OVERVIEW Using Similarity and Proving Triangle Theorems G.SRT.4

G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. / This objective introduces what is sometimes known as the ‘side splitting theorem’ or proportional parts theorem. Look at the impact that parallel lines have on dividing the lines up proportionally. /
Proportional values can be found in other places in the triangle when parallel lines are formed.
(1) The student will prove (the side splitting theorem) that a line parallel to one side of a triangle divides the other two proportionally.
(2) The student will prove (the angle bisector theorem) that an angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.
(3) The student will prove the Pythagorean Theorem using similarity and the geometric mean. / Focus on student outcome 1 more than 2. The angle bisector has very limited use outside of its own basic problems. Emphasize the side splitting theorem and its use in multiple settings. / 1 – The most common error in this unit is found in this objective –Students often form a scale factor between two proportional pieces and then relate it to two proportional sides. These two scale factors are not equal and this causes many errors.
2 – The angle bisector theorem while quite pretty to prove and quite easy to use has very little application beyond itself. The main focus here is the ‘side splitting theorem.’

g.srt.4 UsingSimilarityProvingTriangleTheoremsPage 7 of 74/29/2014

NOTES Using Similarity and Proving Triangle Theorems G.SRT.4

CONCEPT 1 – Prove theorems about triangles – The Side Splitting Theorem - A line parallel to one side of a triangle divides the other two proportionally.

This relationship is sometimes called the ‘Side Splitting” theorem.

Given:
Prove: /
Given that
by Corresponding Ð @
by Corresponding Ð @
Thus DADE ~ DABC by AA. /
Similarity brings proportional sides….

/ Express the distance AB as the sum of its two pieces,
AB = AD + DB (using segment addition)
and AC is the sum of its two pieces,
AC = AE + EC (using segment addition)
So by substitution, it follows that:

AD (AE + EC) = AE (AD + DB) by cross multiplication
AD(AE) + AD(EC) = AE(AD) + AE(DB) by distribution
Subtract AD(AE) from both sides leaving
AD(EC) = AE(DB)

CONCEPT 2 – Prove theorems about triangles – The Angle Bisector Theorem – An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

Given: DABC where is an angle bisector of ÐB.
Prove: /
Create an auxiliary parallel line to through Point A while also extending side until the two meet at point E. /
because Corresponding Angles @, and
because Alternate Interior Angles @ /
EB = AB because of the Isosceles Triangle Theorem
Thus using the side splitting theorem
and then using substitution .

CONCEPT 3 – Prove theorems about triangles. Prove the Pythagorean Theorem using triangle similarity.

Given: A right triangle with an altitude (height) draw from the right angle to the hypotenuse.
Prove: /
When comparing the left inner triangle to the entire triangle, notice that both triangles have a common angle and both have a right angle. Thus they are similar by AA.
Knowing that they are similar allows for the establishment of the proportion between the sides….
/
When comparing the right inner triangle to the entire triangle, notice that both triangles have a common angle and both have a right angle. Thus they are similar by AA.
Knowing that they are similar allows for the establishment of the proportion between the sides….
/
Now the pretty part!!!
It is now known that
and that
The Pythagorean Theorem is beginning to appear…. Next, by adding both equations together.

Very close to the Pythagorean Theorem… Just factor out a c.
from the diagram notice that length c = d + e and substitute that in….. WOW!!!

g.srt.4 UsingSimilarityProvingTriangleTheoremsPage 7 of 74/29/2014

ASSESSMENT Using Similarity and Proving Triangle Theorems G.SRT.4

1. In the given diagram, which of the following statement is NOT true:
A) ÐADE @ ÐACB B) ÐADE @ ÐAED
C) D) /
2. In the given diagram, which of the following statement is NOT true:
A) B)
C) D) /
3. Which of the following would not solve for the correct value of x:
A) B)
C) D) /

4. is the same proportion as . T or F

5. Tim claims that because DADE ~ DACB, that the proportion is valid.
Jennifer disagrees with Tim. Who is correct? Why?
If Jennifer is correct, rewrite the proportion so that it is true. /
6. Complete the proportions.
a) / b) / c) /
/ /
d) / e) / f)
/ /
7. Find the values for the missing variables.
a) / b) / c) / d)
/ / /
x = ______/ x = ______/ x = ______/ x = ______
8. Determine the values for the variables, x and y. /

Answers:

1)  B

2)  B

3)  A

4)  T

5)  Jennifer. The two similar triangles are DADE ~ DACE, which makes the correct proportion. The error that Tim made was that the measurements were all full lengths of sides except, it is only a portion of the side thus is an incorrect proportion.

6)  a) EB

b) AF

c) AB

d) AC

e) EB

f) CG

7)  a) x = 6.4

b) x = 11.25

c) x = 4

d) x = 16

8)  so x = 9.6. Use the Pythagorean Theorem to solve for y. therefore m = 26. Finally to solve for y using, results in y = 15.6.

g.srt.4 UsingSimilarityProvingTriangleTheoremsPage 7 of 74/29/2014