Topics 8 and 9
Summary
TOPIC 8
KEY IDEA
RELATE
CONCEPTS
Are any two
congruent figures
also similar? YES
What is the scale
factor? k = 1
Corresponding Parts of Similar Polygons
In the diagram below, ∆ABC is similar to ∆DEF. You can write “∆ABC
is similar to ∆DEF ” as ∆ABC ∼ ∆DEF. A similarity transformation
preserves angle measure. So, corresponding angles are congruent. A
similarity transformation also enlarges or reduces side lengths by a
scale factor k. So, corresponding side lengths are proportional.
Corresponding angles
A ≅ D, B ≅ E, C ≅ F
Ratios of corresponding side lengths
DE EF FD
=
=
=k
AB BC CA
KEY IDEA
Finding Corresponding Lengths (L) in Similar Polygons
Corresponding Lengths in Similar Polygons
READING
Corresponding lengths
in similar triangles
include side lengths,
altitudes, medians, and
midsegments.
If two polygons are similar, then the ratio of any two
corresponding lengths in the polygons is equal to the scale
factor (k) of the similar polygons.
Corresponding
lengths
𝑰𝒎𝒂𝒈𝒆
𝑳𝒊
k=
=
𝑷𝒓𝒆𝒊𝒎𝒂𝒈𝒆 𝑳𝒑𝒊
Finding Perimeters and Areas of Similar Polygons
THEOREM
8.1 Perimeters (P) of Similar Polygons
RELATE
CONCEPTS
When two similar
polygons have a
scale factor of k,
what is the ratio of
their perimeters?
If two polygons are similar, then the ratio of
their perimeters is equal to the ratios of their
corresponding side lengths.
PQ + QR + RS + SP
If KLMN ~ PQRS, then
=
KL + LM + MN + NK
PQ QR RS SP
=
=
=
.
KL LM MN NK
𝑰𝒎𝒂𝒈𝒆
𝑳𝒊
𝑷𝒊
k=
=
=
𝑷𝒓𝒆𝒊𝒎𝒂𝒈𝒆 𝑳𝒑𝒊
𝑷𝒑𝒊
THEOREM
8.2 Areas (A) of Similar Polygons
If two polygons are similar, then the ratio
of their areas is equal to the squares of the
ratios of their corresponding side lengths.
RELATE
CONCEPTS
When two similar
polygons have a
scale factor of k,
what is the ratio of
their areas?
If KLMN ~ PQRS, then
Area of PQRS
PQ 2
QR 2
RS 2
SP 2
=
=
=
=
.
KL
LM
MN
NK
Area of KLMN
Image 2
Li 2
Ai
𝒌² =
=
=
Preimage
Lpi
Api
Using the Angle-Angle Similarity Theorem
THEOREM
8.3 Angle-Angle (AA) Similarity
Theorem
If two angles of one triangle are congruent
to two angles of another triangle, then the
two triangles are similar.
If A ≅ D and B ≅ E, then
∆ABC ~ ∆DEF.
Using the Side-Side-Side Similarity Theorem
In addition to using congruent corresponding angles to show that two triangles
are similar, you can use proportional corresponding side lengths.
THEOREM
8.4 Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of
two triangles are proportional, then
the triangles are similar.
If
AB BC CA
=
=
, then ∆ABC ∼ ∆RST.
TR
RS ST
Using the Side-Angle-Side Similarity Theorem
THEOREM
8.5 Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an
angle of a second triangle and the lengths
of the sides including these angles are
proportional, then the triangles are similar.
ZX
XY
If X ≅ M and
=
, then ∆XYZ ∼ ∆MNP.
PM MN
CONCEPT SUMMARY
Triangle Similarity Theorems
AA Similarity Theorem
SSS Similarity Theorem
If A ≅ D and
B ≅ E, then
∆ABC ∼ ∆DEF.
AB BC AC
If
=
=
,
DE EF DF
then ∆ABC ∼ ∆DEF.
SAS Similarity Theorem
AB AC
If A ≅ D and
=
,
DE DF
then ∆ABC ∼ ∆DEF.
Using the Triangle Proportionality Theorem
THEOREMS
8.6 Triangle Proportionality Theorem
If a line parallel to one side of a triangle
intersects the other two sides, then it divides
RT RU
If TU ∥ QS, then
=
.
TQ US
the two sides proportionally.
8.7 Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle
proportionally, then it is parallel to the
third side.
If
RT RU
=
, then TU ∥ QS.
TQ US
TOPIC 9
THEOREM
9.1 Pythagorean Theorem
In a right triangle, the square of the length of the
hypotenuse is equal to the sum of the squares of
the lengths of the legs.
c2 = a2 + b2
A Pythagorean triple is a set of three positive integers a, b, and c
that satisfy the equation c2 = a2 + b2.
KEY IDEA
Common Pythagorean Triples and Some of Their
Multiples
STUDY TIP
You may find it helpful
to memorize the basic
Pythagorean triples,
shown in bold, for
standardized tests.
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
6, 8, 10
10, 24, 26
16, 30, 34
14, 48, 50
9, 12, 15
15, 36, 39
24, 45, 51
21, 72, 75
3x, 4x, 5x
5x, 12x, 13x
8x, 15x, 17x
7x, 24x, 25x
The most common Pythagorean triples are in bold. The other
triples are the result of multiplying each integer in a
boldfaced triple by the same positive integer.
Identifying Similar Triangles
When the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are
similar to the original triangle and to each other.
THEOREM
9.6 Right Triangle Similarity Theorem
If the altitude is drawn to the hypotenuse of a
right triangle, then the two triangles formed are
similar to the original triangle and to each other.
∆CBD ∼ ∆ABC, ∆ACD ∼ ∆ABC, and ∆CBD ∼ ∆ACD.
Using the Converse of the Pythagorean Theorem
The converse of the Pythagorean Theorem is also true. You can use it to
determine whether a triangle with given side lengths is a right triangle.
THEOREM
9.2 Converse of the Pythagorean Theorem
If the square of the length of the longest side of a
triangle is equal to the sum of the squares of the lengths
of the other two sides, then the triangle is a right
triangle.
If c2 = a2 + b2, then ∆ABC is a right triangle.
Classifying Triangles
The Converse of the Pythagorean Theorem is used to determine whether a triangle is a right
triangle. You can use the theorem below to determine whether a triangle is acute or obtuse.
THEOREM
9.3 Pythagorean Inequalities Theorem
For any ∆ABC, where c is the length of the longest side, the following statements are true.
If c2 < a2 + b2, then ∆ABC is acute.
If c2 > a2 + b2, then ∆ABC is obtuse.
c2 < a2 + b2
c2 > a2 + b2
∆ABC is acute.
∆ABC is obtuse.
Lesson 9.2
Finding Side Lengths in Special Right Triangles
A 45°- 45°- 90° triangle is an isosceles right triangle that can be formed by cutting a
square in half diagonally.
THEOREM
9.4 45°-45°-90° Triangle Theorem
In a 45°- 45°- 90° triangle, the hypotenuse is 2 times as
long as each leg.
hypotenuse = leg • 2
THEOREM
9.5 30°-60°-90° Triangle Theorem
In a 30°-60°-90° triangle, the hypotenuse is
twice as long as the shorter leg, and the longer
leg is 3 times as long as the shorter leg.
hypotenuse = shorter leg • 2
longer leg = shorter leg • 3
SOH
CAH
TOA
Using the Tangent Ratio
A trigonometric ratio is a ratio of the lengths
of two sides in a right triangle. All right triangles
with a given acute angle are similar by the AA
Similarity Theorem.
KL JL
So, ∆JKL ∼ ∆XYZ, and you can write
=
.
YZ XZ
KL YZ
This can be rewritten as
=
, which is a
JL XZ
trigonometric ratio. So, trigonometric ratios are
constant for a given angle measure.
The tangent ratio is a trigonometric ratio for
acute angles that involves the lengths of the
legs of a right triangle.
△JLK∼△XZY
opp
KL
=
sin J =
hyp
JK
adj
JL
=
cos J =
hy𝑝
JK
tan J =
opp
adj
KL
=
JL
=
𝑌𝑍
opp
=
sin X =
𝑋𝑌
hyp
=
adj
= 𝑋𝑍
cos X =
hy𝑝
XY
=
YZ
opp
=
tan X =
XZ
adj
KEY IDEA
Tangent Ratio
READING
Remember the following
abbreviations.
tangent → tan
opposite → opp
adjacent → adj
Let ∆ABC be a right triangle with acute
A. The tangent of A (written as tan A)
is defined as follows.
tan A =
length of leg opposite ∠A
BC
=
length of leg adjacent to ∠A AC
In the right triangle above, A and B are complementary. So, B is acute.
You can use the same diagram to find the tangent of B. Notice that the leg
adjacent to A is the leg opposite B and the leg opposite A is the leg
adjacent to B.
Solving Real-Life Problems
An angle of elevation is an
angle formed by a horizontal
line and a line of sight up to an
object.
An angle of depression is
an angle formed by a
horizontal line and a line of
sight down to an object.
KEY IDEA
Sine and Cosine of Complementary Angles
The sine of an acute angle is equal to the cosine of its complement. The
cosine of an acute angle is equal to the sine of its complement.
Let A and B be complementary angles. Then the following statements are
true.
sin A = cos(90° − A) = cos B
sin B = cos(90° − B) = cos A
cos A = sin(90° − A) = sin B
cos B = sin(90° − B) = sin A
KEY IDEA
To find the measure of an acute angle in a right triangle.
READING
The expression “tan−1 x”
is read as “the inverse
tangent of x.”
Inverse Trigonometric Ratios
Let A be an acute angle.
Inverse Tangent If tan A = x, then tan−1 x = mA.
Inverse Sine If sin A = y, then sin−1 y = mA.
Inverse Cosine If cos A = z, then cos−1 z = mA.
BC
−1
tan
= mA
AC
BC
−1
sin
= mA
AB
AC
−1
cos
= mA
AB
Topics 8 & 9 Test
(2024 – 2025)
Good luck!