Lesson 8 ~25 min Unit 3 · Geometry +85 XP

Parallelograms and Rectangles

A parallelogram has two pairs of parallel sides. From that ONE rule, a whole list of properties follow, equal opposite sides, equal opposite angles, diagonals that bisect each other. Add four right angles and you get a rectangle, which also has equal diagonals.

Today's hook: If I only tell you "this shape has two pairs of parallel sides", you instantly know SIX more things about it. How?

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Think First

warm-up

Draw a parallelogram on grid paper (slanted, NOT a rectangle). Measure its two pairs of opposite sides. What do you notice? Now measure both diagonals and the angles at each end. Write down everything that looks "equal" or "symmetric".

Record your answer in your workbook.

What Is a Parallelogram?

+5 XP

A parallelogram is a quadrilateral with two pairs of parallel sides. That's the defining property. From it, four other properties automatically follow: opposite sides are equal in length, opposite angles are equal, co-interior angles are supplementary (add to $180^{\circ}$), and the diagonals bisect each other (cut each other in half).

A parallelogram $ABCD$ has $AB \parallel DC$ and $AD \parallel BC$. From this we get: $AB = DC$ and $AD = BC$ (opposite sides equal), $\angle A = \angle C$ and $\angle B = \angle D$ (opposite angles equal), and the diagonals $AC$ and $BD$ meet at a point $M$ where $AM = MC$ and $BM = MD$.

Defining rule: 2 pairs of parallel sides ⇒ 5 other properties follow

Opposite sides equal

$AB = DC$ and $AD = BC$, mark with single/double arrows on a diagram.

Opposite angles equal

$\angle A = \angle C$ and $\angle B = \angle D$. Adjacent angles always sum to $180^{\circ}$.

Diagonals bisect each other

They cross at one point $M$ that's the midpoint of BOTH diagonals.

What You'll Master

objectives

Know

The 5 properties of a parallelogram
The extra 2 properties of a rectangle
How rectangles fit inside the parallelogram family
Standard angle-property names ("opp. angles of parallelogram", "co-int. angles")

Understand

Why opposite sides of a parallelogram must be equal
Why the diagonals always bisect each other
Why a rectangle is a parallelogram but a parallelogram isn't always a rectangle

Can Do

Find missing side lengths in a parallelogram using opposite-sides equal
Find missing angles using opposite-angles or co-interior
Solve diagonal problems using the bisection property

Words You Need

vocabulary

ParallelogramQuadrilateral with both pairs of opposite sides parallel.

RectangleParallelogram with four right angles.

Opposite sidesSides that don't share a vertex (e.g. $AB$ and $DC$).

Opposite anglesAngles at vertices that don't share a side (e.g. $\angle A$ and $\angle C$).

Co-interior anglesTwo adjacent angles between parallel lines, sum to $180^{\circ}$.

BisectTo cut into two equal halves.

DiagonalLine segment joining non-adjacent vertices.

$\parallel$ symbolRead as "is parallel to". $AB \parallel DC$.

Spot the Trap

heads-up

Wrong: "The diagonals of a parallelogram are equal." NO, only in a rectangle. In a general parallelogram, the diagonals usually have DIFFERENT lengths.

Right: The diagonals of a parallelogram bisect each other, but they aren't equal in length unless the parallelogram is a rectangle.

Wrong: "All four angles of a parallelogram are equal." NO, only the OPPOSITE angles. Adjacent angles are usually different (they sum to $180^{\circ}$).

Right: $\angle A = \angle C$ and $\angle B = \angle D$. If $\angle A = 70^{\circ}$, then $\angle B = 110^{\circ}$ (co-interior).

Book notes · Card 4

Diagonals of a parallelogram BISECT each other, they do NOT have to be equal.
Only OPPOSITE angles of a parallelogram are equal; adjacent angles are co-interior (sum $180^{\circ}$).
If diagonals ARE equal ⇒ the parallelogram is a rectangle.

True or False: In every parallelogram, the two diagonals have equal lengths.

The Rectangle, A Special Parallelogram

+5 XP

A rectangle is a parallelogram with one extra rule: all four angles are right angles. Because a rectangle is a parallelogram, it inherits ALL five parallelogram properties. Plus it gets two extras: (i) all angles equal $90^{\circ}$, and (ii) the diagonals are EQUAL in length (and still bisect each other).

In rectangle $PQRS$: opposite sides equal & parallel (parallelogram property), $\angle P = \angle Q = \angle R = \angle S = 90^{\circ}$, and $PR = QS$ (diagonals equal length AND bisect each other at midpoint $M$).

Rectangle: parallelogram + 4 right angles ⇒ equal diagonals

Inherits everything

Opposite sides parallel and equal, carried over from parallelogram.

Plus 4 right angles

$90^{\circ}$ at every vertex, that's the rectangle's defining extra.

Equal diagonals

The two diagonals of a rectangle have the SAME length. Useful for finding lengths.

Book notes · Card 5

Rectangle = parallelogram + 4 right angles ⇒ gets equal diagonals as a bonus.
All 5 parallelogram properties + 2 rectangle extras (all $90^{\circ}$ angles, equal diagonals).
Midpoint $M$ of the diagonals satisfies $PM = QM = RM = SM$.

Which extra property does a rectangle have that a non-rectangular parallelogram does NOT have?

Using the Properties

+5 XP

When you see a parallelogram or rectangle in a question, scan the diagram and ask: which property gives me the unknown? Match the situation to the right rule and ALWAYS write a reason in brackets, e.g. (opp. sides of parallelogram), (opp. angles of parallelogram), (co-int. angles, $AB \parallel DC$), (diagonals of rectangle equal).

Decision flow for parallelograms:
• Need an opposite side → opp. sides equal
• Need an opposite angle → opp. angles equal
• Need an adjacent angle → co-int. angles sum $180^{\circ}$
• Need a half-diagonal → diagonals bisect
• Rectangle only: full diagonal → diagonals equal

Always write the reason next to your answer.

Mark the diagram

Tick marks on equal sides, arrow marks on parallel sides, saves errors.

Reason in brackets

"(opp. angles of parallelogram)", markers expect this short justification.

Check the sum

All four angles should add to $360^{\circ}$ at the end.

Book notes · Card 6

Match the unknown to the right property: side → opp. sides; angle → opp. or co-int.
Co-interior angles in a parallelogram always sum to $180^{\circ}$.
Always state a reason like "(opp. angles of parallelogram)" in brackets.

In parallelogram $ABCD$, if $\angle A = 65^{\circ}$, then $\angle B = $ $^{\circ}$ because adjacent angles are co-interior.

Watch Me Solve It · 3 examples

Watch Me Solve It · Missing angle in parallelogram

+15 XP per step

PROBLEM

In parallelogram $ABCD$, $\angle A = 75^{\circ}$. Find $\angle B$, $\angle C$ and $\angle D$.

1
Opposite angles are equal
$\angle C = \angle A = 75^{\circ}$ (opp. angles of parallelogram)
2
Adjacent angles are co-interior
$\angle B = 180 - 75 = 105^{\circ}$ (co-int. angles, $AB \parallel DC$)
3
Final angle
$\angle D = \angle B = 105^{\circ}$ (opp. angles of parallelogram)
Check: $75 + 105 + 75 + 105 = 360^{\circ}$ ✓

Answer$\angle B = 105^{\circ}, \angle C = 75^{\circ}, \angle D = 105^{\circ}$.

Watch Me Solve It · Diagonals bisect each other

+15 XP per step

PROBLEM

The diagonals of parallelogram $PQRS$ meet at $M$. $PM = 9$ cm and $QM = 6$ cm. Find $PR$ and $QS$.

1
Use bisection of $PR$
$M$ is the midpoint of $PR$ (diagonals of parallelogram bisect), so $MR = PM = 9$.
2
Find $PR$
$PR = PM + MR = 9 + 9 = 18$ cm.
3
Find $QS$
$M$ is the midpoint of $QS$ too, so $MS = QM = 6$, giving $QS = 12$ cm.
Note: $PR \ne QS$ in general, only EQUAL if it's a rectangle.

Answer$PR = 18$ cm, $QS = 12$ cm.

Watch Me Solve It · Rectangle diagonals

+15 XP per step

PROBLEM

Rectangle $WXYZ$ has diagonals meeting at $M$. $WY = 26$ cm. Find $MX$.

1
Diagonals of rectangle equal
$XZ = WY = 26$ cm (diagonals of rectangle equal).
2
Diagonals bisect
$M$ is the midpoint of $XZ$, so $MX = \tfrac{1}{2} \times XZ$.
3
Compute
$MX = \tfrac{26}{2} = 13$ cm.
All four "half diagonals" $WM, MY, XM, MZ$ equal $13$ cm in a rectangle.

Answer$MX = 13$ cm.

Common Pitfalls

heads-up

Saying "all angles equal"

In a parallelogram, only OPPOSITE angles are equal. Adjacent angles are usually different.

Fix: Use co-interior ($180^{\circ}$) for an adjacent angle, opposite-equal for the diagonal corner.

Assuming diagonals are equal

Diagonals are equal ONLY when the parallelogram is a rectangle. Otherwise they bisect but differ in length.

Fix: "Bisect each other" $\ne$ "equal in length". Two separate properties.

Forgetting the reason

Writing just "$x = 75^{\circ}$" without "(opp. angles of parallelogram)" loses a mark in proofs.

Fix: Every line should end with the property in brackets.

Copy Into Your Books

Parallelogram (5)

2 pairs parallel sides
Opposite sides equal
Opposite angles equal
Co-int. angles $= 180^{\circ}$
Diagonals bisect each other

Rectangle extras (2)

All 4 angles $= 90^{\circ}$
Diagonals are equal in length
(still bisect each other)

Reasons

(opp. sides of parallelogram)
(opp. angles of parallelogram)
(co-int. angles, $AB \parallel DC$)
(diagonals of rectangle equal)

Family

Rectangle is a parallelogram
Parallelogram is NOT always a rectangle
If diagonals equal ⇒ rectangle

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Properties Practice

4 problems

Four quick drills using the parallelogram and rectangle properties. Solve, then reveal.

1 In parallelogram $ABCD$, $\angle A = 110^{\circ}$. Find $\angle B$.

Co-int. with $\angle A$: $180 - 110$.$\angle B = 70^{\circ}$
2 Parallelogram $PQRS$ has $PQ = 12$ cm. Find $SR$.

Opp. sides equal.$SR = 12$ cm
3 Rectangle $WXYZ$ has $WY = 20$ cm. Find $XZ$.

Diagonals of rectangle are equal.$XZ = 20$ cm
4 Diagonals of parallelogram meet at $M$ with $AM = 5$ cm. Find $MC$.

Diagonals bisect each other.$MC = 5$ cm

Complete in your workbook.

Labeling Properties with Conventions

+5 XP

Diagrams use standard symbols so properties can be read quickly. Parallel sides are marked with matching arrow heads. Equal sides are marked with matching tick marks. Right angles are marked with a small square. Diagonals are drawn between opposite vertices and labelled with capital letters such as $AC$ and $BD$.

In parallelogram $ABCD$, sides $AB$ and $DC$ carry one arrow each to show $AB \parallel DC$. Sides $AD$ and $BC$ carry two arrows to show $AD \parallel BC$. Equal opposite sides may also carry matching tick marks. Diagonals $AC$ and $BD$ cross at $M$, and the tick marks on the segments show $AM = MC$ and $BM = MD$.

Arrows = parallel; ticks = equal lengths; squares = right angles; capitals = vertices/diagonals

What to write in your book

Parallel sides: matching arrow heads.
Equal sides: matching tick marks.
Right angles: small square symbol.
Diagonals: line segments joining opposite vertices, labelled by their endpoints.

What symbol shows that two sides of a quadrilateral are parallel?

Quick Check · 5 questions

Which statement is TRUE for every parallelogram?

+10 XP

Parallelogram $ABCD$ has $\angle A = 68^{\circ}$. What is $\angle B$?

+10 XP

Which extra property does a rectangle have over a general parallelogram?

+10 XP

Diagonals of parallelogram $ABCD$ meet at $M$. $AM = 7$ cm. What is $MC$?

+10 XP

Parallelogram $ABCD$ has $\angle A = 105^{\circ}$. What is $\angle C$?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 3 MARKS

Q6. Parallelogram $EFGH$ has $\angle E = 55^{\circ}$ and $EF = 9$ cm.
(a) Find $\angle F$, giving a reason.
(b) Find $\angle G$, giving a reason.
(c) Find $HG$, giving a reason.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. The diagonals of rectangle $JKLM$ meet at $N$. $JL = 24$ cm.
(a) Find $KM$, giving a reason.
(b) Find $KN$, giving a reason.
(c) Explain why all four "half-diagonals" $JN, KN, LN, MN$ are equal.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. In parallelogram $ABCD$, $\angle A = (3x + 10)^{\circ}$ and $\angle B = (5x - 30)^{\circ}$.
(a) Set up an equation using a parallelogram property.
(b) Solve for $x$.
(c) State all four angles.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C Opposite sides are equal in length.

2. B$180 - 68 = 112^{\circ}$ (co-interior).

3. D Diagonals are equal in length.

4. A$MC = 7$ cm (diagonals bisect each other).

5. B$\angle C = 105^{\circ}$ (opp. angles of parallelogram).

Show Your Working Model Answers

Q6 (3 marks): (a) $\angle F = 180 - 55 = 125^{\circ}$ (co-int. angles, $EF \parallel HG$) [1]. (b) $\angle G = \angle E = 55^{\circ}$ (opp. angles of parallelogram) [1]. (c) $HG = EF = 9$ cm (opp. sides of parallelogram) [1].

Q7 (3 marks): (a) $KM = JL = 24$ cm (diagonals of rectangle equal) [1]. (b) $KN = \tfrac{KM}{2} = 12$ cm (diagonals bisect each other) [1]. (c) Rectangle has equal diagonals AND they bisect, so all four halves measure $12$ cm [1].

Q8 (3 marks): (a) $\angle A + \angle B = 180^{\circ}$ (co-int. angles): $(3x + 10) + (5x - 30) = 180$ [1]. (b) $8x - 20 = 180 \Rightarrow 8x = 200 \Rightarrow x = 25$ [1]. (c) $\angle A = 85^{\circ}, \angle B = 95^{\circ}, \angle C = 85^{\circ}, \angle D = 95^{\circ}$ [1].

Stretch Challenge · +25 XP, +10 coins

Prove It

In parallelogram $ABCD$, the diagonals meet at $M$. (a) Using alternate angles and congruent triangles, explain why $AM = MC$. (b) A particular parallelogram has $AC = 14$ cm and $BD = 18$ cm. Could it be a rectangle? Why or why not? (c) If you're told $AC = BD$, what kind of parallelogram must it be?

Reveal solution

(a) Triangles $AMB$ and $CMD$: $AB = CD$ (opp. sides), $\angle BAM = \angle DCM$ (alt. angles, $AB \parallel DC$), $\angle ABM = \angle CDM$ (alt. angles). So $\triangle AMB \equiv \triangle CMD$ (AAS), giving $AM = CM$. (b) NO, a rectangle has EQUAL diagonals, but $14 \ne 18$. So this is a non-rectangular parallelogram. (c) Equal diagonals in a parallelogram force all four angles to be $90^{\circ}$, so it must be a rectangle.

Quick Review

Parallelogram

2 pairs of parallel sides.

Opp. sides equal

$AB = DC$ and $AD = BC$.

Opp. angles equal

$\angle A = \angle C$, $\angle B = \angle D$.

Co-int. angles

Adjacent angles sum to $180^{\circ}$.

Diagonals bisect

They cut each other in half (but not always equal).

Rectangle extras

All angles $90^{\circ}$ AND equal diagonals.

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