Rectangles and Special Parallelograms

Lesson 2 of 2

In this lesson:

• Prove rectangles have congruent diagonals — and the converse
• Classify parallelograms as rectangles, rhombi, or squares
• Apply five methods to prove a quadrilateral is a parallelogram

Lesson 2 Goals: Rectangles and Classification

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

4. Prove that a rectangle has congruent diagonals, and that a parallelogram with congruent diagonals is a rectangle
5. Classify parallelograms as rectangles, rhombi, or squares and apply their properties in multi-step proofs

What Makes a Rectangle Special?

• Rectangle: all four angles are — diagonals look equal
• Non-rectangular parallelogram: angles not — diagonals visibly unequal

Question: Does every parallelogram have equal diagonals?

Rectangle: A Parallelogram with Right Angles

• A rectangle is a parallelogram with all four angles
• Inherits all Deck 1 properties:
  • Opposite sides congruent: ,
  • Diagonals bisect each other

Question: Are diagonals and congruent in a rectangle?

Forward Proof: Rectangle Diagonals Are Congruent

Given: is a rectangle. Prove: .

Consider and :

Converse: Congruent Diagonals → Rectangle

Given: is a parallelogram with . Prove: is a rectangle.

Consider and :

Converse Proof: Equal and Supplementary → Right Angle

Continuing from the previous slide:

Check: Does a Parallelogram Become a Rectangle?

Parallelogram has diagonals of equal length: .

Is a rectangle? Which theorem justifies your answer, and what additional condition is required?

State the theorem name and the required condition before answering.

Answer: Congruent Diagonals Identify a Rectangle

Yes — is a rectangle.

Theorem: A parallelogram with congruent diagonals is a rectangle.

Key condition: must already be a parallelogram — congruent diagonals alone do not guarantee a rectangle.

(An isosceles trapezoid has congruent diagonals but is not a parallelogram.)

The Parallelogram Family Tree: Who Inherits What

Every special parallelogram inherits all properties of the level above it.

Which Properties Belong to Which Type

How to Read Classification Conditions

Given parallelogram with , :

• All sides equal → rhombus condition ✓
• Right angle → rectangle condition ✓
• Both conditions met → square

When Only the Rectangle Condition Is Met

Given parallelogram with , , :

• Sides differ: → not a rhombus
• Right angle: → all four angles → rectangle ✓

When Only the Rhombus Condition Is Met

Given parallelogram with , :

• All sides equal → rhombus ✓
• Angle → no right angles → not a rectangle, not a square

Check: Classify This Parallelogram Precisely

Parallelogram has , , and .

What is the most specific classification? State all properties that apply.

Work through: check sides first, then angles.

Answer: Equal Sides Without Right Angles → Rhombus

is a rhombus.

• All sides : opposite sides congruent +
• → no right angles → not a rectangle

Inherits all parallelogram properties, plus: all sides ≅ ✓, diagonals ⊥ ✓

Five Sufficient Conditions to Prove a Parallelogram

Show any one of:

1. Both pairs of opposite sides parallel
2. Both pairs of opposite sides congruent
3. One pair of sides both parallel and congruent
4. Both pairs of opposite angles congruent
5. Diagonals bisect each other

Any single condition closes the argument.

Selecting the Right Method for Your Given Information

Given: In quadrilateral , and .

• Method 3 applies: one pair of sides is both parallel and congruent
• No need to check the other pair

Watch Out: Perpendicular Diagonals Do Not Imply Parallelogram

Common error: "Diagonals are perpendicular → parallelogram."

Counterexample: the kite — perpendicular diagonals, not a parallelogram:

• No pair of opposite sides is parallel
• Only one diagonal bisects the other

Rule: Perpendicular diagonals identify rhombi within the parallelogram family — the parallelogram condition must be established first.

Key Takeaways: Rectangles and Special Parallelograms

✓ Rectangle diagonals: (SAS proof)

✓ Converse: parallelogram + congruent diagonals → rectangle

✓ Hierarchy: square ⊂ {rectangle, rhombus} ⊂ parallelogram

✓ Five methods to prove parallelogram — match method to given conditions

Congruent diagonals: rectangles only

Perpendicular diagonals ≠ parallelogram (kite counterexample)

Where Parallelogram Theorems Appear Next

HSG.GPE.B.4: Coordinate proofs using parallelogram properties

• Midpoint test → formula check; slope conditions verify parallel sides

Connections to earlier work:

• Alternate interior angles (HSG.CO.C.9) powered every parallelogram proof
• Triangle congruence (HSG.CO.B.7, B.8) was the engine throughout