Math Problem Statement

steps to prove that triangle ABD is congruent to triangle CDB

Prove that triangle ABD is congruent to triangle CDB in a parallelogram, where AB || CD and AD || BC.

Solution

To prove that ABDCDB\triangle ABD \cong \triangle CDB, we can use the ASA (Angle-Side-Angle) postulate or any other congruence criterion depending on the properties of the parallelogram. Here's the proof step by step:


Given:

  1. ABCD\overline{AB} \parallel \overline{CD}
  2. ADBC\overline{AD} \parallel \overline{BC}

To Prove:

ABDCDB\triangle ABD \cong \triangle CDB.


Proof:

1. Opposite angles of a parallelogram are congruent:

  • Since ABCD\overline{AB} \parallel \overline{CD} and ADBC\overline{AD} \parallel \overline{BC}, the quadrilateral ABCDABCD is a parallelogram.
  • Therefore:
    • ABD=CDB\angle ABD = \angle CDB (alternate interior angles are equal).

2. Diagonal BD\overline{BD} is shared:

  • BD\overline{BD} is the common side of ABD\triangle ABD and CDB\triangle CDB.

3. Opposite angles of a parallelogram are congruent:

  • Similarly:
    • ADB=CBD\angle ADB = \angle CBD (alternate interior angles are equal).

4. Conclusion:

  • By the ASA postulate:
    • ABDCDB\triangle ABD \cong \triangle CDB because:
      1. ABD=CDB\angle ABD = \angle CDB,
      2. BD\overline{BD} is common,
      3. ADB=CBD\angle ADB = \angle CBD.

This completes the proof.


Do you want further clarifications or have any questions?

Here are 5 additional related questions:

  1. How can you use the properties of parallelograms to prove congruence in other triangles?
  2. Why does sharing a common side matter in triangle congruence proofs?
  3. Can you identify other pairs of congruent triangles in parallelograms?
  4. What are other congruence postulates that could work for this proof (e.g., SAS or SSS)?
  5. How do the parallel lines ABCDAB \parallel CD and ADBCAD \parallel BC directly affect the congruence?

Tip: Always check for shared sides or angles in geometric figures, as they often simplify proofs!

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Math Problem Analysis

Mathematical Concepts

Congruent Triangles
Properties of Parallelograms
Angle-Side-Angle (ASA) Postulate

Formulas

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Theorems

Alternate Interior Angles Theorem
ASA (Angle-Side-Angle) Congruence Postulate

Suitable Grade Level

Grades 8-10