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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5?

Standards• 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

• 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?

AB

CD

Vertical Angles are ALWAYS equal

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

• Angle 1 is the complement of angle 2• Angle B is the complement of Angle A. What conclusion can

we draw?– Angle B is 30 degrees

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B1

2

A

Angle Pairs –Adjacent Angles

• Adjacent Angles – Two coplanar angles with one common side, one common vertex, and no common interior points

A B

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Common Side

Common Vertex

Angle Pairs –Supplementary Angles

• Supplementary Angles –Two angles whose measures have a sum of 180 degrees

• Each angle is called the supplement of the other

• The angles do not have to be touching, or share a vertex, to be supplementary. They just have to sum 180 degrees.

A B

135

45

These are also known as “Linear Pairs” because they make a line

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

• 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?

120

A

B

C

DE

Example

• Solve for X

• Since they are equal in measure, we set them equal to each other: 4X = 3X + 35

• Therefore X = 35

4X

3X+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 Angles2.Adjacent Angles3.Supplementary Angles4.Complementary Angles5.Linear Angles