Geometry 2206
Unit 2:
Mrs. Bondi
Unit 2 Right Triangles - Lesson Topics:
Lesson 1: Ratio, Proportion and Similarity (PH text 7.1-2)
Lesson 2: Proving Triangles Similar: AA, SAS, and SSS (PH text 7.3)
Lesson 3: Similarity in Right Triangles (PH text 7.4)
Lesson 4: Proportions in Triangles (optional introduction) (PH text 7.5)
Extra - Proof practice
Lesson 5: The Pythagorean Theorem and Its Converse (PH text 8.1)
Lesson 6: Special Right Triangles (PH text 8.2)
Lesson 7: The Tangent Ratio (PH text 8.3)
Lesson 8: The Sine and Cosine Ratios (PH text 8.3)
Lesson 9: Angles of Elevation and Depression (PH text 8.4)
Extension 1: Fractals (PH text p.448-449)
Extension 2: The Golden Ratio (PH text p.468-469)
Extension 3: Inclinometer (PH text p.515)
1
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 1: Ratio, Proportion and Similarity (PH text 7.1-2)
Objective:
to find how to use ratio and proportion with similar polygons
Ratio – a comparison of two numbers by division – often looks like a fraction
a
– can be written in three forms => a to b, a:b,
(or a/b)
b
Example: The ratio of ice cream flavors (vanilla to chocolate) sold at the school fair is 4/3. If they hope
to sell 500 ice cream cups, how many of each flavor should they buy?
4x  3x  500

Extended Ratio – a comparison of three or more numbers – a:b:c, then a:b, b:c, and a:c
Example: The ratio of ice cream flavors (vanilla, chocolate, and mixed) sold at the school fair is 3:2:4.
If they hope to sell 500 ice cream cups, how many of each flavor should they buy?
Proportion – an equation that states that two ratios are equal
a c

b d
a and d are extremes, and b and c are means
a:b=c:d
Cross-Product Property –
a c
 , then
If
b d
Properties of Proportions:
a c
 is equivalent to:
b d
Examples:
5 x

8 96
(1) ad = bc
(2)
b d

a c
x  2 3 x

7
5
(3)
a b

c d
(4)
ab cd

b
d
4
x3

x2
3
2
Geometry 2206
Unit 2:
Mrs. Bondi
Similar Figures – have the same shape but not necessarily the same size
– the symbol for “similar to” is
In similar polygons, corresponding angles are _________________
and corresponding sides are ______________________
Consider these triangles.
5
4
3
We call these triangles similar.
Corresponding angles are congruent.
Corresponding sides are proportional.
10
8
6
Extended Proportion – written from three or more equal ratios
Example: from the triangles above,
Scale Factor – the ratio of corresponding linear measurements of two similar figures
Example: from the triangles above,
Determining Similarity – Prove it by asking yourself two questions.
1. Are corresponding angles congruent?
2. Are corresponding sides proportional?
Examples:
3
Geometry 2206
Unit 2:
Mrs. Bondi
Scale Drawing – a drawing made to represent a real item in a smaller way
All angles are congruent to actual angles.
All lengths are proportional to their corresponding actual lengths.
The scale is the ratio that compares the length in the scale drawing to the length of the actual item.
Most often, the scale will involve two different units of measure.
For example: 1cm : 50 m or
1in : 30 mi.
Examples:
HW: p.437 #37-44; p.444 #10-32 even, 37-38, 43-46
4
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 2: Proving Triangles Similar: AA, SAS, and SSS (PH text 7.3)
Objective:
to prove two triangles similar using AA ~ Postulate, SAS ~ Theorem and SSS ~ Theorem
to use similarity in indirect measurement to find distances
Angle-Angle Similarity Postulate (AA ~ Postulate)
Postulate 7-1
If two angles of one triangle are congruent to two angles of another triangle, then
the triangles are similar. (note: not congruent!)
E
B
C
A
D
F
Side-Angle-Side Similarity Theorem (SAS ~ Theorem)
Theorem 7-1
If an angle of one triangle is congruent to an angle of a second triangle, and the
sides including the two angles are proportional, then the triangles are similar.
K
H
I
G
J
L
Side-Side-Side Similarity Theorem (SSS ~ Theorem)
Theorem 7-2
If the corresponding sides of two triangles are proportional, then the triangles are
similar.
Q
N
M
O
P
R
A
Example:
B
Explain why the triangles are similar. Write a similarity statement.
E
C
D
5
F
Geometry 2206
Unit 2:
Mrs. Bondi
Proof: Side-Angle-Side Similarity Theorem
AB AC
Given:
, A   Q

QR QS
Prove: ABC QRS
Examples:
4)
Explain why the triangles are similar. Write a similarity statement. Find NO.
24
M
13.4
P
6
N
12
O
6
Geometry 2206
Unit 2:
Mrs. Bondi
Indirect Measurement – using similarity to find a length of something too difficult to measure
Examples:
5)
To find the height of a fire tower, Latisha places a mirror on the ground 40 ft from the base of the
tower. Latisha’s eyes are 5 ½ feet above the ground. When Latisha stands 4 ft from the mirror,
she can see the top of the tower in the mirror. How tall is the fire tower?
6)
The sun shining on a lamppost casts a 9-ft shadow at the same time a person 6 ft tall casts a 4-ft
shadow. Use similar triangles to find the height of the lamppost.
Proof Practice:
HW: p.455 #8-28 even, choose two from 32-36
(Mid-Chapter Quiz p.459)
7
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 3: Similarity in Right Triangles (PH text 7.4)
Objective:
to find the relationships in similar right triangles
Similar Polygons: Two figures that have the same shape are said to be similar. When two figures are similar, the
corresponding angles are congruent and the ratios of the lengths of their corresponding sides are equal.
Altitude – a segment from a ________________, that is ____________________ to the opposite side
Theorem 7-3
The altitude to the hypotenuse of a right triangle divides the triangle into two
triangles that are similar to the original triangle and to each other.
C
C
B
B
A
C
D
A
a c

b d
Geometric Mean
D
b and c are means, a and d are extremes
a x

x d
Geometric Mean of 6 and 20
Corollary 1
B
B
A
Geometric Mean:
D
of 15 and 20
The length of the altitude to the hypotenuse of a right triangle is the geometric
mean of the lengths of the segments of the hypotenuse.
C
C
B
B
A
A
D
A
C
AD BD

BD CD
8
B
D
B
D
Geometry 2206
Unit 2:
Mrs. Bondi
Example 1
A
C
C
3
B
D
4
y
B
A
x
B
A
D
B
D
C
Corollary 2
The altitude to the hypotenuse of a right triangle intersects it so that the lengths of
each leg are the geometric mean of the length of the entire hypotenuse and the
length of its adjacent segment of the hypotenuse.
C
B
C
B
A
A
B
A
D
B
D
C
D
AD AB

AB AC
Example 2
and
DC BC

BC AC
C
C
A
B
4
D
A
x
A
y
B
D
B
D
B
12
C
Example 3
A road from Tanya’s house and a road from Sonya’s house meet at their school at a right
angle. The public pool is 2 mi east of the school and 4 mi north of Sonya’s house. (hint – draw a diagram)
a)
What is the distance from Tanya’s house to the pool?
b)
What is the distance from Sonya’s house to school?
9
Geometry 2206
Unit 2:
Mrs. Bondi
Find the geometric mean of each pair of numbers.
17.
16 and 64
18. 5 and 25
19.
10
12 and 16
Geometry 2206
Unit 2:
Mrs. Bondi
HW: p.464 #8, 10, 22-40 even, 31 (#30 is optional)
11
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 4: Proportions in Triangles (PH text 7.5)
Objective:
Theorem 7-4
to recognize the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem
Side-Splitter Theorem
If a line is parallel to one side of a triangle
and intersects the other two sides, then it
divides those sides proportionally.
Example 1: Find the value of x.
Example 2: Find the value of x.
Corollary to Side-Splitter Theorem If three parallel lines
intersect two transversals, then the
segments intercepted on the transversals
are proportional.
Example 3: Find the value of x and y.
12
Geometry 2206
Unit 2:
Mrs. Bondi
Theorem 7-5
Triangle-Angle-Bisector Theorem
If a ray bisects an angle of a triangle,
then it divides the opposite side into
two segments that are proportional to
the other two sides of the triangle.
Examples:
4. Find the value of x.
9
5. Find the value of x.
5
6
X
3
We will use these triangle proportions in greater depth in Unit 3.
HW: p.487 #1-13
13
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 5: The Pythagorean Theorem and Its Converse (PH text 8.1)
Objective:
to use the Pythagorean Theorem and its converse
B
Vocabulary:
Hypotenuse – 1) _____________________________________
Legs – ____________________________________________
Theorem 8-1
c
a
2) _____________________________________
C
b
A
Pythagorean Theorem
If a triangle is a __________________ triangle, then the sum of the
______________________ of the lengths of the legs is equal to the square of the
length of the hypotenuse.
a 2  b2  c 2
2
2
(leg1) + (leg2) = (hypotenuse)
2
a
c
b
Pythagorean Triple: is a set of nonzero whole numbers, a, b, and c that satisfy the equation
a 2  b 2  c 2 . List examples below.
Theorem 8-2
Converse of the Pythagorean Theorem
If the sum of the __________________ of the lengths of two sides of a triangle is
_________________ to the square of the length of the third side, then the triangle
is a _______________________ triangle.
Examples:
1.
The hypotenuse of a right triangle has length 16 cm and one of its legs has length 10 cm. Find
the length of the other leg. Leave your answer in simplest radical form.
2.
A city park department rents paddle boats at docks near the entrances to the park. Suppose paths
connecting the docks are 175m and 200m. About how far is it to paddle from one dock to the
other? Round your answer to the nearest tenth.
14
Geometry 2206
Unit 2:
Mrs. Bondi
Use the Pythagorean Theorem to write an equation for the length of a leg of a right triangle.
Use the Pythagorean Theorem to write an equation for the length of the hypotenuse of a right triangle.
Theorem 8-3
If the square of the length of the longest side of a triangle is
_____________________ than the sum of the squares of the lengths of the other
two sides, then the triangle is _______________________________.
Theorem 8-4
If the square of the length of the longest side of a triangle is ______________
than the sum of the squares of the lengths of the other two sides, then the triangle
is _____________________.
Example 3)
a.
Give an angle name to the type of triangle that has sides of the given lengths.
18, 22, 39
b.
24, 45, 51
c. 11, 7, 4
HW: p. 495 #6, 22-25, 31-33, 34-50 even
15
Geometry 2206
Unit 2:
Mrs. Bondi
Proof Practice:
Objective:
To use proofs to write convincing arguments
E
Two Column Proof for Theorem 4-4
Given: EFG is a right triangle with the right angle F
Prove: mE and mG are complementary
G
F
Statements
1.) F is a right angle
1.)
Reasons
2.) mF = 90
2.) Def. of
3.) mE  mF  mG  180
3.)
4.) mE  90  mG  180
4.) S
5.) mE  mG  90
5.) S
6.) mE and mG are complementary
6.) Def. of
P
of E
Two Column Proof for Theorem 4-6
Given: A and B are right angles
Prove: A  B
B
A
Statements
1.) A and B are right angles
1.)
Reasons
2.) mA  90 and mB  90
2.)
3.) mA  mB
3.)
4.)
4.)
16
Geometry 2206
Unit 2:
Mrs. Bondi
Two-Column Proof to Theorem 4-7
Given: X and Y are congruent and supplementary
X
Prove: X and Y are right angles
Statements
Y
Reasons
1.) X and Y are congruent and
supplementary angles
1.)
2.) X  Y or mX  mY
2.)
3.)
3.)
4.) mX  mX  180
4.)
5.) 2mX  180
5.)
6.)
6.)
7.) mY  90
7.)
8.) X and Y are right angles
9.)
Complete this paragraph proof.
Given: LMN , with a right angle L
Prove: M and N are complementary.
We are given that L is a right angle, so by the definition of a right angle, mL = _____. By the
______________________, L  mM  mN  180 . Then, 90  mM  mN  180 by
___________________________. By the Subtraction Property of Equality, ______________
____________________. Finally, by the ____________________________, M and N are
complementary.
For practice for similarity proofs, try online at http://regentsprep.org/Regents/math/geometry/GP11/PracSimPfs.htm.
For interactive practice with proofs, try online at http://feromax.com/cgi-bin/ProveIt.pl.
17
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 6: Special Right Triangles (PH text 8.2)
Objective:
to use the properties of 45̊ - 45̊ - 90̊ and 30̊ - 60̊ - 90̊ triangles.
Explore: Given a square with sides of x, what is the length of the hypotenuse?
Theorem 8-5
45o – 45o – 90 o Triangle Theorem
Theorem 8-6
30o – 60o – 90 o Triangle Theorem
Example: What is the height of an equilateral triangle with sides that are 12 cm long?
Try more practice on the next two pages. You are given the length of one side of the
special right triangles. You need to find the lengths of the other two sides using these two
theorems.
HW: p. 503 #5, 14 – 30 even, 33
18
Geometry 2206
Unit 2:
Mrs. Bondi
45o – 45o – 90 o Triangle Theorem
Hypotenuse =
1)
2)
2  leg
3)
8
4 2
6
4)
5)
6)
13
10
7)
8)
6 3
9)
9
7
3 2
10)
11)
12)
12
2
5
19
Geometry 2206
Unit 2:
Mrs. Bondi
30o – 60o – 90 o Triangle Theorem
Hypotenuse = 2  shorter leg
3  shorter leg
Longer Leg =
1)
2)
3)
14
6
4 3
4)
5)
6)
3.4
6 3
6
7)
8)
9)
17
4
2 3
10)
11)
12)
10
2 3
60
20
Geometry 2206
Unit 2:
Mrs. Bondi
Special Right Triangle Practice
21
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 7: The Tangent Ratio (PH text 8.3)
Objective:
to calculate tangents of acute angles in right triangles.
to use tangents to determine unknown measures in right triangles.
If you measure and compare relationships of the sides and angles of hundreds of right triangles, you may
notice a pattern. The ratios of corresponding sides will always simplify to the same relationship.
Trigonometry (means “triangle measurement”) is the branch of mathematics dealing with the relations of
the sides and angles of triangles (and with the relevant functions of any angles). We will begin to look at
these relationships in right triangles.
The Tangent Ratio:
The tangent of A 
leg opposite A
.
leg adjacent to A
We abbreviate that to… tan A 
B
5
3
opp
.
adj
C
A
4
Write the tangent ratios for A and B .
A =
B =
All similar right triangles will share the same ratios of sides, so it allows us to use the tangent
relationship to find measurements that would be difficult to measure directly. If we know that the angle
measurements are the same, we can use proportions to find the measure of the corresponding sides.
The tangent ratio of sides opposite and adjacent to an angle of a specific measure will always be the
same. A table of Trigonometric Ratios can be found in the back of your text, p.688. Our calculators
can help us, also. When we know the ratio but do not know the angle, the inverse tangent (tan-1) is
used.
Examples:
1)
4)
tan 58 
x
25
2)
tan A 
2
3
tan 75 
3)
5)
9
15
x
6
A
25
12
x
22
Geometry 2206
Unit 2:
Mrs. Bondi
HW: p.510 #8, 16, 19, 20, 23, 27,
23
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 8: The Sine and Cosine Ratios (PH text 8.3)
Objective:
to calculate sines and cosines of acute angles in right triangles.
to use sines and cosines to unknown measures in right triangles.
Sine and Cosine relationships are similar to tangent relationships in right triangles.
The sine of A 
leg opposite A
.
hypotenuse
We abbreviate that to… sin A 
The cosine of A 
B
opp
.
hyp
C
leg adjacent A
.
hypotenuse
We abbreviate that to… cos A 
5
3
A
4
adj
.
hyp
The table of Trigonometric Ratios and calculator is useful with sine and cosine ratios also.
The inverse sine (sin-1) and inverse cosine (cos-1) will determine angle measures for us when the ratio
of sides is known.
To remember the side ratios associated with each trigonometric ratio, we often refer to the pneumonic
“SOHCAHTOA” .
SOH
CAH
TOA
Examples:
1)
4)
sin 58 
9
x
25
2)
cos A 
2
5
3)
sin 75 
5)
15
15
x
x
A
25
6
24
Geometry 2206
Unit 2:
Mrs. Bondi
HW: Part 1: p. 510 #11 – 28
25
Geometry 2206
Unit 2:
Mrs. Bondi
Review Trigonometric Ratios
Right Triangle ABC, with right angle C
opp leg
BC
AC
sin  
sin A 
sin B 
hyp
AB
AB
adj leg
AC
BC
cos 
cos A 
cos B =
hyp
AB
AB
opp leg
BC
AC
tan  
tan A 
tan B 
adj leg
AC
BC
A
B
C
1. If BC = 8 and m  C = 36, find AB.
Example 1:
A
B
2. If AB = 10 and m
 A = 54, find BC.
3. If BC = 15 and m
 A = 48, find AC.
C
Example 2: Use diagram above.
1. Find m  A, if AB = 41 and BC = 100.
2. Find m  C, if BC = 8 and AC = 13.
HW: Part 2: p. 511 # 29, 33-38
26
Geometry 2206
Unit 2:
Mrs. Bondi
Lesson 9: Angles of Elevation and Depression (PH text 8.4)
Objective:
to use angles of elevation /depression and trigonometric ratios to solve problems
Angle of Elevation: angle formed from the horizontal upwards
Because we can make an angle of depression or an angle of elevation into one of the acute angles in a
right triangle, we can use our trigonometric ratios to help us determine unknown measures.
Example: A ship is out at sea. The angle of elevation from the ship to the top of a cliff is 15.3 . If the
vertical height of the cliff is 680m, how far away from the foot of the cliff is the ship. (sketch first)
Angle of Depression: angle formed from the horizontal downwards
Example: A man is standing on top of a cliff and looking out to sea at a sailboat. The angle of
depression from the man to the boat is 36.8 . If the boat is 1220m away from the foot of the cliff, how
high is the cliff?
27
Geometry 2206
Unit 2:
Mrs. Bondi
Since the two horizontal lines are parallel, what is the relationship of the angles of elevation/depression?
horizontal line
angle of depression
angle of elevation
horizontal line
Examples:
13. A person is standing 40 ft from a flagpole and can see the top of the pole at a 35° angle of elevation. The
person’s eye level is 4 ft from the ground. What is the height of the flagpole to the nearest foot?
28
Geometry 2206
Unit 2:
Mrs. Bondi
14. An eagle perched 40 ft up in a tree looks down at a 35° angle and spots a vole. How far is the vole from the
eagle to the nearest tenth of a foot?
15. You stand 40 ft from a tree. The angle of elevation from your eyes (5 ft above the ground) to the top of the
tree is 47°. How tall is the tree? Round your answer to the nearest foot.
20. An airplane is flying at an altitude of 10,000 ft. The airport at which it is scheduled to land is 50 mi away.
Find the average angle at which the airplane must descend for landing. Round your answer to the nearest degree.
21. A lake measures 600 ft across. A lodge stands on one shore. From your point on the opposite shore, the angle
of elevation to the top of the lodge is 4°. How high above the lake does the lodge stand? Round your answer
to the nearest foot.
22. A library needs to build an access ramp for wheelchairs. The main entrance to the library is 8 ft above
sidewalk level. If the architect designs the slope of the ramp in such a way that the angle of elevation is 5°,
how long must the access ramp be? Round your answer to the nearest foot.
The angle of elevation e from A to B and the angle of depression d from B to A are given. Find the
measure of each angle.
23. e: (3x + 6)°, d: (x + 20)
24. e: (6x + 3)°, d: 3(x + 6)
HW: p. 519 # 9 – 26, 29, 33, 34
29
Geometry 2206
Unit 2:
Extension 1:
Mrs. Bondi
Fractals (PH text p.448-449)
Fractal - A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and
surfaces that cannot be represented by classical geometry. Fractals are used especially in computer
modeling of irregular patterns and structures in nature.
Three Important Properties of Fractals:
1–
2–
3–
Find an example of a fractal in nature or on-line (not from the textbook). Sketch a magnification of a
small portion to demonstrate the pattern. (Be sure to include enough to clearly demonstrate the repeated
pattern.)
Create a fractal for display in the classroom.
30
Geometry 2206
Unit 2:
Mrs. Bondi
Extension 2:
The Golden Ratio (PH text p.468-469)
In lesson 1, we looked at the extreme and mean of a ratio. In approx. 300 BC, a Greek mathematician
named Euclid published the first known reference to the golden ratio. He referred to it as the extreme
and mean ratio.
Draw a line AB, so that C divides it in such a way that the geometric mean is formed by AC .
AB AC

AC CB
then the ratio formed by
AC
is known as the golden ratio.
CB
Golden Ratio approximation –
Golden Rectangle – a rectangle in which the ratio of the length to the
width is the golden ratio
A golden rectangle can always be divided into a square and a smaller
rectangle that is similar to the original.
Fibonacci sequence –
Explain the pattern of the Fibonacci sequence.
Explore the Fibonacci sequence to discover how this sequence of numbers relates to the golden ratio.
Find information on and at least three examples of the golden ratio, golden rectangle, or Fibonacci
sequence. Examples should come from a variety of categories: nature, art, architecture, fashion, music,
etc. (not from the textbook). Be prepared to share your findings with the class (physically,
electronically or on paper), preferably in a form suitable for display in the classroom.
31
Geometry 2206
Unit 2:
Extension 3:
Mrs. Bondi
Inclinometer (PH text p.515)
Inclinometer or Clinometer - a surveying instrument used for measuring angles of elevation, slope, or
incline, as of an embankment.
Explain the reason this tool was created.
Name at least three occupations that would likely make use of a clinometer.
Create a clinometer, or find a picture of one, for display in the classroom.
32