1. Lesson 7-1: Using Proportions
Ch 7
Similarity
Lesson 7-1: Using Proportions

2. Lesson 5-1: Using Proportions

3. Lesson 5-1: Using Proportions
Ratio
A ratio is a comparison of two numbers such as a : b.
Ratio:
When writing a ratio, always express it in simplest form.
Example: A B C D
3.6 6 8
4.8 10
Now try to reduce the fraction.
Type notes here
Lesson 5-1: Using Proportions

4. Lesson 5-1: Using Proportions
Example……….
A baseball player goes to bat 348 times and gets 107 hits. What is the players batting average?
Solution:
Set up a ratio that compares the number of hits to the number of times he goes to bat.
Ratio:
Convert this fraction to a decimal rounded to three decimal places.
Type notes here
Decimal:
The baseball player’s batting average is which means he is getting approxiamately one hit every three times at bat.
Lesson 5-1: Using Proportions

5. Lesson 5-1: Using Proportions
Proportion: An equation that states that two ratios are equal.
Terms
First Term
Third Term
Second Term
Fourth Term
Type notes here
To solve a proportion, cross multiply the proportion:
Extremes: a and d
Means : b and c
Lesson 5-1: Using Proportions

6. Proportions- examples….
84 yards
2 ft x
356 yards
Find the value of x.
Example 2:
Solve the proportion.
Type notes here
8x = 30
8 • x =
6 • 5
x = 3.75
8x = 30
Lesson 5-1: Using Proportions

7. Lesson 5-2: Similar Polygons

8. Lesson 5-2: Similar Polygons
Definition: Two polygons are similar if:
1. Corresponding angles are congruent.
2. Corresponding sides are in proportion.
Two polygons are similar if they have the same shape not necessarily have the same size.
Type notes here
The scale factor is the ratio between a pair of corresponding sides.
Scale Factor:
Lesson 5-2: Similar Polygons

9. Naming Similar Polygons
When naming similar polygons, the vertices (angles, sides) must be named in the corresponding order. P Q A B D C S R
Lesson 5-2: Similar Polygons

10. Lesson 5-2: Similar Polygonsz x 10 y 15 30 20 B C A D F G E H
Example-
The two polygons are similar.
Solve for x, y and z.
Step1: Write the proportion of the sides.
Step 2: Replace the proportion with values.
Step 3: Find the scale factor between the two polygons.
Note: The scale factor has the larger quadrilateral in the numerator and the smaller quadrilateral in the denominator.
Type notes here
Step 4: Write separate proportions for each missing side and solve.
Lesson 5-2: Similar Polygons

11. Lesson 5-2: Similar Polygons
Example:
If ABC ~ ZYX, find the scale factor from ABC to ZYX.
Scale factor is same as the ratio of the sides. Always put the first polygon mentioned in the numerator. 7 9 5 18 10 14 C A B Z Y X
The scale factor from ABC to ZYX is 2/1.
What is the scale factor from ZYX to ABC?
Lesson 5-2: Similar Polygons

12. Proving Triangles Similar
Lesson 7-3
Proving Triangles Similar
(AA, SSS, SAS)
Lesson 5-3: Proving Triangles Similar

13. AA Similarity (Angle-Angle)
If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.
Given:
and
Conclusion:
Lesson 5-3: Proving Triangles Similar

14. SSS Similarity (Side-Side-Side)
If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar. 5 11 22 8 16 10
Given:
Conclusion:
Lesson 5-3: Proving Triangles Similar

15. SAS Similarity (Side-Angle-Side)
If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar. 5 11 22 10
Given:
Conclusion:
Lesson 5-3: Proving Triangles Similar

16. Similarity is reflexive, symmetric, and transitive.
Proving Triangles Similar
Similarity is reflexive, symmetric, and transitive.
Steps for proving triangles similar:
1. Mark the Given.
2. Mark …
Shared Angles or Vertical Angles
3. Choose a Method. (AA, SSS , SAS)
Think about what you need for the chosen method and be sure to include those parts in the proof.
Lesson 5-3: Proving Triangles Similar

17. Lesson 5-3: Proving Triangles Similar
Problem #1
Step 1: Mark the given … and what it implies
Step 2: Mark the vertical angles AA
Step 3: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more?
Statements
Reasons C D E G F
Given
Alternate Interior <s
Alternate Interior <s
AA Similarity
Lesson 5-3: Proving Triangles Similar

18. Lesson 5-3: Proving Triangles Similar
Problem #2
Step 1: Mark the given … and what it implies
SSS
Step 2: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more?
Statements
Reasons
1. IJ = 3LN ; JK = 3NP ; IK = 3LP
Given
Division Property
Substitution
SSS Similarity
Lesson 5-3: Proving Triangles Similar

19. Lesson 5-3: Proving Triangles Similar
Problem #3
Step 1: Mark the given … and what it implies
Step 2: Mark the reflexive angles
SAS
Step 3: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Next Slide………….
Step 5: Is there more?
Lesson 5-3: Proving Triangles Similar

20. Lesson 5-3: Proving Triangles Similar
Statements
Reasons
G is the Midpoint of
H is the Midpoint of
Given
2. EG = DG and EH = HF
Def. of Midpoint
3. ED = EG + GD and EF = EH + HF
Segment Addition Post.
4. ED = 2 EG and EF = 2 EH
Substitution
Division Property
Reflexive Property
SAS Postulate
Lesson 5-3: Proving Triangles Similar

21. Lesson 5-4: Proportional Parts

22. Lesson 5-4: Proportional Parts
Similar Polygons
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. D E F A B C
Lesson 5-4: Proportional Parts

23. Lesson 5-4: Proportional Parts
Side Splitter Theorem
If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional length. 1 2 3 4 A B C D E
Converse:
If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.
Lesson 5-4: Proportional Parts

24. Lesson 5-4: Proportional Parts
Examples………
Example 1:
If BE = 6, EA = 4, and BD = 9, find DC. A B C D E 6 4 9 x
6x = x = 6
Example 2:
Solve for x.
4x + 3 9 A B C D E
2x + 3 5
Lesson 5-4: Proportional Parts

25. Lesson 5-4: Proportional Parts
Midsegment Theorem
A segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. R M L T S
Lesson 5-4: Proportional Parts

26. Lesson 5-4: Proportional Parts
Extension of Side Splitter
If three or more parallel lines have two transversals, they cut off the transversals proportionally.
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. F E D A B C
Lesson 5-4: Proportional Parts

27. Lesson 5-4: Proportional Parts
Forgotten Theorem
An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. C A B D
Lesson 5-4: Proportional Parts

28. If two triangles are similar:
(1) then the perimeters are proportional to the measures of the corresponding sides.
(2) then the measures of the corresponding altitudes are proportional to the measure of the corresponding sides..
(3) then the measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides.. B C A E F D H G I J
Lesson 5-4: Proportional Parts

29. Lesson 5-4: Proportional Parts
Example:
Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4. Find the perimeter of ΔDEF.
The perimeter of ΔABC is = 60.
Side DF corresponds to side AC, so we can set up a proportion as: A B C D E F 15 20 25 4
Lesson 5-4: Proportional Parts