Math Problem Statement

Write a proof to show that triangle ABD is congruent to triangle CDB given AB || CD and angle CBD ≅ angle ADB.

Solution

To prove ABDCDB\triangle ABD \cong \triangle CDB, we will use the given information and apply the appropriate congruence criteria. Here's the step-by-step proof:

Given:

  1. ABCD\overline{AB} \parallel \overline{CD} (AB is parallel to CD)
  2. CBDADB\angle CBD \cong \angle ADB (angles are congruent)

To Prove:

ABDCDB\triangle ABD \cong \triangle CDB


Proof:

  1. Given Information:

    • ABCD\overline{AB} \parallel \overline{CD} implies that these segments form parallel sides of a trapezoid.
    • CBDADB\angle CBD \cong \angle ADB.
  2. Angles formed by parallel lines:

    • Since ABCD\overline{AB} \parallel \overline{CD} and BD\overline{BD} is a transversal, the alternate interior angles ABD\angle ABD and CDB\angle CDB are congruent: ABDCDB\angle ABD \cong \angle CDB
  3. Shared side:

    • BD\overline{BD} is shared by both ABD\triangle ABD and CDB\triangle CDB, so: BDBD(Reflexive Property of Congruence)\overline{BD} \cong \overline{BD} \quad (\text{Reflexive Property of Congruence})
  4. Congruence Criterion:

    • Now, we have:
      • ABDCDB\angle ABD \cong \angle CDB (alternate interior angles)
      • BDBD\overline{BD} \cong \overline{BD} (shared side)
      • CBDADB\angle CBD \cong \angle ADB (given).
    • Using the ASA (Angle-Side-Angle) criterion, ABDCDB\triangle ABD \cong \triangle CDB.

Conclusion:

ABDCDB(ASA Congruence Criterion).\triangle ABD \cong \triangle CDB \quad (\text{ASA Congruence Criterion}).

Let me know if you'd like further clarifications! Here are some related questions for practice:

  1. How can you prove two triangles are congruent using the SAS criterion?
  2. Why are alternate interior angles equal for parallel lines and a transversal?
  3. What is the reflexive property of congruence?
  4. Can this proof also work with other congruence criteria like AAS? Why or why not?
  5. How does proving congruence help in solving real-world geometry problems?

Tip: Always identify shared sides or angles in geometric figures—they are key to many proofs.

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Math Problem Analysis

Mathematical Concepts

Triangle Congruence
Parallel Lines and Transversals
Geometry Proofs

Formulas

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Theorems

Angle-Side-Angle (ASA) Congruence Theorem
Alternate Interior Angles Theorem
Reflexive Property of Congruence

Suitable Grade Level

Grades 9-10