Math Problem Statement

Prove that quadrilateral ABCD is a parallelogram given that AD ≅ BC and AB ≅ DC.

Solution

In the image, you are working on a proof that quadrilateral ABCDABCD is a parallelogram. Here's the breakdown of the proof, including the missing statements and reasons:

Given:

  • AD‾≅BC‾\overline{AD} \cong \overline{BC}
  • AB‾≅DC‾\overline{AB} \cong \overline{DC}

To Prove:

  • ABCDABCD is a parallelogram.

Proof:

StatementsReasons
Quadrilateral ABCDABCD with AD‾≅BC‾\overline{AD} \cong \overline{BC}, AB‾≅DC‾\overline{AB} \cong \overline{DC}Given
AC‾≅AC‾\overline{AC} \cong \overline{AC}Reflexive Property
△DAC≅△BCA\triangle DAC \cong \triangle BCASAS (Side-Angle-Side Postulate)
∠DAC≅∠BCA\angle DAC \cong \angle BCA, ∠DCA≅∠BAC\angle DCA \cong \angle BACCPCTC (Corresponding Parts of Congruent Triangles are Congruent)
DA∥BCDA \parallel BC, AB∥DCAB \parallel DCConverse of Alternate Interior Angles Theorem
ABCDABCD is a parallelogramDefinition of Parallelogram

This proof demonstrates that opposite sides of the quadrilateral are parallel, which satisfies the definition of a parallelogram.

Would you like more details on any of the steps or explanations?

Here are 5 related questions to deepen understanding:

  1. How does the SAS postulate help in proving triangle congruence?
  2. Why is the Reflexive Property important in geometric proofs?
  3. What is the importance of the Converse of the Alternate Interior Angles Theorem in this proof?
  4. How would the proof change if the quadrilateral was not a parallelogram?
  5. Can you think of another method to prove that ABCDABCD is a parallelogram?

Tip: When proving properties about parallelograms, always look for ways to establish congruence between triangles formed by diagonals.

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Math Problem Analysis

Mathematical Concepts

Geometry
Congruent Triangles
Parallelograms

Formulas

-

Theorems

SAS (Side-Angle-Side Postulate)
CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Converse of Alternate Interior Angles Theorem
Reflexive Property

Suitable Grade Level

Grades 9-10

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