notes 5.5-5.6 November 14, 2016
chapter 5
5.1 Indirect proof.
G: DB AC
F is the midpt. of AC
P: AD == CDA B C
D
F
Based on work from pages 178-179, complete
In an isosceles triangle, the ___________ &
_________________ & ______________&
________________ drawn from the vertex angle of an isosceles triangle are the _______!
notes 5.5-5.6 November 14, 2016
G: BD bisects <ABC,
<ADB is acute
P: AB = BC
notes 5.5-5.6 November 14, 2016
G: ABC
P: BCD > B
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.
notes 5.5-5.6 November 14, 2016
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.
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
_______________
Theorems 31-36
notes 5.5-5.6 November 14, 2016
A
B
C
D
E
1 2
3
G: <1 comp. to <2
<3 comp. to <2
P: CA // DB
notes 5.5-5.6 November 14, 2016
G: <1 supp. to <2
<3 supp. to <2
P: FLOR is a parallelogram
notes 5.5-5.6 November 14, 2016
notes 5.5-5.6 November 14, 2016
notes 5.5-5.6 November 14, 2016
5.3 Congruent angles associated with parallel lines
Through point P, how many lines are parallel to line k?
a // b, Find <1:x + 2x
4x + 36
Look at the theorems numbered 37-44...
notes 5.5-5.6 November 14, 2016
FH
G
K
M1
2J
G: FH // JM, <1 = <2
JM = FH
P: GJ = HK
notes 5.5-5.6 November 14, 2016
A B
C Y
Z
G: CY AY, YZ // CA
P: YZ bis. <AYB
notes 5.5-5.6 November 14, 2016
THE famous crook problem
50 deg
132 deg.x deg
notes 5.5-5.6 November 14, 2016
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?
notes 5.5-5.6 November 14, 2016
examine, list properties
examine, list properties
examine, list properties
examine, list properties
notes 5.5-5.6 November 14, 2016
examine, list properties
find the area of the trapezoid4
13
215
notes 5.5-5.6 November 14, 2016
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
notes 5.5-5.6 November 14, 2016
notes 5.5-5.6 November 14, 2016
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
notes 5.5-5.6 November 14, 2016
notes 5.5-5.6 November 14, 2016
Prove that the diagonals of a kite are perpendicular
notes 5.5-5.6 November 14, 2016
parallelogramsquarerectangle
kite
trapezoid
rhombus
quadrilateral
isosc. trapezoid
notes 5.5-5.6 November 14, 2016
What am I ?
notes 5.5-5.6 November 14, 2016
5.6 Proving that a quadrilateral is a parallelogram
given BCDF is a kite with BC=3x+4y,
CD=20, BF=12 and FD=x+2y, find the perimeter.
B
CF
D
Prove that if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.
notes 5.5-5.6 November 14, 2016
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
(x^2-25)
Show that the figure above is a parallelogram
notes 5.5-5.6 November 14, 2016
notes 5.5-5.6 November 14, 2016
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
notes 5.5-5.6 November 14, 2016
G: AB // CD, <ABC <ADC
AB AD
P: ABCD is a rhombus
A
B C
D
notes 5.5-5.6 November 14, 2016
G: FR bisects ED,
FE RE
P: FRED is a kiteF R
E
D
notes 5.5-5.6 November 14, 2016
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
notes 5.5-5.6 November 14, 2016
