Name Date

Geometry Pd. 7-7 Notes

Lesson 7-7 quadrilateral Proofs with Parallelograms and Rectangles

Today's Goal: How can we use our knowledge of the properties of rectangles to prove quadrilaterals to be rectangles?

Example 1:

Given: ABCD is a quadrilateral withDB drawn. ITU<DBA <BDC

AB CD

Prove: ABCD is a parallelogram

I) How is this proof different than the coordinate geometry parallelogram proofs we have practiced in this unit?

2) Is there anything missing that would help us be successful?

3) What do you expect this proof to look like (in structure)?

Z columøs

4) How can you prove a quadrilateral is a parallelogram?

stdQs

Let'S PO this'

oo

Let's Try it! Example 1: Diagram:

Given: ABCD is a quadrilateral withDB drawn.

<DBA <BDC

AB CD cprqve: ABCD is a parallelogram

a) Which property of a parallelogram will make sense to prove here?

Before we begin)

We must draw and then

mark our diagram'

Tips for drawing:

-draw

opp sìck2sb) Plan: SAS B CC

Statements

l. ABCD is a quadrilateral with DB drawn, <D

BDC,AB CD

CC

Given:

2. One More! You take this one!

prove:

Statements

1. XYRS is a rectangleM is the mid oint of YR

2.

4. YR L SR and YR XY

SCu8,

Reasons

s)

XYRS is a rectangle

M is the midpoint of YR

XM SM

Reasons1. Given

x

s

2. Rectangles have two sets of opposite

con ruent sides3.

4. Adjacent sides of rectangles are

5. Definition of Per endicular lines

6. All ri htan les are con ruent

7. SAS8. uc-y c

DateName

Class work 7-3 Page BGeometry Pd.

Let's Practice!

Remember, we need to know our parallelogram and rectangle properties need to know how to prove that

quadrilaterals are parallelograms or rectangles.

Don't forget to label!!

Also, if a given piece of information is that we have a "quadrilateral" it does not mean we have any parallelogram

properties yet!

S1) Given: PQRS isa quadrilateral 2

3

4

1

Prove: PQRS is a parallelogram

MetihGJ

svoc) t,

sell f'œs, hsn

cecTC

(z seb ov

In rectangle QRST, diagonals CS .

Prove that -RTS

s

Statements Reasonsl. Rectangle QRST, diagonals QS, RT

2. QTZRS 2.

3. TSZTS 3.

4. QS=RT 4.

S. AQTS ARST 5.

6. zQST CRTS 6.

3. parallelogram ABCD s— diagonals AC and BD intersect at E.

C

Prove: acro ZCAB

4) Given. PQRS is a parallelogram pThink transitive property'

Prove: ZS=ZTs

S

l, PQRS

bOSCDlLS

3, ðPæ

kQRË&S

Procfh

Problem 1-There are multiple ways to this proof:-Either start by proving that ASRP z- AQPR, and then showing both pairs of opposite sides congruent.

Or

-Show how alternate interior angles give us 2 pairs of opposite sides parallel.

Problem 2-What 2 overlapping triangles should you prove congruent based on the prove statement?

-What do you know about the opposite sides ofa rectangle?-What do you know about the diagonals?

Start by marking your diagram with the parallelogram properties.-What type of triangle is AQRT?

-What do you know about opposite angles in parallelograms?

-We must show that <S <QRT (think about the parallel lines), then we must use the transitive property

at some point!