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Angles and Triangles
Module 5 – Lesson 2

Types of Angles

Types of angles

How to measure angles

The little square in the corner of the triangle is a symbol for a right angle that measures 90°
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Now solve the equation
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

When you translate a figure, whether a line, angle, or object, every part of the figure moves in the same
direction for the same distance
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In a translation, a new figure is shown with the same letters as the original, but you add an apostrophe (') after each
letter, which is read as "prime
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 Triangle ABC is translated down and to the right
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 If Polygon LMNOP is translated up and to the right, the new polygon would be called L'M'N'O'P', which is
read as L prime M prime N prime O prime P prime
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The original figure is called the pre-image
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Pre-image: The original figure prior to a transformation
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
The same letters are used for each with the "prime"
marks to remind you that they are related to one another
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

When two figures are exactly the same, they are called congruent
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

This means that their angles are also exactly the same and are also congruent
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

All lines, angles, and segments remain exactly the same, just placed in a new location
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Reflections: A type of transformation in which one figure is a direct mirror image of another
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Rigid transformation: Moving a figure so that it is in a new location but has the same size, shape, and area
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

Imagine a line halfway between you and your reflection
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Line of reflection: The line over which a pre-image is reflected to create a new image
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Each piece of the pre-

image is exactly the same in the image
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The length of a line segment will be the same in a reflection
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The angles of the image are congruent to the angles in the pre-image
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Rotation: A transformation that turns a figure a given angle and direction around a fixed point
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45°

90°

180°

270°

360°

Clockwise and Counterclockwise
Rotations can be either clockwise or counterclockwise in direction
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

Think of the middle of a clock
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

Often called the origin

Congruency
Module 5 – Lesson 7
Having the same size and shape; expressed with the symbol ≅




After a rigid transformation has been performed, the pre-image and the image are congruent
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Rigid Transformation: Movement that changes the position of a figure but not its size; translations, reflections, and
rotations
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

Dilations are not rigid transformations because the size does not stay the same
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Triangle HJK
has been shrunk to form Triangle H'J'K'
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Scale factor: A ratio of two corresponding lengths that determines the change in size from a pre-image to an image
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

A scale factor that is less than 1 represents a decrease in the size of the pre-image
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Set up a ratio with the image measurement on top and the pre-image measurement in the denominator
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5 units/
10 units

 Simplify if possible
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

If a figure goes across the graph at a diagonal, don't try to estimate the lengths of the sides
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Vertex A has coordinates (2, 1)
Vertex A' has coordinates (6, 3)

 Set up a ratio
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Remember to put the image coordinate on top and pre-image on the
bottom
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6/ = 6÷2/
3
2
2÷2 = /1 = 3

The scale factor for this dilation is 3
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

They have the same angle measurements and are the same size
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Transversals and Angles
Module 5 – Lesson 9
Parallel Lines and Transversals
When two lines are crossed by a third line, the third
line is called a transversal
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


These corresponding angles are congruent
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The corresponding angles are as follows:
m∠1 ≅ m∠5

m∠2 ≅ m∠6

m∠3 ≅ m∠7

m∠4 ≅ m∠8

Alternate interior angles lie on alternate (or
opposite) sides of the transversal and inside the
parallel lines
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So are Angle 5 and Angle 4
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Alternate exterior angles lie on alternate (or opposite)
sides of the transversal and outside the parallel lines



Angle 1 and Angle 8 are alternate exterior angles
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When parallel lines are intersected by a transversal, alternate exterior angles are congruent
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It shows parallel lines m and n intersected
by transversal t
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The figure shows that one
of the obtuse angles measures 150 degrees
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STEP 1
 Start by marking the angles that you know are congruent to
the 150-degree angle
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


Each of these angles will be congruent to 150 degrees
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 Now think about the missing angle to the right of 150
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Since these adjacent angles form a
straight line, they are supplementary—they add up to 180 degrees
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 The missing angle next to 150 is 30 degrees
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So you can say that

x + 20 = 78


Then, solve for x
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Exterior Angles: Angles that are located outside the triangle are called exterior angles
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

There is a special relationship between an exterior angle and the two remote interior angles across from it
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Exterior Angle Theorem:
The measure of an exterior angle is equal to the sum of the two remote interior
angles

Triangle Similarity
Corresponding Angles
Congruent angles: Angles with the same measure
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But they can be different sizes
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Matching the number of arcs will tell you which angles match up or correspond to one another
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You can tell because they both have one arc marked
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

Angles that have the same measure in similar triangles are called congruent angles
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If the angles are corresponding, they can

be in different positions but still have the same measure
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There are several
ways to prove similarity, but a simple one is the angle-angle criterion
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The triangles shown are similar
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 The angle-angle criterion states that when two pairs of
angles are congruent, the triangles are similar

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