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November 11, 2014
chapter 5
Based on work from pages 178-179, complete
In an isosceles triangle, the ___________ &
_________________ & ______________&
________________ drawn from the vertex
angle of an isosceles triangle are the _______!
5.1 Indirect proof.
G: DB
D
AC
F is the midpt. of AC
P: AD == CD
A
BF
C
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G: BD bisects <ABC,
<ADB is acute
P: AB = BC
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G:
ABC
P:
BCD > B
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draw median from A, through seg. BC, at M, such that AM = MP
What is true about
^ABM and ^PCM ?
what is true about <1, <3?
explain how the Prove
statement may be conclude.
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5.2 Proving that lines are parallel
The measure of an exterior angle of a triangle is greater than either
of the two remote interior angles.
Theorems 31-36
If two lines are cut by a transversal such that two
• alternate interior angles are congruent OR
• alternate exterior angles are congruent OR
• corresponding angles are congruent OR
• same-side interior angles are supplementary OR
• same-side exterior angles are supplementary
THEN the lines are parallel
If two coplanar lines are parallel to a third line then the lines
_______________
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E
G: <1 comp. to <2
C
<3 comp. to <2
P: CA // DB
D
1
2
A
3
B
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G: <1 supp. to <2
<3 supp. to <2
P: FLOR is a parallelogram
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5.3 Congruent angles associated with parallel lines
Through point P, how many lines are parallel to line k?
x + 2x
a // b, Find <1:
4x + 36
Look at the theorems numbered 37-44...
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G: FH // JM, <1 = <2
JM = FH
P: GJ = HK
K
F
2
J
H
1
M
G
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G: CY AY, YZ // CA
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C
Y
P: YZ bis. <AYB
A
Z
B
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THE famous crook problem
50 deg
x deg
132 deg.
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5.4 Four sided polygons
BE able to define the basic quadrilaterals as
described on page 236.
What does convex mean? Can you draw a convex polygon?
What does concave mean? Can you draw a concave
polygon?
examine carefully, what are
some properties?
examine carefully, what are
some properties?
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examine, list
properties
examine, list
properties
examine, list
properties
examine, list
properties
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examine, list
properties
13
find the area of the trapezoid
4
21
5
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A S N
1) a square is a rhombus
2) a rectangle is a square
3) a parallelogram has at least two sides parallel
4)the diagonals of a square are congruent
5)a trapezoid has at most two sides parallel
6)a kite is a trapezoid
7)the diagonals of a trapezoid are congruent
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5.5
Properties of quadrilaterals
Prove that (1) the opposite sides of a parallelogram
are congruent
(2)the opposite angles of a parallelogram are
congruent
(3) the diagonals of a parallelogram bisect each
other
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Prove that the diagonals of a kite are
perpendicular
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quadrilateral
rectangle
parallelogram
square rhombus
kite
isosc. trapezoid
trapezoid
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What am I ?
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5.6
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Proving that a quadrilateral is a parallelogram
B
given BCDF is a kite with BC=3x+4y,
CD=20, BF=12 and FD=x+2y, find the
perimeter.
F
C
D
Prove that if both pairs of opposite sides of a quadrilateral are
congruent, then it is a parallelogram.
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Prove that if the diagonals
of a quadrilateral bisect
each other then it is a
parallelogram
(x^5)(x^2)
(x-5)(x+5)
x^7
Show that the figure above is a parallelogram
(x^2-25)
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5.7 Proving that figures are special quadrilaterals
How do you prove that a figure is
>>Rectangle
parallelogram with at least one right angle
parallelogram with congruent diagonals
quadrilateral with 4 right angles
>>Kite
2 disjoint pairs of consecutive sides of quadrilateral are
congruent
1 diagonal is the perpendicular bisector of the other diagonal
>>Rhombus
parallelogram contains a pair of consecutive sides congruent
either diagonal of a parallelogram bisects two angles
the diagonals of a quadrilateral are perpendicular bisectors
of each other
>>Square
quadrilateral is both a rhombus and a rectangle
>>Isosceles Trapezoid
non-parallel sides of a trapezoid are congruent
lower or upper pair of base angles of a trapezoid are congruent
diagonals of a trapezoid are congruent
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G: AB // CD, <ABC
AB
B
<ADC
C
AD
P: ABCD is a rhombus
A
D
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E
G: FR bisects ED,
FE
RE
R
F
P: FRED is a kite
D
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Prove that the segments joining the midpoints of the sides of a
rectangle form a rhombus. Use coordinate geometry.
The distance formula is d= (x2-x1)^2 + (y2-y1)^2