This lesson flips the script from Lesson 16. Instead of starting with a shape you know is a parallelogram, you'll be given a general quadrilateral. Your mission is to prove that it is a parallelogram by showing it passes one of four specific tests.

1. The Four Official Tests for a Parallelogram

To prove a quadrilateral is a parallelogram, you only need to prove that it satisfies ONE of the following conditions. I've organized them in a table for easy reference.

2. How to Approach a Proof (Rider)

Most "Prove this is a parallelogram" questions follow a similar pattern. The key is usually to prove a pair of triangles are congruent to get the information you need.

Your Step-by-Step Game Plan:

1. Analyze the "Given": Read the problem and mark all the information you are given directly onto your diagram.

2. Choose Your Test: Look at the "Given" information and decide which of the four tests above will be easiest to prove.

   • Given info on side lengths? Aim for Test 1 or 4.

   • Given info on diagonals? Aim for Test 3.

   • Given info on parallel lines? Aim for Test 4.

3. Find the Triangles: Identify two triangles that you can prove are congruent to get the information you need (e.g., to get a pair of equal sides or equal alternate angles).

4. Prove Congruence: Write out the proof that the triangles are congruent (using SAS, AAS, SSS, or RHS).

5. Connect to the Test: Use the result from your congruent triangles (e.g., "corresponding sides are equal" or "corresponding angles are equal") to satisfy all parts of the test you chose in Step 2.

6. Write the Final Conclusion: State your final answer clearly using the theorem.

3. A Detailed Worked Example

• Given: In quadrilateral ABCD, the diagonals intersect at O. The midpoint of BC is E. DE is produced to meet AB produced at Q.

• To Prove: BCED is a parallelogram. (Note: A diagram would be provided in the exam)

• Game Plan: We are given a midpoint E on the diagonal BC of the shape BCED. We are also told DE is produced. This suggests the diagonals BC and DQ might bisect each other. Let's aim for Test 3.

• Proof:

  • Step 1 (Find Triangles): We need to show DE = EQ. Let's compare ΔDCE and ΔQBE.

  • Step 2 (Prove Congruence):

    • CE = EB (Given that E is the midpoint of BC) -> Side

    • ∠DEC = ∠QEB (Vertically opposite angles) -> Angle

    • Since ABCD is a parallelogram (from Lesson 16 context, often implied), DC || AQ. Therefore, ∠DCE = ∠QBE (Alternate angles) -> Angle

  • Step 3 (Congruence Statement):

    • Therefore, ΔDCE ≅ ΔQBE (AAS).

  • Step 4 (Connect to the Test):

    • From the congruence, we know DE = EQ (Corresponding sides of congruent triangles).

    • We were also given that CE = EB.

    • This means the diagonals DQ and BC of the quadrilateral BCDQ bisect each other at E.

• Conclusion:

  • Therefore, since the diagonals bisect each other, BCDQ is a parallelogram.

4. Important Points & Common Traps

• Lesson 16 vs. 17:

  • Lesson 16: You know it's a parallelogram. You use the properties.

  • Lesson 17: You prove it's a parallelogram. You use one of the four tests.

• The "One Pair" Trap (Test 4): Be careful with Test 4. It requires the same pair of sides to be both equal and parallel. You cannot prove a shape is a parallelogram if one pair of sides is equal (AB=DC) and the other pair of sides is parallel (AD||BC).

• State the Full Reason: In your conclusion, don't just say "it's a parallelogram". You must state which test you used.

  • Correct: "Since both pairs of opposite sides are equal, ABCD is a parallelogram."