Prove Theorems About Parallelograms

Lesson 1 of 2

In this lesson:

• Prove opposite sides and angles of a parallelogram are congruent
• Prove the diagonals of a parallelogram bisect each other
• Prove the converse: bisecting diagonals imply a parallelogram

Parallelogram Theorems: Three Goals for Today

By the end of this lesson, you should be able to:

1. Prove opposite sides of a parallelogram are congruent (ASA + alternate interior angles)
2. Prove opposite angles are congruent (CPCTC + co-interior angles)
3. Prove diagonals bisect each other — and the converse

What Can Parallel Lines Tell Us?

Today we combine three tools you already know:

• Alternate interior angles: parallel lines → congruent angles
• ASA congruence: two angles + included side → triangles congruent
• CPCTC: corresponding parts of congruent triangles are congruent

Question: Two pairs of parallel sides — what follows?

Setting Up the Parallelogram Proof

The Diagonal Splits Into Two Triangles

• divides into and ; (reflexive)

Since : (alt. interior angles)

Since : (alt. interior angles)

Alternate Interior Angles on Diagonal AC

• Teal angles: (from )
• Red angles: (from )
• Shared side: (reflexive)

ASA Proof: Opposite Sides Congruent

Given: is a parallelogram. Prove: and .

Result: Opposite Sides Are Congruent

• The diagonal divides the parallelogram into two congruent triangles
• This result becomes a building block for every subsequent proof

Strategy: One diagonal → two triangles → alternate interior angles → ASA → CPCTC

Check: Apply the Opposite Sides Theorem

In parallelogram , if and , find and .

Name the theorem before computing, then find the values.

Answer: Opposite Sides Are Congruent

By the Opposite Sides Theorem: opposite sides are congruent.

• No calculation needed — the theorem applies directly
• Knowing two adjacent sides tells you all four sides

Proving Opposite Angles Are Congruent

Approach 1 (congruent triangles):

• by CPCTC

Approach 2 (co-interior angles):

Consecutive Angles in a Parallelogram Are Supplementary

Check: Find All Four Angle Measures

In parallelogram , . Find , , and .

Use the relationship table — name each reason before writing the value.

Answer: Using Opposite and Supplementary Angles

• (opposite angles congruent)
• and (consecutive angles supplementary)

Watch out: One pair of congruent sides alone ≠ parallelogram.

Diagonals: Setting Up the Next Proof

Now draw both diagonals of parallelogram . They intersect at point .

Observation: Measure , , , . What do you find?

The diagonals appear to bisect each other. Our job: prove it.

Both Diagonals Intersecting at E

Forward direction: Given is a parallelogram → prove and

Forward Proof: Diagonals Bisect Each Other

Given: is a parallelogram. Diagonals intersect at .
Prove: and .

Understanding What Bisect Means Here

• : is the midpoint of diagonal
• : is the midpoint of diagonal

Watch out: means the diagonal is cut in half — not that equals a side. Half-diagonals ≠ sides.

Converse: If Diagonals Bisect → Parallelogram

Given: Quadrilateral with diagonals intersecting at , where and .
Prove: is a parallelogram.

Key tool: vertical angles at

Converse Proof: Second Pair of Parallel Lines

Continuing from the previous slide:

Quick Check: Converse in Action

Quadrilateral has diagonals and that intersect at midpoint , with and .

Is a parallelogram? Which theorem justifies your answer?

Identify the theorem, then state the conclusion.

Answer: Diagonals Bisecting Proves Parallelogram

Yes — is a parallelogram.

→ bisects . → bisects . Both diagonals are bisected at .

By the converse: bisecting diagonals → parallelogram. ✓

Midpoint Test: Proving Parallelogram by Coordinates

Vertices: , , , .

Same midpoint → diagonals bisect each other → parallelogram ✓

Key Takeaways — Deck 1

✓ Opposite sides congruent (ASA + CPCTC)

✓ Opposite angles congruent; consecutive angles supplementary

✓ Diagonals bisect each other (forward: ASA; converse: SAS + vertical angles)

One pair of congruent sides is not enough — need both

Half-diagonals ≠ sides; each converse needs its own proof

What Comes Next: Deck 2

In Deck 2 — Rectangles and Special Parallelograms:

• Rectangles: additional property of congruent diagonals — forward (SAS) and converse (SSS)
• Classification hierarchy: rectangle, rhombus, square — which properties belong where?
• Five methods to prove a quadrilateral is a parallelogram