Download Geometry Fall 2015 Lesson 058 _Proportions involving line

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Lesson Plan #58
Class: Geometry
Date: Friday February 26th, 2016
Topic: Proportions involving line segments
Aim: How do we solve proportions involving line segments?
Objectives:
1) Students will be able to solve proportions involving line segments in a triangle.
HW #58:
Pg. 272 #’s 3-11
Do Now
̅̅̅̅ .
2) In ∆𝐴𝐵𝐶, ̅̅̅̅
𝐷𝐸 ∥ 𝐵𝐶
Set up the ratio of 𝐴𝐷 𝑡𝑜 𝐷𝐵.
Set up the ratio of 𝐴𝐸 𝑡𝑜 𝐸𝐶.
What do you notice about those two ratios?
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Theorem (See Page 5): If a line is parallel to one side of a triangle and intersects the
other two sides, the line divides those sides proportionally.
Let’s list the proportions that can be set:
Example #1:
̅̅̅ intersecting 𝑅𝑆
̅̅̅̅ in 𝐾 and 𝑅𝑇
̅̅̅̅ in 𝐿. If 𝑅𝐾 = 5 𝑖𝑛., 𝐾𝑆 = 10𝑖𝑛., and 𝑅𝑇 = 18𝑖𝑛.,
In triangle RST, a line is drawn parallel to ̅𝑆𝑇
find 𝑅𝐿.
Note: In the interest of time and space, we’ll accept the converse of the above theorem without proof.
Postulate: If a line divides two sides of a triangle proportionally, the line is parallel to the third side
Example #2:
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Assignment #1:
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Assignment #2:
̅̅̅̅ is drawn ∥to 𝐵𝐶
̅̅̅̅ .
𝐺𝐻
What is the ratio of 𝐴𝐷 to 𝐷𝐺?
What is the ratio of 𝐴𝐸 to 𝐸𝐻?
G
H
What is the ratio of 𝐷𝐺 to 𝐺𝐵?
What is the ratio of 𝐸𝐻 to 𝐻𝐶?
What can we say if three parallel lines intersect two transversals?
Theorem: If three parallel lines intersect two transversals, then they divide the
transversals proportionally
Example:
Find the value of x in the figure at right.
Find the value of x in the figure at right
Online Activity:
http://www.mathwarehouse.com/geometry/similar/triangles/side-splitter-theorem.php
Online Activity:
http://www.geogebra.org/en/upload/files/english/nebsary/AngleBisectorTheorem/AngleBisectorFinal.html
Theorem:
If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other side.
Online Activity:
http://www.mathwarehouse.com/geometry/similar/triangles/angle-bisector-theorem.php
Proof – but need to know similar triangles first. https://www.khanacademy.org/math/geometry/triangleproperties/angle_bisectors/v/angle-bisector-theorem-proof
Assignment #3:
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If enough time:
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Proof of Theorem – If a line is parallel to one side of a triangle and intersects the other two sides, the line divides those sides
proportionally
Given:
ABC with DE || BC
Prove:
Statements
1.
ABC with DE || BC
2.  1  2 ,
 3  4
3.
ADE
ABC
4.
AD AE

AB AC
Reasons
1. Given
2. If 2 parallel lines are cut by a transversal, the corresponding
angles formed are congruent. (1)
3. If two angles of one triangle are congruent to two angles of
another triangle, then the two triangles are similar. (2)
4. Corresponding sides of similar triangles are in proportion.
(3)
5. AB  AD  DB
5. Partition Postulate
6. AC 
6. Partition Postulate
AE  EC
AD
AE

AD  DB AE  EC
8. AD( AE  EC )  AE ( AD  DB)
7.
9.
AD  AE  AD  EC  AE  AD  AE  DB
10. AD  AE  AD  AE
7. Substitution Postulate (4, 5, 6)
8.Multiplication Postulate (7)
9. Distributive Property of multiplication over addition (8)
10. Reflexive Property of equality
11.
AD  EC  AE  DB
11. Subtraction Postulate (9, 10)
12.
EC  DB  EC  DB
12. Reflexive Property of equality
13.
AD AE

DB EC
13. Division Postulate (11, 12)