Download Geometry Fall 2011 Lesson 17 (S.A.S. Postulate)

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Lesson Plan #61
Date: Monday February 27th, 2017
Class: Geometry
Topic: Similar Polygons
Aim: What are the properties of similar polygons?
Objectives:
1) Students will students will know the properties of polygons that are similar.
HW # 61:
Pg. 250 #’s 2-26 (Even number exercises only)
Do Now
1)
B
E
D
2) If < 𝐵𝐷𝐸 ≅< 𝐵𝐴𝐶, prove that 𝐵𝐸: 𝐸𝐶 = 𝐵𝐷: 𝐷𝐴
C
Statements
A
Reasons
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Assignment #1: Examine the two polygons at right. What can you state about the
two polygons?
Assignment #2: Examine the two pairs of polygons at below. How do these polygons compare to the pair in assignment #1?
What makes each pair of polygons in assignment #2 dissimilar?
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Definition: Two polygons are similar if their vertices can be paired so that
1) Corresponding angles are congruent
2) Corresponding sides are in proportion
The symbol for similarity is ~.
What is the ratio of the lengths of any two corresponding sides in the similar
polygons at right?
Definition: The ratio of similitude of two similar polygons is the ratio of the lengths of any two corresponding sides; sometimes
also referred to as the scale factor.
Question:
Is similarity of polygons an equivalence relation? Why?
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Online Interactive Activity: Let’s go to
http://www.mathopenref.com/similartriangles.html
Online Interactive Activity: Let’s go to
http://www.mathopenref.com/similaraaa.html
Theorem:
Two triangles are similar if two angles of one triangle are congruent to two corresponding angles of the other.
Example #1:
Statements
Example #2:
Example #3:
Reasons
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Example #4:
̅̅̅̅
Given: ∆𝐴𝐵𝐶 and ∆𝐷𝐵𝐸 ̅̅̅̅
𝐷𝐸 ∥ 𝐴𝐶
Prove: ∆𝐴𝐵𝐶~∆𝐷𝐵𝐸
Statements
Reasons
Theorem: A line that is parallel to one side of a triangle and intersects the other two sides in different points cuts off a triangle
similar to the given triangle.
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Proof of the Angle Angle Similarity Theorem:
Given:
ABC and DEF with A  D
and B  E
Prove:
ABC
1)
2)
3)
DEF
Statements
DEF
A   D , B   E
C  F
ABC and
4) Let G be a point on DE so that
AB  DG
( s.  s.)
5) Let H be a point on DF so that
AC  DH
( s.  s.)
ABC  DGH
B  G
E  G
9) GH EF
6)
7)
8)
DG DH

DE DF
11) AB  DG , AC  DH
AB AC

12)
DE DF
10)
…Do same to produce proportion with the other side
If enough time:
1)
Reasons
1) Given
2) Given
3) If two angles of one triangle are congruent, respectively to
two angles of another triangle, then the third angles are
congruent. (1, 2)
4) A line segment may be copied.
5) A line segment may be copied
6) s.a.s.  s.a.s
(4.2,5)
7) Corresponding parts of congruent triangles are congruent (6)
8) Transitive property of congruence (2,7)
9) If two coplanar lines are cut by a transversal forming
congruent corresponding angles, then the lines are
perpendicular. (8)
10) If a line is parallel to one side of a triangle and intersects the
other two sides, then line divides the sides proportionally. (9)
11) Definition of congruent line segments. (4,5)
12) Substitution Postulate (10, 11)
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3)
4)
5)