Download Lesson 1 – Triangle Inequalities When considering triangles, two

UNIT 4B NOTES
GEOMETRY B
Lesson 1 – Triangle Inequalities
17. I can apply the triangle inequalities theorems
When considering triangles, two basic questions arise:
▪ Can any three sides form a triangle?
▪ What is the relationship between the angles
and sides of a triangle?
Examples:
1. Write the angles in order from
smallest to largest
2. Write the sides in order from
shortest to longest
3. Determine whether a triangle can have the given side lengths. Show work or explain your
reasoning.
▪ 7, 10, 19
▪ 2.3, 3.1, 4.6
▪ 8, 13, 21
4. A triangle has side lengths 7 and 12. What is the range of possible side lengths for the other
side?
UNIT 4B NOTES
GEOMETRY B
UNIT 4B NOTES
GEOMETRY B
Lesson 2 – Ratios and Proportions
1. I can express ratios in multiple formats
2. I can solve proportions
A ratio compares two numbers by division. It can be written in three ways:
A proportion is an equation stating two ratios are equal.
Cross Products Property:
Other Properties of Ratios
The proportion
a c
 is equivalent to:
b d
Examples:
1. Solve the following proportions:
7 56
a. 
x 72
c.
x4
20

5
x4
b.
2x  3 x

5
4
d.
x 3
8

2
x 3
UNIT 4B NOTES
2. Given that 6 x  5 y , then
GEOMETRY B
x
=?
y
3. According to a recent study, 5 out of 6 high school students have a smart phone. If there are
900 students at Gull Lake High School, approximately how many of them have a smart phone?
4. Let x be the length of the model in centimeters. The
rectangular model of the racing car is similar to the rectangular
racing car, so the corresponding lengths are proportional.
Find the length of the model to the nearest tenth of a
centimeter.
5. The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the
length of the shortest side?
UNIT 4B NOTES
GEOMETRY B
Lesson 3 – Similarity in Figures
3. I can state the properties of similarity
4. I can find the similarity ratio of similar triangles
6. I can find missing angles of similar triangles
10. I can write a similarity statement
11. I can verify that triangles are similar
We have already learned what it means for two figures to be congruent.
We are going to see what it means for two figures to be similar.
Two figures are congruent if and only if
they have the same
and
.
Two figures are similar if and only if they
have the same
, but not
necessarily the same
.
If two figures are congruent, then their
corresponding angles are
and their corresponding side lengths are
.
If two figures are similar, then their
corresponding angles are
and their corresponding side lengths are
.
Congruence Transformations:
Similarity Transformations:
The similarity ratio is
.
The similarity ratio of ABCD to EFGH is
.
The similarity ratio of EFGH to ABCD is
.
When figures are congruent to each other their similarity ratio is
.
UNIT 4B NOTES
GEOMETRY B
Examples:
Determine if each pair of figures is similar to each other.
If so, identify the corresponding sides and angles and determine the similarity ratio and similarity
statement.
If not, explain why they are not similar.
UNIT 4B NOTES
GEOMETRY B
Lesson 4 – Triangle Similarity
5. I can find the missing side lengths of similar triangles
7. I can show triangles are similar using the AA Postulate
8. I can show triangles are similar using the SAS Theorem
9. I can show triangles are similar using the SSS Theorem
10. I can write a similarity statement
11. I can verify that triangles are similar
Just like we learned that there are shortcuts to proving triangles congruent to each other, there
are shortcuts to proving triangles similar to each other. They are:
______ ______ ______
Examples:
Prove or explain why the triangles are similar and write a similarity statement.
UNIT 4B NOTES
Refer to the diagram to the right.
Explain why ABE ~ ACD .
Now that you know the triangles are similar, determine CD.
Refer to the diagram at the right.
Explain why ABE ~ ACD .
Now that you know the triangles are similar, determine BE and CD.
GEOMETRY B
UNIT 4B NOTES
GEOMETRY B
Lesson 5 – Properties of Similar Triangles
12. I can apply the Side-Splitter Theorem
13. I can apply the Converse of the Side-Splitter Theorem
14. I can apply the Two-Transversal Proportionality Corollary
15. I can apply the Triangle Angle Bisector Theorem
Using the theorems involving proportional relationships, you can complete the following
examples.
Examples:
In each example, find what’s asked for and NAME the theorem you used.
Find PN
Find SU
Given that AC  36 , what value of BC would
make DE AB ?
Verify that DE BC
UNIT 4B NOTES
GEOMETRY B
Find LM and MN
Find PS and SR
Find AC and CD
UNIT 4B NOTES
GEOMETRY B
Lesson 6 – Indirect Measurement
16. I can solve real-world problems using similar triangles
Indirect measurement is any method that uses formulas, similar figures, and/or proportions to
measure an object.
The following examples use indirect measurement to find a missing measure. For each of the
following, draw a sketch of the situation and use similar triangles to determine the missing
length.
Using an object’s shadow
Follow along: Tyler wants to find the height of a telephone pole. He measured the pole’s shadow
and his own shadow and then made a diagram. What is the height h of the pole?
You Try: A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole.
What is LM?
Using a scale drawing
Follow along: On a Wisconsin road map, Kristin measured a distance of 11 in. from Madison to
Wausau. The scale of this map is 1inch: 13 miles. What is the actual distance between Madison
and Wausau to the nearest mile?
You Try: The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide.
Make a scale drawing of the floor of the chamber using a scale of 1 in.:20 ft.
UNIT 4B NOTES
GEOMETRY B
What is the relationship between Similar Figures, Perimeter, and Area?
The following figures are similar squares, similar triangles, and similar rectangles. Find the
similarity ratio, and perimeter and area of each figure.
A. Similar Squares
B. Similar Triangles
C. Similar Rectangles
P:
P:
P:
P:
P:
P:
A:
A:
A:
A:
A:
A:
Similarity Ratio:
Perimeter Ratio:
Area Ratio:
Follow along: Given that ∆LMN ∆QRT, find the perimeter P and area A of ∆QRS.
You try: ∆ABC ~ ∆DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm2 for ∆DEF,
find the perimeter and area of ∆ABC.