ANGULAR GEOMETRIC PRINCIPLES

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UNIT 21

• ANGULAR GEOMETRIC PRINCIPLES

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NAMING ANGLES

• Angles are named by a number, a letter, or three
  letters. When an angle is named with three
  letters, the vertex must be the middle letter.
  For example, the angle shown below can be called
  ?1, ?B, ?ABC, or ?CBA
• Note In cases where a point is the vertex of
  more than one angle, a single letter cannot be
  used to name an angle

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TYPES OF ANGLES

• An acute angle is an angle that is less than 90
• A right angle is an angle of 90
• An obtuse angle is an angle greater than 90 and
  less than 180
• A straight angle is an angle of 180
• A reflex angle is an angle greater than 180 and
  less than 360
• Two angles are adjacent if they have a common
  vertex and a common side

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ANGLES FORMED BY A TRANSVERSAL

• Transversal A line that intersects (cuts) two or
  more lines. Line l in the figure on the next
  slide is a transversal
• Alternate interior angles Pairs of interior
  angles on opposite sides of the transversal. The
  angles have different vertices. Examples Angles
  3 and 5 and angles 4 and 6 (shown on the next
  slide)
• Corresponding angles Pairs of angles, one
  interior and one exterior. Located on same side
  of the transversal, but with different vertices.
  Examples ?1?5, ?2?6, ?3?7, and ?4?8 on the
  following slide

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ILLUSTRATION OF A TRANSVERSAL
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THEOREMS AND COROLLARIES

• A statement in geometry that can be proved is
  called a theorem
• A corollary is a statement based on a theorem. It
  is often a special case of the theorem
• The following theorems and corollaries will be
  used throughout the text. They are numbered for
  easier reference
• 1. If two lines intersect, the opposite, or
  vertical angles are equal.
• 2. If two parallel lines are intersected by a
  transversal, the alternate interior angles are
  equal
• 3. If two lines are intersected by a transversal
  and a pair of alternate interior angles are
  equal, the lines are parallel.

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THEOREMS AND COROLLARIES

• 4. If two parallel lines are intersected by a
  transversal, the corresponding angles are equal
• 5. If two lines are intersected by a transversal
  and a pair of corresponding angles are equal, the
  lines are parallel
• 6. Two angles are either equal or supplementary
  if their corresponding sides are parallel
• 7. Two angles are either equal or supplementary
  if their corresponding sides are perpendicular

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ANGULAR MEASURE EXAMPLE

• Determine the measure of all the missing angles
  in the figure below given that l ?? m, p ?? q, ?1
  110, and ?2 80

• ?3 110? because it is vertical to ?1 (Theorem
  1)

q
1
2
l

• ?4 110? because it is alternate interior to
• ?3 (Theorem 2) and corresponding to ?1
• (Theorem 4)

5
3

• ?5 70? (180 110) because it is
• supplementary to both ?1?3

4
8
m
6

• ?6 70? because it is corresponding to ?5
• (Theorem 4)

p

• ?8 ?2 80? because two angles are either
• equal or supplementary if their corresponding
• sides are parallel (Theorem 6)

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PRACTICE PROBLEMS

• Define the terms in problems 16
• 1. Obtuse angle
• 2. Reflex angle
• 3. Corresponding angles
• 4. Transversal
• 5. Straight angle
• 6. Name ?1 in the figure below in three
  additional ways

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PRACTICE PROBLEMS

• 7. Determine the measure of angles 28 in the
  figure below given that l ?? m and that ?1 50

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PROBLEM ANSWER KEY

• 1. An angle greater than 90? and less than 180?
• 2. An angle greater than 180 and less than 360
• 3. A pair of angles, one interior and one
  exterior. Both angles are on the same side of the
  transversal with different vertices
• 4. A line that intersects (cuts) two or more
  lines
• 5. An angle of 180
• 6. ?D, ?CDE, ?EDC
• 7. ?2 130?, ?3 50?, ?4 130?, ?5 130?, ?6
  50?,
• ?7 50?, and ?8 130?