geometry -chapter 5 parallel lines and related … -chapter 5 parallel lines and related figures ......
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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.