Math Problem Statement

Given segment AB is parallel to segment CD and segment AD is parallel to segment BC. Prove triangle ABD and triangle CDB are congruent.

Solution

To prove that triangles ABD\triangle ABD and CDB\triangle CDB are congruent, we can use the following logical steps, applying geometry and congruence postulates:


Proof:

Given:

  1. ABCDAB \parallel CD
  2. ADBCAD \parallel BC

To prove: ABDCDB\triangle ABD \cong \triangle CDB


Steps:

  1. Identify the transversal lines:

    • Since ABCDAB \parallel CD, segment BDBD acts as a transversal.
    • Similarly, since ADBCAD \parallel BC, segment ACAC acts as a transversal.
  2. Establish corresponding angles:

    • From ABCDAB \parallel CD, the alternate interior angles ABD\angle ABD and CDB\angle CDB are equal: ABD=CDB(Alternate Interior Angles)\angle ABD = \angle CDB \quad \text{(Alternate Interior Angles)}
    • From ADBCAD \parallel BC, the alternate interior angles DAB\angle DAB and BCD\angle BCD are equal: DAB=BCD(Alternate Interior Angles)\angle DAB = \angle BCD \quad \text{(Alternate Interior Angles)}
  3. Establish congruent sides:

    • Segment BDBD is common to both triangles ABD\triangle ABD and CDB\triangle CDB: BD=BD(Reflexive Property of Equality)BD = BD \quad \text{(Reflexive Property of Equality)}
  4. Apply the ASA Congruence Postulate:

    • In ABD\triangle ABD and CDB\triangle CDB:
      • DAB=BCD\angle DAB = \angle BCD (Step 2)
      • BD=BDBD = BD (Step 3)
      • ABD=CDB\angle ABD = \angle CDB (Step 2)
    • By the ASA Congruence Postulate, ABDCDB\triangle ABD \cong \triangle CDB.

Conclusion:

Therefore, ABD\triangle ABD and CDB\triangle CDB are congruent.


Would you like me to illustrate this proof or elaborate further on any steps?


Related Questions:

  1. How do parallel lines influence congruent triangles in other scenarios?
  2. Can you prove congruence using other postulates, such as SSS or SAS?
  3. What are some common applications of congruent triangles in real-world problems?
  4. How does the Reflexive Property help in other triangle congruence proofs?
  5. What other properties of parallel lines are essential for geometric proofs?

Tip:

Always look for transversal angles and shared sides when working with parallel lines and proving triangle congruence. They are key elements in most geometric proofs.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Congruence
Parallel Lines

Formulas

Alternate Interior Angles Theorem
Reflexive Property

Theorems

ASA Congruence Postulate
Alternate Interior Angles Theorem

Suitable Grade Level

Grades 9-10