Download Cornell Notes-Chapter 6 - Kenwood Academy High School

Honors Geometry
Chapter 6: Proportions and Similarity
Objective:
6.1 Proportions
-write ratios
-use properties of proportions
ratio
A ratio is a comparison of two quantities. The ratio a to b, where b is not
a
zero, can be written as or a:b.
b
Examples a. In 2002, the Chicago Cubs baseball team won 67 games out of 162.
Write a ratio for the number of games won to the total number of games played.
b. A doll house that is 15 inches tall is a scale model of a real house
with a height of 20 feet. What is the ratio of the height of the doll house to the
height of the real house?
proportion
A statement that two ratios are equal is called a proportion.
Solve for x.
Examples a.
6x 4

27 3
b.
x2 8

3
9
c.
x2 x4

4
2
d. Find the measures of the angles in a triangle where the ratio of the measures of
the angles is 4:5:6.
Assignment 6.1 page
285 #18-35
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6.2 Similar
Polygons
(Agilemind: Topic 14)
Objective:
-Identify similar figures
-solve problems involving scale factors
Consider the table below. Analyze the pairs of shapes in the Similar column and
the pairs of shapes in the Not Similar column. Then, think of a definition for
similarity based on your observations.
similar polygons
Example Is polygon WXYZ  polygon PQRS?
Scale factor
2
Example
If ABCD  EFGH, find each of the following.
1. scale factor of ABCD to EFGH =
2. EF =
3. FG =
5. perimeter of ABCD =
4. GH =
6. perimeter of EFGH =
7. ratio of perimeter of ABCD to perimeter of EFGH =
Assignment 6.2
Page 293 #11-20,
27-33, 35, 37, 39
3
6.3 Similar
Triangles
Objective:
-Identify similar triangles
-use similar triangles to solve problems
There are three ways to determine whether two triangles are similar.
AA Similarity • Show that two angles of one triangle are congruent to two
angles of the other triangle. (AA Similarity)
SSS Similarity • Show that the measures of the corresponding sides of the triangles are
proportional. (SSS Similarity)
SAS Similarity
• Show that the measure of two sides of a triangle are proportional to the measures
of the corresponding sides of the other triangle and that the included angles are
congruent.
(SAS Similarity)
Examples
Determine whether each pair of triangles is similar. Give a
reason for your answer.
1.
4
Identify the similar triangles in each figure. Explain why they
are similar and find the missing measures.
2.
3. Find x and y
4. Find x and y
Indirect Example:
Measurement A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3
inches casts an 8-foot shadow.
a. Write a proportion that can be used to determine the height of the lighthouse.
b. What is the height of the lighthouse?
Assignment 6.3
page 302 #10-21 all
32, 41,42
5
6.4 Parallel Lines
and Parts
(Agilemind: Topic 15)
Objective:
-use proportional parts of a triangle
-divide a segment into parts
Triangle If a line is parallel to one side of a triangle and intersects the other two sides in
Proportionality two distinct points, then it separates these sides into segments of proportional
Theorem lengths
Converse of the
Triangle If a line intersects two sides of a triangle and separates the sides into
Proportionality corresponding segments of proportional lengths, then the line is parallel to the
Theorem third side.
Midsegment A segment whose endpoints are the midpoints of two sides of the triangle
Triangle Midsegment a midsegment is parallel to the third side and is half its length.
Theorem
Corollary 6.1 If three or more parallel lines intersect two transversals, then they cut off the
transversals proportionally.
Corollary 6.2 If three or more parallel lines cut off congruent segments on one transversal, then
they cut off congruent segments on every transversal.
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Example
In ▲ABC, EF || CB. Find x.
Example
Find x and y
Example Triangle ABC has vertices A(-2,2), B(2,4), and C(4,-4). DE is a midsegment of
▲ABC.
a. Find the coordinates of D and E.
b. Verify that BC  DE.
c. Verify that DE = ½ BC.
Assignment 6.4
Part I: page 312 #1525 odd
Assignment 6.4
Part II: page 313 27,
29, 30, 33, 34
7
6.5 Parts of
Similar Triangles
Objective:
-recognize and use proportional relationships of corresponding perimeters of similar
triangles
-recognize and use proportional relationships of corresponding angle bisectors, altitudes,
and medians of similar triangles
Proportional If two triangles are similar, then the perimeters are proportional to the measures of
Perimeters Theorem the corresponding sides.
Example Each pair of triangles is similar. Find the perimeter of the indicated triangle.
Special Segments of
Similar Triangles
a. If two triangles are similar, then the measures of the corresponding
altitudes are proportional to the measures of the corresponding sides.
b. If two triangles are similar, then the measures of the corresponding angle
bisectors of the triangles are proportional to the measures of the
corresponding sides.
c. If two triangles are similar, then the measures of the corresponding
medians are proportional to the measures of the corresponding sides.
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Example
Find x for each pair of similar triangles.
a.
b.
c.
Angle Bisector
Theorem
An angle bisector in a triangle separates the opposite side into segments that have
the same ratio as the other two sides.
Example
Find x for each pair of similar triangles.
Assignment 6.5 page
319 #3-7, 17, 21, 23,
25, 26
9
10