1-6: Exploring Angles

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1-6 Exploring Angles

• Expectations
• You will be able to use the definitions of rays
  and angles.
• You will be able to solve problems involving
  congruent angles.

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Ray

• Defn A ray with endpoint D containing E and F,
  denoted DE or DF, is the union of DE and all
  points F, such that E is between D and F.

F
E
D
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What do you mean?

• AB The distance between A and B.
• AB The segment with endpoints A and B.
• AB The ray with endpoint A, containing B
• AB The line through A and B.

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Opposite Rays

• Opposite Rays AB and AC are opposite rays iff A,
  B and C are _____________ and A is between B and
  C.

C
A
B
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Angles

• Defn An angle is the union of two _____ with the
  same endpoint, but no other points in common. The
  common endpoint is the __________ of the angle
  and all other points on the rays make up the
  _______ of the angle.

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Angle
Y is the vertex and the sides are YX and YZ.
X
Y
Z
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• _____ is the symbol for an angle.

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3 ways to name an angle

• 1. If there is only one angle, we may name the
  angle by its vertex.

A
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3 ways to name an angle

• 2. We can place the vertex between a point from
  each side of the angle.

Z
X
Y
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3 ways to name an angle

• 3. If there are 2 or more angles, we can use
  numbers.

2
1
3
4
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Angles and the Plane

• Angles split the plane into 3 parts

interior
exterior
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Measure of an Angle

• Defn The measure of ?ABC, written _______, is a
  real number between 0 and 180.
• Technically, an angle ________ have measure of 0
  or 180.

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Protractor Postulate

• Given AB and any real number r, s.t. 0ltrlt180,
  there are exactly 2 rays AC such that m?CAB r
  (one on each side of AB).

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Given AB and let r 45
A
B
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Angle Addition Postulate(Somewhat obvious, but
really important!)

• If R is in the interior of ?ABC, then m?ABR
  m?RBC __________.
• If m?ABR m?RBCm?ABC, then R is in the interior
  of _________.

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• gsp angle addition

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A
R
B
C
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Determine the value of a if m?WXZ 59.
Y
W
Z
38
a
X
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Determine the value of x if m?WXZ 102.
Y
W
Z
(3x 9)
(7x 7)
X
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Acute, Right, Obtuse Angles

• Defn Acute, Right, Obtuse Angles Let a be the
  measure of ?A. Then angle A is
• a. Acute iff ________.
• b. Right iff ________.
• c. Obtuse iff ________.
• d. Straight Angle iff _______.
• e. Zero Angle if ________.

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Congruent Angles

• Defn ?A is congruent to ?B, written ?A ? ?B,
  iff ___________________.

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Angle Bisector

• Remember to bisect means to cut exactly in half,
  so
• Defn BD is the bisector of ?ABC iff D is in the
  interior of ?ABC and _____________.

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Solve for x and the measure of all three angles
if XY bisects ?WXZ.
Y
Z
Z
15x-37
8x12
X
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• Which of the following is true of a ray KL?
• It extends infinitely in both directions.
• Its endpoint is L.
• It has length.
• It contains exactly 2 points, K and L.
• It is a portion of a line beginning with point K.

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• The notation FG represents
• The length of a line
• The length of a segment
• The length of a ray
• Two points
• A plane

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Assignment

• Pages 50 51,
• 17 25 (odds), 33 and 35