Math Problem Statement

PROVE TRIANGLE DBC AND APQ CONGRUENT

Solution

To prove that triangles DBC\triangle DBC and APQ\triangle APQ are congruent, let's analyze the given parallelogram ABCDABCD and the lines PP and QQ that are drawn inside it.

Analyzing the Parallelogram and Lines:

  1. Parallelogram Properties: Since ABCDABCD is a parallelogram, opposite sides are equal and parallel:

    • ABCDAB \parallel CD and AB=CDAB = CD
    • ADBCAD \parallel BC and AD=BCAD = BC
  2. Triangles to Prove Congruence:

    • We need to prove DBCAPQ\triangle DBC \cong \triangle APQ.
  3. Lines PP and QQ:

    • PP and QQ are points on ADAD and BCBC, respectively, suggesting that PQABPQ \parallel AB and PQCDPQ \parallel CD because both PQPQ and ABAB are parallel to the base lines of the parallelogram.

Proving Congruence:

To prove DBCAPQ\triangle DBC \cong \triangle APQ, we can use the following approach based on the properties of parallelograms and parallel lines:

  1. Parallel Lines:

    • PQABPQ \parallel AB and PQCDPQ \parallel CD, implies APQ=DBC\angle APQ = \angle DBC because they are corresponding angles.
    • Similarly, PAQ=BDC\angle PAQ = \angle BDC because they are corresponding angles formed by the transversal PQPQ.
  2. Sides Equal Due to Parallelogram:

    • AP=BDAP = BD because AD=BCAD = BC and APAP and BDBD are parts of these equal lengths.
    • PQ=BCPQ = BC because they are between the same parallel lines ABAB and CDCD.
  3. Angle-Side-Angle (ASA) Congruence:

    • In APQ\triangle APQ and DBC\triangle DBC:
      • APQ=DBC\angle APQ = \angle DBC (corresponding angles due to parallel lines),
      • PAQ=BDC\angle PAQ = \angle BDC (corresponding angles),
      • PQ=BCPQ = BC (parallel lines property).

    With two angles and the included side equal, triangles APQ\triangle APQ and DBC\triangle DBC are congruent by the Angle-Side-Angle (ASA) postulate.

Conclusion:

Thus, DBC\triangle DBC and APQ\triangle APQ are congruent based on ASA congruence criteria.


Would you like more details or have any questions?

Related Questions:

  1. What other properties of parallelograms could help in proving triangle congruence?
  2. How does the property of corresponding angles apply to non-parallel lines?
  3. What are other ways to prove triangles congruent besides ASA?
  4. Can you explain the properties of a parallelogram that make it a special type of quadrilateral?
  5. What role do transversals play in the congruence of triangles?

Tip:

In any geometric figure involving a parallelogram, always look for properties like equal opposite sides, parallel lines, and angle relationships to simplify proofs.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Congruence

Formulas

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Theorems

Angle-Side-Angle (ASA) Congruence

Suitable Grade Level

High School