Download M2 Module 6 Newsletter_Similarity and Right Triangle Trig

PUSD Math News – Mathematics 2
Module 6: Similarity and Right Triangle Trigonometry
Module 6 Overview – Similarity and
Right Triangle Trigonometry
(Standards: G.SRT.1, G.SRT.2, G.SRT.3, G.SRT.4,
G.SRT.5, G.SRT.6, G.SRT.7, G.SRT.8, G.CO.9, G.CO.10,
G.GPE.5, G.GPE.6, F.TF.8)
Student and Teacher materials can be found at
Mathematics Vision Project
http://www.mathematicsvisionproject.org/
(Curriculum>Secondary Mathematics
Two>Module 6: Similarity and Right Triangle
Trigonometry)
In Module 6 students develop and formalize the
definition of dilations and proportional reasoning.
With dilations and proportions in mind, a formal
development of similarity follows. They then identify
whether figures are similar and use the proportions
within similar figures to solve problems. Students
analyze relationships formed by transversals and
parallel lines by identifying and using the similar
triangles. Then students narrow their focus to derive
the location of a point between two points at a given
proportion and to use proportions of similar triangles
to prove the Pythagorean Theorem. Finally, similarity
is used to develop an understanding of sine, cosine, and
tangent leading to a study of trigonometry. They learn
how to find unknown angles and sides, and how to
solve a right triangle.
Students continue to gain proficiency in the Standards
for Mathematical Practice. Module 6 includes an
extensive use of MP8, look for and express regularity in
repeated reasoning, as student repeatedly go back to
the concepts of similarity learned at the beginning of
the unit and use them to develop each new
concept. Students also focus on MP7, look for and
make use of structure, as they continually use
proportions throughout the unit. MP3, construct
viable arguments and critique the reasoning of others,
is used in several of the tasks as students are required
to make or examine conjectures and then verify
them. The trigonometry at the end requires MP4,
Model with mathematics, as students use trigonometry
to find measures in real world contexts.
Scan the QR code below to take you
directly to the PUSD Secondary Math
Resources webpage for
Mathematics 2, Module 6: Similarity
and Right Triangle Trigonometry
You will find the student text, newsletter,
standards for the module, homework help
video links and more!
https://goo.gl/1D5T4e
Vocabulary and Major Mathematical
Concepts
Note: Section numbers followed by an H will be
addressed in the Honors Mathematics 2 course.
Prerequisite Concepts and Skills:
 Angle relationships with parallel lines
 Transformations (Translations,
rotations, and reflections)
 Pythagorean Theorem
 Slope
 Sum of the angles of a triangle
 Solving equations
 Congruent triangles
 Arithmetic mean
 Geometric mean
PUSD Math News – Mathematics 2
Module 6: Similarity and Right Triangle Trigonometry
AA Similarity postulate (6.3) – if two angles of
one triangle are congruent to two angles of
another triangle, then the triangles are similar.
Example:
Image A- http://www.mathwords.com/a/aa_similarity.htm
Angle of depression (6.10) – from a horizontal,
look down, the angle from the horizontal to your
line of sight is called the angle of depression (see
Image B).
Angle of elevation (6.10) – from a horizontal,
look up, the angle from the horizontal to your
line of sight is called the angle of elevation (see
Image B).
Example:
Image Bhttp://bscstudent.buffalostate.edu/kajfkr79/web/Masters_Project/Angle
s_of_Elevaation_and_Depression.html
Arithmetic Mean (6.7) – the arithmetic average
of a set of numerical values, calculated by adding
them together and dividing by the number of
terms in the set. In section 6.7 this process may
be used to find missing terms in a sequence.
Center of dilation (6.1) – a fixed point in the
plane about which all points are expanded or
contracted. A dilation stretches or shrinks the
original figure according to a scale factor and the
center of dilation (see point O in Image D).
Congruent (6.3) – two figures are congruent if
they have exactly the same size and shape.
Angles and line segments are congruent if they
have the exact same measurement.
Cosine (6.9) – the trigonometric function that is
equal to the ratio of the length of a side adjacent
to an acute angle (in a right triangle) to the
length of the hypotenuse (see Image J).
Dilation (6.1-6.3) – a transformation of the
plane that produces an image that is the same
shape as the original, but is a different size. If O is
the center of the dilation and a non-zero number
k is the scale factor, then A’ is the image of point
A if O, A and A’ are collinear and
Example:
𝐴′
𝐴
=k
Angle of reference (6.8) – the angle of a right
triangle used to determine the trigonometric
ratios sine, cosine and tangent. The example
below uses angle A as the angle of reference. If
Angle C is the angle of reference opposite and
adjacent sides would be reversed.
Example:
Image Dhttp://www.regentsprep.org/regents/math/geometry/gt3/ldilate2.htm
Image C
PUSD Math News – Mathematics 2
Module 6: Similarity and Right Triangle Trigonometry
Geometric mean (6.7) – the nth root of the
product of n numbers. For example, the
geometric mean of 2, 4 and 8 is the cube root of
64 or 4. In section 6.7 this process may be used
to find missing terms in a sequence.
Hypotenuse (6.8) – the longest side of a right
triangle, it is the side opposite the right angle
(see Image C).
Proportional (6.1) – when two ratios are
equivalent. In this module corresponding sides
of similar shapes are proportional, which means
each set of corresponding sides has the same
ratio.
Example:
Image (6.1) – the final shape and position of an
object after a transformation is performed on a
pre-image (see triangle A’B’C’ in Image D).
Inverse trigonometric functions (6.10) – in
this module, inverse trigonometric functions
(sin-1, cos-1 and tan-1) are used to find a missing
angle when two sides of a right triangle are given
(see video link Inverse trigonometric functions on
a graphing calculator at the end of this
newsletter).
Example:
Image F- http://math-problems.math4teaching.com/theorems-onproportional-sides-of-triangles/
Proportionality statement (6.2) – an equation
that equates two ratios.
Example:
In the diagram,
∆BTW ~ ∆ ETC,
Image E- https://www.mathsisfun.com/algebra/trig-inverse-sin-costan.html
Pre-image (6.1) – the original shape and
position of an object prior to a transformation
(see triangle ABC in Image D).
Write the statement of
proportionality.
𝐸𝑇
𝑇𝐶
𝐶𝐸
=
=
𝐵𝑇 𝑇𝑊 𝑊𝐵
Image G
Ratio (6.1) – the quotient or comparison of two
quantities (for example a and b) measured in the
𝑎
same units. In this module it is written as , but a
ratio can also be written as a : b.
𝑏
PUSD Math News – Mathematics 2
Module 6: Similarity and Right Triangle Trigonometry
Scale factor (6.1) – the ratio of lengths of two
corresponding sides of two similar polygons.
Similar figures (6.1) – Two figures are similar if
one is the image of the other under a
transformation from the plane into itself that
multiplies all distances by the same positive
scale factor, k. That is to say, one figure is a
dilation of the other.
Trigonometric ratios (6.8) – the ratios defined
below as sine, cosine and tangent.
Example:
Similar polygons (6.3) – two polygons are
similar if all corresponding angles are congruent
and all corresponding pairs of sides are
proportional.
Example:
Image J- http://precalc0.weebly.com/41.html
Facts about Dilation
Image H- http://www.dummies.com/how-to/content/how-to-identifyand-name-similar-polygons.html
Sine (6.9) – the trigonometric function that is
equal to the ratio of the side opposite an acute
angle (in a right triangle) to the hypotenuse (see
Image J).
Tangent (6.9) – the trigonometric function that
is equal to the ratio of the side opposite an acute
angle (in a right triangle) to the side adjacent to
the same acute angle see Image J).
Transversal (6.4) – a line that intersects two or
more lines in the same plane at different points.
Example:
Image Ihttps://gcps.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/6605/11
031/Angle_Pairs_print.html
 Lines are taken to lines, and line segments to
line segments of proportional length in the
ratio given by the scale factor.
 Angles are taken to angles of the same
measure.
 A line not passing through the center of a
dilation is taken to a parallel line, and lines
passing through the center of dilation are
unchanged.
 To describe a dilation, specify a center of
dilation and a scale factor. The center of
dilation is a fixed point in the plane about
which all points are expanded or contracted.
It is the only invariant (unchanging) point
under a dilation.
 Dilations create similar figures – the image
and pre-image are the same shape, but
different sizes (unless the scale factor is 1,
then the image and pre-image are
congruent).
PUSD Math News – Mathematics 2
Module 6: Similarity and Right Triangle Trigonometry
Theorems used in this module
 Vertical angles are congruent.
 Measures of interior angles of a triangle sum
to 180 degrees.
 When transversals cross parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent.
 A line parallel to one side of a triangle divides
the other two sides proportionally.
 If an altitude is drawn to the hypotenuse of a
right triangle, the length of the altitude is the
geometric mean between the lengths of the
two segments formed on the hypotenuse.
 If an altitude is drawn to the hypotenuse of a
right triangle, the length of each leg of the
right triangle is the geometric mean between
the length of the hypotenuse and the length
of the segment on the hypotenuse adjacent to
the leg.
Main Topics
Note: Section numbers followed by an H will be
addressed in the Honors Mathematics 2 course.
Section in student text – Task done in class
Related Homework Help Videos
6.1 – Describing the essential features of a
dilation
Comparing side lengths after dilation
https://goo.gl/mavaKs
6.2 – Examining proportionality relationships
in triangles that are known to be similar to
each other based on dilations
How to perform a dilation
https://goo.gl/c6wmtD
6.3 – Comparing definitions of similarity
based on dilations and relationships between
corresponding sides and angles
Solving proportions
https://goo.gl/FrPAat
6.4 – Examining proportional relationships of
segments when two transversals intersect
sets of parallel lines
How to find a missing measure with similar
triangles
https://goo.gl/WGzoTH
Angles formed by parallel lines and transversals
https://goo.gl/iyJm5p
Figuring out angles between transversals and
parallel lines
https://goo.gl/SbsYag
6.5 – Applying theorems about lines, angles
and proportional relationships when parallel
lines are crossed by multiple transversals
See video suggestion for sections 6.3 and 6.4
6.6 – Applying understanding of similar and
congruent triangles to find midpoint or any
point on a line segment that partitions the
segment in a given ratio
Finding the mean
https://goo.gl/DWuXf
Finding a point part way between two points
https://goo.gl/EoE1Ld
PUSD Math News – Mathematics 2
Module 6: Similarity and Right Triangle Trigonometry
6.7 – Using similar triangles to prove the
Pythagorean theorem and theorems about
geometric means in right triangles
6.10 – Solving for unknown values in right
triangles using trigonometric ratios
Pythagorean theorem proof using similarity
https://goo.gl/ylShwW
Tutorial Activity: Angles of elevation and
depression
https://goo.gl/hP2dCD
Finding arithmetic means (missing terms)
https://goo.gl/1b3RMq
Using trigonometry to solve a right triangle
https://goo.gl/N2jqEz
Finding geometric means (missing terms)
https://goo.gl/Yslbqy
Inverse Trigonometric functions on a graphing
calculator
https://www.youtube.com/watch?v=FOKJn1wU
krI
6.8 – Developing an understanding of right
triangle trigonometric relationships based on
similar triangles
Triangle similarity and the trigonometric ratios
https://goo.gl/72dCZK
Finding the trigonometric ratios in a right
triangle
https://goo.gl/YJrUVG
How to solve a quadratic by factoring (to find xintercepts)
https://goo.gl/5G9a5i
6.9 – Finding relationships between the sine
and cosine ratios for right triangles, including
the Pythagorean identity
Match trigonometric values and side ratios
https://goo.gl/S2K5e6
Introduction to the Pythagorean trigonometric
identity
https://goo.gl/rqBE6O
6.11 – Practicing setting up and solving right
triangles to model real world context
How to solve a right triangle word problem with a
missing angle
https://goo.gl/wVbgRl