Math 310

1
Math 310

• Section 9.3
• More on Angles

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Linear Pair

• Def
• Two angles forming a line are called a linear
  pair.

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Ex.
Not a linear pair ltABC ltFDE
Linear pairs ltABC ltDBC ltBDE ltFDE
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Question

• What can we say about the sum of the measures of
  the angles of a linear pair?

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Vertical Angles

• Def
• When two lines intersect, four angles are
  created. Taking one of the angles, along with
  the other angle which is not its linear pair,
  gives you vertical angles. (ie it is the angle
  opposite of it)

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Ex.
Vertical angles ltABC ltEBD ltCBE ltDBA
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Vertical Angle Theorem

• Thrm
• Vertical angles are congruent.

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Ex.
If mltABC 95 find the other three angle
measures.
mltEBD 95 mltCBE 85 mltDBA 85
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Supplementary Angles

• Def
• Supplementary angles are any two angles whose sum
  of their measures is 180.

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Ex.
ltABC ltCBE ltABC ltFED ltABC ltBEG ltDEB ltFED
ltDEB ltCBE ltDEB ltBEG ltGEF ltFED ltGEF ltCBE
ltGEF ltBEG
Given ltABC is congruent to ltFEG Find all pairs
of supplementary angles.
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Complementary Angles

• Def
• Complementary angles are any two angles whose sum
  of their measures is 90.

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Ex.
Given ray BC is perpendicular to line AE. Name
all pairs of complementary angles.
ltCND ltDBE
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Ex.
Name all pairs of complementary angles.
ltABC ltGHI ltDEF ltGHI
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Transversal

• Def
• A line, crossing two other distinct lines is
  called a transversal of those lines.

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Ex.
Name two lines and their transversal.
Lines JK QO Transversal OK
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Transversals and Angles

• Given two lines and their transversal, two
  different types of angles are formed along with 3
  different pairs of angles
• Interior angles
• Exterior angles
• Alternate interior angles
• Alternate exterior angles
• Corresponding angles

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Interior Angles
ltJKO ltMKO ltQOK ltNOK
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Exterior Angles
ltJKL ltMKL ltQOP ltNOP
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Alternate Interior Angles
ltJKO ltNOK ltMKO ltQOK
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Alternate Exterior Angles
ltJKL ltNOP ltMKL ltQOP
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Corresponding Angles
ltJKL ltQOK ltMKL ltNOK ltQOP ltJKO ltNOP ltMKO
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Parallel Lines and Transversals

• Thrm
• If any two distinct coplanar lines are cut by a
  transversal, then a pair of corresponding angles,
  alternate interior angles, or alternate exterior
  angles are congruent iff the lines are parallel.

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Ex.
Given Lines AB and GF are parallel. Name all
congruent angles.
ltABC ltGFB ltDBC ltEFB ltGFH ltABF ltEFH ltDBF
ltABC ltEFH ltDBC ltGFH ltDBF ltGFB ltABF ltEFB
ltABC ltGFB
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Triangle Sum

• Thrm
• The sum of the measures of the interior angles of
  a triangle is 180.

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Angle Properties of a Polygon

• Thrm
• The sum of the measures of the interior angles of
  any convex polygon with n sides is 180n 360 or
  (n 2)180.
• The measure of a single interior angle of a
  regular n-gon is (180n 360)/n or (n 2)180/n.

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Ex.

• What is the sum of the interior angles of a
  heptagon? A dodecagon?
• Heptagon (7 2)180 (5)180 900
• Dodecagon (10 2)180 (8)180 1440

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Exterior Angle Theorem

• Thrm
• The sum of the measures of the exterior angles
  (one at each vertex) of a convex polygon is 360.

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Proof

• Given a convex polygon with n sides and vertices,
  lets say the measure of each interior angles is
  x1, x2, ., xn. Then the measure of one exterior
  angle at each vertices is 180 xi. Adding up
  all the exterior angles
• (180 x1) (180 x2) (180 xn)
• 180n (x1 x2 xn)
• 180n (180n 360 )
• 180n 180n 360 360

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Ex.

• Pg 610 12a
• Pg 610 - 7