \cong \triangle D F B.
Step Statement AE‾∥FD‾\overline{A E} \| \overline{F D} 1 BF‾∥EC‾\overline{B F} \| \overline{E C} AC‾≅BD‾\overline{A C} \cong \overline{B D}
Reason
Given try Type of Statement

Studdy Solution

STEP 1

1. AE‾∥FD‾\overline{AE} \parallel \overline{FD}
2. BF‾∥EC‾\overline{BF} \parallel \overline{EC}
3. AC‾≅BD‾\overline{AC} \cong \overline{BD}

STEP 2

1. Identify pairs of alternate interior angles.
2. Use the properties of parallel lines to establish angle congruence.
3. Apply the Side-Angle-Side (SAS) Congruence Postulate.

STEP 3

Identify pairs of alternate interior angles formed by the parallel lines and the transversal:
- Since AE‾∥FD‾\overline{AE} \parallel \overline{FD} and AC‾\overline{AC} is a transversal, ∠AEC≅∠DFB\angle AEC \cong \angle DFB (alternate interior angles).
- Since BF‾∥EC‾\overline{BF} \parallel \overline{EC} and BD‾\overline{BD} is a transversal, ∠AEC≅∠DFB\angle AEC \cong \angle DFB (alternate interior angles).

STEP 4

Use the properties of parallel lines to establish angle congruence:
- ∠AEC≅∠DFB\angle AEC \cong \angle DFB by the Alternate Interior Angles Theorem.

STEP 5

Apply the Side-Angle-Side (SAS) Congruence Postulate:
- We have AC‾≅BD‾\overline{AC} \cong \overline{BD} (given). - ∠AEC≅∠DFB\angle AEC \cong \angle DFB (from Step 2). - EC‾≅FB‾\overline{EC} \cong \overline{FB} (since they are corresponding parts of parallel lines).
Thus, by the SAS Congruence Postulate, △AEC≅△DFB\triangle AEC \cong \triangle DFB.
The triangles △AEC\triangle AEC and △DFB\triangle DFB are congruent.

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