1. Chapter Seven Similar Polygons
Ruby Weiner & Leigh Zilber

2. 7.1 Ratios and Proportions
Ratio: the quotient of 2 values D A 60 60 10 E 90 90
Find the ratio of AE to BE
10: 5x 2:x
Find the ratio of largest > of triACE to smallest > of triBDE
90:30 3:1 30 C 5x 30 B

3. Ratio Practice Problems
A telephone pole 7 meters tall snaps into 2 parts. The ratio of the 2 parts is 3:2. Find the length of each part.
A teams best hitter has a life time batting average of He has been at bat 325 times.
how many hits has he made?

4. workout 1) 3x + 2x = 7 5x = 7 --> 7/5 3(7/5) = 21/5 meters

5. 7.2 Properties of Proportions
Proportion: equation stating that 2 ratios are equal
Properties (given a/b = c/d) :
b/a = d/c
ad = bc
a/c = b/d
a+b/b = c+d/d
examples: (given a/b = 3/5)
1. 5a = 3b
2. 5/b = 3/a
3. a+b/b = 3+5/5 --> 8/5
4. 5/3 = b/a

6. Proportion Practice Problems
Choose yes or no
given: 10/20 = a/b
is 10 x b = 20 x a ?
is 10/20 = b/a ?
is 30/20 = a+b/b ?
is 20/10 = b/a ?
10/a = b/20?

7. ANSWERS YES b.c ad = bc NO b.c a/b no= d/c YES b.c a+b/b = c+d/d
YES b.c b/a = d/c
NO b.c a/c no= d/b

8. 7.3 Similar Polygons -Corresponding angles are congruent
recall congruent triangles
corresponding angles --> congruent
corresponding sides --> congruent
Similar triangles B E
-Corresponding angles are congruent
-Corresponding sides are in proportion
-AB/DE = BC/EF = AC/DF A C D F

9. Examples: find length of EF if triABC is similar to triDEF2 4 4 x A C 3 D F 6
2/4 = 4/x
2x = 16
x = 8

10. 7-4 A Postulate for Similar Triangles
Postulate 15: AA Similarity Postulate
- If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Example: Are these triangle similar? How?
Conclude: yes, AA Similarity (AA~) B A C

11. Practice Problems Determine if the triangles are similar and how. 1)
2) Given: Both Triangles are Isosceles 50 40 5 5 5 5

12. Answers
x = x = 50
x = 180 x = 40
they are similar by AA similarity

13. 7-5 Theorems for Similar Triangles
Theorem 7-1 (SAS Similarity Theorem)
If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.
Example: Are these
Triangles congruent?
Why?
Answer: Yes, SAS~ A 1 x y B C F E
Given: Angle A is congruent to Angle D
AB/DE = AC/DF

14. 7-5 Continued Theorem 7-2 (SSS Similarity Theorem)
If the sides of two triangles are in proportion, then the triangles are similar.
Example:
Answer: Yes, SSS~ A D 1 Y X
Given:
AB/DE = BC/EF = AC/DF C E F B

15. Practice Problems 1. 2. E 10 6 C B 15 D 9 A P L 65 8 5 O 7.5 N 65
K M

16. Answers 1. Triangle BAC ~ Triangle EDC; SAS~
2. Triangle LKM ~ Triangle NPO; SAS~

17. 7-6 Proportional Lengths
Theorem 7-3 (Triangle Proportionality Theorem)
If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
Example:
Find the numerical value of
A) TN/ NR
B) TR/NR
C) RN/RT R 6 M N
Answer:
tn/nr = sm/mr = 3/6 = ½
Tr/nr = sr/mr = 9/6 = 3/2
Rn/rt = rm/rs = 6/9 = 2/3 3 S T

18. 7-6 Continued
Corollary
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Theorem 7-4 (Triangle Bisector Theorem)
If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.

19. 7-6 Continued Example: F G D E2 1 D E 3
Given: Triangle DEF; Ray DG bisects Angle FDE
Prove: GF/ GE = DF/DE
Answer: (Plan for Proof)
Draw a line through E parallel to Ray DG
And intersecting Ray FD at K.
Apply Triangle Proportionality Theorem
To Triangle FKE. Triangle DEK is isosceles
With DK = DE. Substitute this into your
Proportion to complete the proof. 4 K

20. Practice Problems
State a proportion for the diagram: n a g b

21. Answer
1. a/n = b/g

22. Practice Proof Given: Angle H and Angle F are right triangles
Prove: HK * GO = FG * KO K
Statements
Reasons H 1 O 2 G F

23. Answer to Proof Statements Reasons Angle 1 is congruent to Angle 2.
Angle H and Angle F are right Triangles.
Angle H = 90 and Angle F = 90
Angle H is congruent to Angle F.
Triangle HKO ~ Triangle FGO
HK/FG = KO/GO
HK*GO = FG*KO
Vertical Triangles are congruent.
Given
Def. of right triangle.
Def. of congruent triangle
AA~
Corr. Sides of ~ Triangles are in proportion.
A property of proportions.