Download ASA and SAS Postulates - Clark Magnet High School

26 Sept 2016 8:00 - 9:30 AM
Geometry Proposed Agenda
1) Bulletin - (remember you can always go to Clarkmagnet.net
to read the night before school)
2) G-CO Symmetries of a Quadrilateral
3) Rules of Transformations
4) SAS Postulate
5) ASA Postulate
6) Homework
G-CO Symmetries of a Quadrilateral I
Suppose ABCD is a quadrilateral for which
there is exactly one rotation, through an angle
larger than 0 degrees and less than 360
degrees, which maps it to itself. Further, no
reflections map ABCD to itself. What shape is
ABCD?
Compare what you have to your neighbor. Be
ready to share your results - with reasons.
Quadrilaterals:
Angles
angles
Quadrilaterals:
4 sides of equal length
2 pairs parallel sides
Opposite angles equal,
but not 90 degrees
2 reflection symmetries
1 rotation - 180 degrees
G-CO Symmetries of a Quadrilateral I
ABCD is a quadrilateral
one rotation maps to itself
No reflections
G-CO Symmetries of a Quadrilateral I
ABCD is a quadrilateral
one rotation maps to itself (180degree)
No reflections - no other symmetry for shape
G-CO Symmetries of a Quadrilateral I
ABCD is a
parallelogram
which is
neither a
rectangle nor a
rhombus.
G-CO Symmetries of a Quadrilateral II
There is exactly one reflection and no rotation
that sends the convex quadrilateral ABCD onto
itself. What shape(s) could quadrilateral ABCD
be? Explain.
Compare what you have to your neighbor. Be
ready to share your results - with reasons.
Convex quadrilateral?
Regular trapezoid
Line of symmetry
kite
Line of symmetry
Rules of Transformations
In previous sections we have
been working with three major
transformations:
Rules of Transformations
In previous sections we have
been working with three major
transformations: translations,
reflections and rotations.
Rules of Transformations
These three transformations
have one major thing in
common.
These three transformations
have one major thing in
common. They will ALWAYS
produce congruent figures.
These are often called rigid
motion transformations.
Rigid Motion
Refers to any transformation that
ALWAYS maintains congruent shapes.
Translations, Reflections and Rotations
will ALWAYS produce congruent figures.
So, these three are considered rigid
motions.
To understand this a little better,
let’s look at the two triangles below.
For this, we will assume that there is a
translation that exists the will map ABC
onto A’B’C’.
A
B
C
A’
B’
C’
What can we say about these 2
congruent triangles?
A
B
C
A’
B’
C’
Postulates
A postulate is something that is assumed
to be true based on reasoning or
observations. For example a fundamental
geometry postulate is written below:
“Between two points, there is exactly
one line.”
“Between two points, there is exactly
one line.”
We don’t need to get expert advice to
prove this is true. We can all agree that
there is only one straight line that exists
between two points. Because it is
something we can all agree on, it is known
as a postulate.
SAS Postulate
To increase our confidence in postulates
we will use in the future and to save time
then, we are going to use rigid motions
here to see how the SAS postulate works.
SAS Postulate
If two sides and the included angle of one
triangle are congruent to two sides and
the included angle of another triangle,
then the triangles are congruent.
Let’s start with the given information
of the postulate.
AB = DE
BC = EF and
∠ABC = ∠DEF.
Mark that on the figures.
AB = DE
BC = EF and
∠ABC = ∠DEF.
D
A
F
C
B
E
1. Translate triangle ABc so that point B maps
to point E. Draw below. Mark the map of A
as A’ and the map of C as C’.
C
D
A’
A
C’
B
F
E
1. Translate triangle ABc so that point B maps
to point E. Draw below. Mark the map of A
as A’ and the map of C as C’.
D
A’
C’
F
E
2. Rotate triangle A’EC’ with center E so that A’
maps onto D. Draw that below. Mark the map
of C’ as C”.
D
A’
C’
F
E
2. Rotate triangle A’EC’ with center E so that A’
maps onto D. Draw that below. Mark the map
of C’ as C”.
C’’
D
F
C’
E
2. Rotate triangle A’EC’ with center E so that A’
maps onto D. Draw that below. Mark the map
of C’ as C”.
C’’
D
F
E
3. For the transformation in 2, how can you be
certain that a rotation with center E would map
A’ exactly onto D?
C’’
D
F
E
3. For the transformation in 2, how can you be
certain that a rotation with center E would map
A’ exactly onto D?
Line segment AB is the same length as line
segment ED (congruent).
C’’
We have only used
D
rigid motions so we
have not changed any
F
lengths. They will fit
on top of each other.
E
On the previous page, you should have ended
up with a figure that looks like the one below.
Mark the corresponding parts of the two
triangles that are congruent based on our given
information and the fact that the two
transformations we did C’’
D
will maintain congruence.
F
E
In order to finish our “proof” investigation of the
SAS postulate, we will do a reflection through
line segment ED. If the two triangles are in fact
congruent the C” should be able to map directly
onto F because of this reflection.
C’’
D
F
E
How do you know that a reflection through DE
exists that maps C” onto F? Explain below.
D
F
E
How do you know that a reflection through DE
exists that maps C” onto F? Explain below.
We are reflecting across DE, A”B” will not
move, it will still be on DE, so that part of
the triangle stays in position.
D
F
E
Also angle ABC is the same as (congruent to)
angle DEF so the two edges of the angles will
map onto each other, so then B”C” will be on
top of EF which will also mean that C” had
D
mapped onto F.
All three parts of triangle
F
ABC have mapped onto
Triangle DEF.
E
As we were told,
the triangles are indeed congruent.
SAS Postulate
SAS
If two sides and the included angle of one
triangle are congruent to two sides and
the included angle of another triangle,
then the triangles are congruent.
ASA Postulate
ASA
If two angles and the included side of one
triangle are congruent to two angles and
the included side of another triangle, then
the triangles are congruent.
To do now:
Optional: Complete diagram and explain how to
show the two triangles are congruent.
(Complete page 1C-12).
Complete the homework problems 1- 6 on
pages 1C-13 and 14.
Draw more diagrams in the space at the bottom
of the page and mark in the congruent parts
CAREFULLY.