Geometry -Chapter 5 Parallel Lines and Related Figures

5.1 Indirect Proof:We’ve looked at several different ways to write proofs. We will look at indirect proofs. An indirect proof is usually helpful when a direct proof would be difficult to use. Example: A D

B C E FGiven:

�

∠A

�

≅

�

∠D, AB

�

≅ DE, AC

�

≅ DFProve:

�

∠B

�

≅

�

∠ EProof: Either ____

�

≅ ____ or _____

�

≅______

Assume _______________.

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

Indirect-Proof Procedures

1. List the __________________ for the conclusion.

2) Assume the ______________ of the desired conclusion is correct.

3) Write a chain ____________until you reach an ______________.

This will be a_______________ of either

a) ________________or b) a theorem, definition or other known

fact.

4) State the remaining ____________ as the desired conclusion.

P 2

Given: RS⊥ PQ S

PR≅ QR R

Prove: RS does not bisect

�

∠ PRQ Q

Either _____________________ or _______________________

Assume____________________________________

Then________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

Homework #2 J

Given: P is not the midpoint of HK

HJ

�

≅ JK

Prove: JP does not bisect

�

∠ HJK H P K

Either _____________________ or _______________________

Assume____________________________________

Then________________________________________________

____________________________________________________

____________________________________________________

___________________________________________________-

_________________________

3

Homework #5

Given: • O

OB is not an altitude o

Prove: OB does not bisect

�

∠AOC A C

B

Either _____________________ or _______________________

Assume____________________________________

Then________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

________________________

Homework # 6

ODEF is a square

In terms of “a”, find y axis

a)Coordinates of F and E

b)Area of Square F E

c) Midpoint of FD

d) Midpoint of OE

o (0,0) D ( 2a , 0) x axis

4

5.2 Proving That Lines Are Parallel:

The exterior angle of a triangle is formed whenever a side of the triangle is extended to form an angle supplementary to the adjacent interior angle.

adjacent exterior interior angle angle

remote interior angles Theorem #30 The measure of_________________________

____________________________________________________

Theorem # 31 If two lines are cut by a transversal_________

____________________________________________________

____________________________________________________( short form: alt inter

�

∠ ’s

�

≅ ⇒ ll lines)

Theorem #32 If two lines cut by a transversal such_________

____________________________________________________

____________________________________________________

( short form: alt ext

�

∠ ’s

�

≅ ⇒ ll lines )

Theorem #33 If two lines are cut by a transversal such ______

____________________________________________________

____________________________________________________

( short form: corres.

�

∠ ’s

�

≅ ⇒ ll lines )

Theorem # 34 If two lines are cut by a transversal such ______

____________________________________________________

____________________________________________________

5

Theorem # 35 If two lines are cut by a transversal such ______

____________________________________________________

____________________________________________________

Theorem # 36 If two coplanar lines are __________________

____________________________________________________

Pg 219

1a) 1b) 1c)

2a) 2b) 2c)

3) list all the pairs of angles that will prove a ll b.

__________________________________________________

6) Q D

Given: QD ll UA 1

Prove :

�

∠ 1

�

≅

�

∠2

2

U A

Either _____________________ or _______________________

Assume____________________________________

Then________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

6

#11

Complete the inequality that shows the restrictions on x.

_______< x < _______

xº

110º

5.3 Congruent Angles Associated with Parallel Lines:

In this section we shall see the converses of many of the theorems in Section 5.2 are also true.

“Parallel Postulate “ #8 Through a point not on a line there is exactly one parallel to the given line.

We will assume that the Parallel Postulate is true. In this section we will learn that the converse is true- that is if we start with parallel lines, then we can conclude that alternate interior angles are congruent. In fact many pairs of congruent angles are determined by parallel lines cut by a transversal.

Theorem # 37 If two parallel lines are cut by a transversal, each

pair of alternate interior angles are congruent.

( Short Form: ll lines ⇒alt int

�

∠ ’s

�

≅)

7

Theorem # 38 If two parallel lines are cut by a

transversal, then any pair of the angles formed are either congruent

or supplementary.

a ll b , and let x be the measure of any one of the angles

line a

xº

line b

Find all the measures of the other seven angles , algebraically, based on

the Theorem #38.

Theorem # 39 If two parallel lines______________________

____________________________________________________

( Short Form:_________________________ )

Theorem # 40 IF two parallel lines are cut________________

____________________________________________________

( Short Form:_________________________ )

Theorem # 41 If two parallel lines are cut by a transversal_____

____________________________________________________

8

Theorem # 42 If two parallel lines are cut by a transversal, each

pair of ______________________________________________

________________________________________

Theorem # 43 In a plane, if a line is _______________ to one

of the _________________, it is _______________ to the other.

Create a drawing that shows this theorem. Then state the givens that

support it.

Given

Can prove_______

Theorem # 44 If two lines are ____________ to a______line,

they are _______________ to each other. ( Transitive Property of

______________lines.)

Summary of if two parallel lines are cut by a transversal, then :

• each pair of _________________ angles are congruent

• each pair of _________________ angles are congruent

• each pair of _________________ angles are congruent

• each pair of _________________ angles on the same side of the

transversal are __________________

• each pair of _________________ angles on the same side of the

transversal are __________________

9

Given AB

�

≅ DC A B

AB ll DC

Prove: AD

�

≅ BC

D C

Statement Reason

Given EF ll GH E G

EF

�

≅ GH J

Prove EJ

�

≅ JH F H

Statements Reasons

10

Given: a ll b , 30º angle as shown- find the seven remaining angles

30º

Homework # 4 R S

Given

�

∠5

�

≅

�

∠ 6 5 6

RS ll NP

Prove:

�

Δ NPR is isosceles N P

Statements Reasons

Homework # 5 ( 2x + 5 ) º

Given: a ll b 1

Find m

�

∠ 1 (3x – 13)º

Do “Crook Problem” Pg 229 #4

11

5.4 Four-Sided Polygons

Polygons are _________ figures. The following are example of

polygons. Decide which are convex and which are not convex.

A B C d

Why is PLAN not a polygon? P

N

L A

Look at bottom pg. 234 and top of pg. 235 to help you.

Define convex polygons. Be complete and specific________________

________________________ _________________________

________________________ _________________________

How do we name polygons?_________________________________

____________________________________________________

Definition # 42 A convex polygon is a polygon in which each

interior angle has a measure less than 180º

Explain why polygon “C “ above is not convex.___________________

__________________________________________________

Diagonals of Polygons- Draw all the diagonals in each polygon.

12

Definition # 43 A diagonal of a polygon is any segment that

connects two nonconsecutive ( nonadjacent) vertices of the polygon.

A quadrilateral

A parallelogram

A rectangle

A rhombus

A kite

A square

A trapezoid

An isosceles trapezoid

13

Turn to page 237-238 Do problems # 1-3 below

1__________________________________________________

2___________________________________________________

3___________________________________________________

7) In the isosceles trapezoid shown ST ll RV

Name: S T

a) The base__________

b) The diagonals___________

c) The legs_____________ R V

d) The lower base angles___________

e) The upper base angles______________

f) All pairs of congruent alternate interior angles_______________

8) Write S-sometines, A-always, N-never for each statement below.

a) A square is a rhombus.__________________________________

b) A rhombus is a square.__________________________________

c) A kite is a parallelogram.________________________________

d) A rectangle is a polygon.________________________________

e) A polygon has the same number of vertices as sides.____________

f) A parallelogram has three diagonals.________________________

g) A trapezoid has three bases._____________________________

14

10) Cut out any size rectangle. Then listen for more directions.

Explain how the formula for a rectangle can be used to find the formula

for a parallelogram.

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

11) If the sum of the measures of the angles of a triangle is 180º, what

is the sum of the measures of all the angles in

a) a quadrilateral__________ b) a pentagon______________

5.5 Properties of Quadrilaterals

Get out the long paper we have begun and we are going to add the

characteristics of parallelograms, rectangles, rhombus, kites, squares,

and isosceles trapezoids.

15

Homework #1 D C

Given ABCD ( is a rectangle)

Conclusion: Δ ABC ≅ Δ CDA

A B

Statement Reason

Homework #2 J H

Given : EFHJ

∠1 ≅ ∠2 K G

Conclusion: KH ≅ EG E 2 F

Statements Reasons

1)EFHJ is a rectangle 1)

2) ∠ J≅ ∠F 2)

3) JH ≅ EF 3)

4) ∠1≅ ∠2 4)

5) Δ KJH≅ Δ GFE 5)

6)KH≅ EG 6)

16

Homework #3 S R

Given: Rectangle MPRS

MO ≅ PO

M O P

Prove: Δ ROS is isosceles

Statement Reason

Homework #4 D C Given: ABCD F

AE ≅ CF

Conclusion: DE≅ BF A E B

Statement Reason

17

5.6 Proving that a Quadrilateral is a Parallelogram

1) If both pairs of _________________ of a quadrilateral are

___________, then the quadrilateral is a ________________(reverse of the definition)

2) If ___________ of opposite sides of a _________________ are

______________, then the _______________ is a parallelogram.(converse of a property)

3) If ___________ of opposite sides of a quadrilateral are both

_________________ and _________________, then the

quadrilateral is a parallelogram.

4) If the _______________ of a quadrilateral _______each other,

then the quadrilateral us a ________________.( converse of a property)

5) If _______ pairs of _____________________ of a quadrilateral

are ______________, then the quadrilateral is a parallelogram,

( converse of a property)

Homework #1 ( Look at the diagrams in your book pg. 251)

a)

b)

c)

d)

e)

18

Homework #2 T V

Given: ∠ XRV ≅ ∠ RST

∠ RSV≅ ∠TVS

Conclusion: RSTV is a S R X

Statements Reasons

Homework #3 S R

Given • O O

Conclusion: SMPR is a M P

Statements Reasons

19

5.7 Proving That Figures Are Special Quadrilaterals

Proving that a quadrilateral is a rectangle E H

1)

F G

2)

3)

Proving that a quadrilateral is a kite. K

1) E J

2)

T

Proving that a quadrilateral is a rhombus. J O

1) K M

2)

3)

20

Proving that a Quadrilateral is a Square. N S

1)

P R

Proving that a Trapezoid is Isosceles. A D

1)

B C

2)

3)

Problem # 6

a) A quadrilateral with diagonals that are perpendicular bisectors of

each other._________________

b) A rectangle that is also a kite.____________

c) A quadrilateral with opposite angles supplementary and consecutive

angles supplementary.________________

d) A quadrilateral with one pair of opposite sides congruent and the

other pair of opposite sides parallel.__________________

21

Problem #13

What is the most descriptive name for each quadrilateral below?

a c e g

b d f h

8 10

Problem #17

a) If a quadrilateral is symmetrical across both diagonals, it is a

_____________?

b) If a quadrilateral is symmetrical across exactly one diagonal, it is

a ___________?

c) Which quadrilateral has four axes of symmetry?_______________

Homework #1

Locate points Q=( 2,4), U=( 2,7), A=( 10,7), and D=(10,4) on a graph.

Then give the most descriptive name for QUAD._______________

Turn to your book and lets begin # 2 – 5 on your own paper.