Download 2 Nov 2015 Banking Day 9:15

2 Nov 2015 Banking Day 9:15 - 10:25
Geometry Agenda
1) Turn in 4-4 Proving Congruence SSS, SAS
Exercises starting on page 230
2) AA Similarity Postulate
3) Properties of Triangles
4) 7-4 Exercises (Homework)
Objectives
Individual :
Students more responsible for learning correcting - reflecting - teaching - editing
Academic:
Angle-Angle Similarity postulate for triangles
AA Similarity Postulate (p. 32)
In this worksheet, we are going to use
transformations to prove the Angle-Angle Similarity
Postulate
AA Similarity Postulate: If two angles of one
triangle are congruent to two angles of
another triangle, then the triangles are similar.
What’s the difference between similar
and congruent?
cont.
In our first unit, we learned that there are 3
transformations that maintain congruent figures.
1) translations T(l/r, u/d)
2) Reflections
rline of reflection
3) Rotations R(0, 90°) counterclockwise rotation of
90 degrees about the origin
cont.
We also learned that Dilations, though they do not
maintain congruent figures they do maintain
similar figures with congruent angles. So our proof
of the AA postulate would go something like this...
(on white board)
cont.
From our transformations, we know that E is on
top of B’’’ and F is ontop of C’’’. What postulate or
theorem that we know allows us to conclude that
the two triangles are congruent (making D on top
or A’’’)? Angle Side Angle postulate.
If this dilation caused the two triangles to be
congruent, then we have proved that if two angles
of one triangle are congruent to two angles of
another triangle, then a dilation exists that maps the
triangles onto each other. In other words.
Angle Angle Similarity Postulate
If two angles of one triangle are congruent to
two angles of another triangle, then the
triangles are similar.
Properties of Triangles
We can use what we know about similar
triangles to look at some other relationships
with triangles.
Let’s look at a triangle with a line inside it that is
parallel to another side.
(on white board)
So, for the picture above let’s assume that line
segment MN is parallel to line segment YZ. On
the picture above, mark the parallel lines and
the angles that we know are congruent.
Use the space below to draw triangle XMN and
triangle XYZ as two separate triangles.
Now, explain how you know that triangle XMN
is similar to triangle XYZ, using the fact that line
segment MN is parallel to line segment YZ.
Now, explain how you know that triangle XMN
is similar to triangle XYZ, using the fact that line
segment MN is parallel to line segment YZ.
Angle X is congruent to Angle X by reflexive
property
Angle X is congruent to Angle X by reflexive
property
Angle XMN is congruent to angle XYZ since
parallel lines cut by a transversal have
corresponding angles that are congruent.
So we know 2 angles are congruent in the
triangles and so by the Angle-Angle Similarity
postulate we can state that the triangles are
similar.
Angle X is congruent to Angle X by reflexive
property
Angle XMN is congruent to angle XYX since
parallel lines cut by a transversal have
corresponding angles that are congruent.
So we know 2 angles are congruent in the
triangles and so by the Angle-Angle Similarity
postulate we can state that the triangles are
similar.
Since the two triangles are similar, when we
look at the original picture, we know that line
segment MN divides the sides of the triangle
proportionally.
Triangle Proportionality Theorem: If a
line parallel to one side of a triangle
intersects the other two sides, then it
divides sides proporitionally.
Triangle Proportionality Theorem: If a
line parallel to one side of a triangle
intersects the other two sides, then it
divides sides proporitionally.
We can write:
MY = NZ
XM = XN
XM XN
MY NZ
Properties of Triangles Page 2
Practice
Use the triangle proportionality theorem, find the
values of the variables. Any lines that appear
parallel are indeed parallel.
6
4
X
8
Y
8 =
6
6
4
X
8
Y
8 = X
6
4
6
4
3
b
12
a
7-4 Parallel lines and Proportional
Parts
Exercises (p.411)
13 - 21 and 23.
Finish for homework, due next time.