Download Circles (Arc Lengths), Types of Angles, Parallel Lines

Chapter
11
Introductory
Geometry
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Congruent Segments and Angles
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Regular Polygons
All sides are congruent and all angles are congruent.
A regular polygon is equilateral and equiangular.
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Triangles and Quadrilaterals
Right triangle
a triangle containing one
right angle
Acute triangle
a triangle in which all the
angles are acute
Obtuse triangle
a triangle containing one
obtuse angle
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Triangles and Quadrilaterals
Scalene triangle
a triangle with no congruent
sides
Isosceles triangle
a triangle with at least
two congruent sides
Equilateral triangle
a triangle with three
congruent sides
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Triangles and Quadrilaterals
Trapezoid
a quadrilateral with one pair
of parallel sides
Kite
a quadrilateral with two pairs
of adjacent sides congruent
and opposite sides not
congruent.
(NO parallel sides)
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Triangles and Quadrilaterals
Isosceles trapezoid
a trapezoid with congruent nonparallel sides & base angles
Parallelogram
a quadrilateral in which each
pair of opposite sides is
parallel
Rectangle
a parallelogram with a right
angle.
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Triangles and Quadrilaterals
Rhombus
a parallelogram with two
adjacent sides congruent
Square
a rectangle with two
adjacent sides congruent
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Hierarchy Among Polygons
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Line Symmetries
Mathematically, a geometric figure has a line of
symmetry ℓ if it is its own image under a reflection
in ℓ.
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Example 11-8
How many lines of symmetry does each drawing
have?
6
1
2
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0
Rotational (Turn) Symmetries
A figure has rotational symmetry, or turn
symmetry, when the traced figure can be rotated
less than 360° about some point, the turn center,
so that it matches the original figure.
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Similar Example 11-9
Determine the amount of the turn for the rotational
symmetries of each figure.
a.
b.
72°, 144°, 216°, 288°
c.
60°, 120°, 180°, 240°, 300°
180°
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Point Symmetry
Any figure that has rotational symmetry 180° is said
to have point symmetry about the turn center.
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11-4 More About Angles
 The Sum of the Measures of the
Angles of a Triangle
 The Sum of the Measures of the
Interior Angles of a Convex Polygon
with n sides
 The Sum of the Measures of the
Exterior Angles of a Convex n-gon
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Vertical Angles
Vertical angles created by intersecting lines are a
pair of angles whose sides are two pairs of
opposite rays.
Angles 1 and 3 are vertical
angles.
Angles 2 and 4 are vertical
angles.
Vertical angles are congruent.
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Supplementary Angles
The sum of the measures of two supplementary
angles is 180°.
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Complementary Angles
The sum of the measures of two complementary
angles is 90°.
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Transversals and Angles
2, 4, 5, 6
Interior angles
Exterior angles
1, 3, 7, 8
Alternate interior angles 2 and 5, 4 and 6
Alternate exterior angles 1 and 7, 3 and 8
Corresponding angles
1 and 2, 3 and 4,
5 and 7, 6 and 8
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Angles and Parallel Lines Property
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 if, and only if, the lines are
parallel.
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Angles formed by a Transversal
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Parallel Lines and Transversals
Theorem:
Alternate
Interior
Angles
If two parallel lines are cut by a transversal, then each pair of
congruent
Alternate interior angles is _________.
1 2
43
56
87
4  6
3  5
Parallel Lines and Transversals
Theorem:
If two parallel lines are cut by a transversal, then each pair of
Consecutive consecutive interior angles is _____________.
supplementary
Interior
Angles
1 2
43
5 6
87
m4  m5  180
m3  m6  180
Parallel Lines and Transversals
Theorem:
Alternate
Exterior
Angles
If two parallel lines are cut by a transversal, then each pair of
alternate exterior angles is _________.
congruent
1 2
43
5 6
87
1  7
2  8
The Sum of the Measures of the
Angles of a Triangle
The sum of the measures of the interior angles of
a triangle is 180°.
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Example 11-10
In the framework for a tire jack, ABCD is a
parallelogram. If ADC of the parallelogram
measures 50°, what are the measures of the other
angles of the parallelogram?
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Example 11-11
In the figure, m || n and k is
a transversal. Explain why
m1 + m 2 = 180°.
Because m || n, angles 1 and 3 are corresponding
angles, so m1 = m3.
Angles 2 and 3 are supplementary angles, so
m2 + m3 = 180°.
Substituting m1 for m3, m1 + m2 = 180°.
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The Sum of the Measures of the Interior
Angles of a Convex Polygon with n sides
The sum of the measures of the interior angles of
any convex polygon with n sides is (n – 2)180°.
The measure of a single interior angle of a regular
n-gon is
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The Sum of the Measures of the Exterior
Angles of a Convex n-gon
The sum of the measures of the exterior angles of
a convex n-gon is 360°.
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Example 11-12
a. Find the measure of each interior angle of a
regular decagon.
The sum of the measures of the angles of a
decagon is (10 − 2) · 180° = 1440°.
The measure of each interior angle is
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Example 11-12
(continued)
b. Find the number of sides of a regular polygon
each of whose interior angles has measure 175°.
Since each interior angle has measure 175°, each
exterior angle has measure 180° − 175° = 5°.
The sum of the exterior angles of a convex polygon
is 360°, so there are
exterior angles.
Thus, there are 72 sides.
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Example 11-13
Lines l and k are parallel, and the angles at A and B
are as shown. Find x, the measure of BCA.
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Example 11-13
(continued)
Extend BC and obtain the
transversal BC that intersects
line k at D.
The marked angles at B and D
are alternate interior angles,
so they are congruent and
mD = 80°.
mACD = 180° − (60° + 80°) = 40°
x = mBCA = 180° − 40° = 140°
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Circumference of a Circle
Example 4A: Finding Arc Length
Find each arc length. Give answers
in terms of  and rounded to the
nearest hundredth.
FG
Use formula for
area of sector.
Substitute 8 for r
and 134 for m.
 5.96 cm  18.71 cm
Simplify.
Example 4B: Finding Arc Length
Find each arc length. Give answers in terms of
 and rounded to the nearest hundredth.
an arc with measure 62 in a circle with radius 2 m
Use formula for
area of sector.
Substitute 2 for r
and 62 for m.
 0.69 m  2.16 m
Simplify.
Find each arc length. Give your answer in terms
of  and rounded to the nearest hundredth.
GH
Use formula for
area of sector.
Substitute 6 for r
and 40 for m.
=
 m  4.19 m
Simplify.
Find each arc length. Give your answer in terms
of  and rounded to the nearest hundredth.
an arc with measure 135° in a circle with
radius 4 cm
Use formula for
area of sector.
Substitute 4 for r
and 135 for m.
= 3 cm  9.42 cm
Simplify.
Find each measure. Give answers in terms of
 and rounded to the nearest hundredth.
1. length of NP
2.5  in.  7.85 in.
2. The gear of a grandfather clock has a radius of 3 in.
To the nearest tenth of an inch, what distance does
the gear cover when it rotates through an angle of
88°?
 4.6 in.