Download Unit 2 Angles

Bellringer
• Your mission:
• Construct a perfect square using the construction
techniques you have learned from Unit 1.
• You may NOT measure any lengths with your ruler.
• You may NOT measure any angles
• All sides must be perfectly perpendicular (90 degree
angle) and all side segments must be congruent (hint
hint ;)
• You have 10 minutes.
Unit 2 Angle Pairs
Unit 2: This unit introduces angles, types of angles, and angle
pairs. It defines complimentary and supplementary angles.
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Standards
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SPI’s taught in Unit 2:
SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and
space.
SPI 3108.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to
write/complete proofs and/or to solve problems.
SPI 3108.3.1 Use algebra and coordinate geometry to analyze and solve problems about geometric
figures (including circles).
SPI 3108.4.2 Define, identify, describe, and/or model plane figures using appropriate mathematical
symbols (including collinear and non-collinear points, lines, segments, rays, angles, triangles,
quadrilaterals, and other polygons).
CLE (Course Level Expectations) found in Unit 2:
CLE 3108.1.1 Use mathematical language, symbols, definitions, proofs and counterexamples correctly
and precisely in mathematical reasoning.
CLE 3108.4.1 Develop the structures of geometry, such as lines, angles, planes, and planar figures, and
explore their properties and relationships.
CFU (Checks for Understanding) applied to Unit 2:
3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems.
3108.4.5 Use vertical, adjacent, complementary, and supplementary angle pairs to solve problems and
write proofs.
Review
• We have already addressed much of what is
covered in the section on angles
• We classify angles in 4 ways:
• Less than 90 degrees:
• Acute Angle
• Equal to 90 degrees:
• Right angle
• Greater than 90, but less than 180:
• Obtuse angle
• Equal to 180 degrees:
• Straight angle
Review
• We define an angle bisector as:
• An angle bisector is a ray that divides an angle
into two congruent coplanar angles. Its
endpoint is the angle vertex.
• You can also say that a ray or segment bisects
the angle.
Angle Pairs –Vertical Angles
• Vertical Angles: Two angles whose sides are
opposite rays
A
D
B
Vertical Angles are
ALWAYS equal
C
• Which angle pairs are vertical angles?
– Angle A and Angle C
– Angle D and Angle B
• What letter in the alphabet always creates vertical
angles?
Angle Pair –Complementary Angles
• Complementary Angles –Two angles whose
measures have a sum of 90 degrees
• Each angle is called the complement of the other
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2
B
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• Angle 1 is the complement of angle 2
A
• Angle B is the complement of Angle A. What conclusion can
we draw?
– Angle B is 30 degrees
Angle Pairs –Adjacent Angles
• Adjacent Angles – Two coplanar angles with
one common side, one common vertex, and
no common interior points
Common
Vertex
A
B
Common
Side
1
2
Angle Pairs –Supplementary Angles
These are also known as “Linear Pairs” because they make a line
• Supplementary Angles –Two angles whose
measures have a sum of 180 degrees
• Each angle is called the supplement of the other
A
B
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135
• The angles do not have to be touching, or share a
vertex, to be supplementary. They just have to sum
180 degrees.
Example
• Identify the given angle pairs
– Complementary Angles
– Supplementary Angles
– Vertical Angles
– Adjacent Angles
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Conclusions
• Given the type of diagram we have seen, you can conclude
that angles are:
– Adjacent Angles
– Vertical Angles
– Adjacent supplementary Angles
• Without congruency marks, you cannot conclude that:
– Angles or segments are congruent
– An angle is a right angle
– Lines are parallel or perpendicular
– Adjacent angles are complementary
Example
• What conclusions can we make about this
diagram?
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Vertical Angle Theorem
• Vertical Angles are Congruent
A
C
120
B
D
E
• If angle ABC = 120 degrees, what is the
measure of angle EBD?
• What is the measure of angle CBD?
• What is the measure of angle ABE?
Example
• Solve for X
4X
3X+35
• Since they are equal in measure, we set them
equal to each other: 4X = 3X + 35
• Therefore X = 35
Assignment
• Text, Page 38-39 problems 7-30, 33-36 (guided
practice)
• Worksheet P 1-5
• Worksheet 2-5
• Angles and Segments Worksheet
• IF YOU DO NOT USE THE ANGLE SYMBOL,
THEN I WILL MARK -3 ON YOUR PAPER. LABEL
PROPERLY!
Unit 2 Bellringer (2 points each)
• In your own words –in other words, don’t
copy your notes word for word- define:
1. Vertical Angles
2. Adjacent Angles
3. Supplementary Angles
4. Complementary Angles
5. Linear Angles