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Section 9.1
Plane Figures
Mathematical Systems
Undefined Terms – point, line, plane
Defined Terms – collinear, half plane, line
segment, ray, angle
Axioms – statements we assume are true and do
not try to prove
Theorems – Statements that can be proven with
axioms, defined terms, undefined terms and
deductive reasoning
Defined Terms
Half-Plane – A line in a plane will partition
the plane into two half planes.
Line Segment -- two points on a line and
all the points between them. The line
segment with endpoints A and B is
denoted by AB . The midpoint C bisects AB
A
C
B
Defined Terms
Ray– A point on a line partitions a line into
two half-lines. A ray is a point on a line
and all the points in one of the half-lines.
Angle – An angle is formed by two rays or
line segments that have a common
endpoint called the vertex.
Angles
Right angle – 90o
Acute angle – Less than 90o
Obtuse angle –More than 90o
and less than 180o
Straight angle – 180o
Reflex angle – More than 180o
8) Place the corner of a sheet of paper on
the angles of the polygons to check for
right angles
a. Which angles, if any, are
acute?
b. Which angles, if any, are
right angles?
c. Which angles if any, are
reflex angles?
More on Angles
Complementary – If the sum of
two angles is 90o, the angles
complementary.
Supplementary – If the sum of
two angles is 180o, the angles
are supplementary.
Adjacent angles – two angles
having the same vertex and
sharing a common side.
More on Angles
Vertical Angles –The nonadjacent angles
formed by two intersecting lines are called
vertical angles. Vertical angles are
congruent.
10) Use the angles in the following
figure to identify the pairs of
angles.
a. Three pairs of
adjacent supplementary
angles
b. Three pairs of vertical
angles
c. Two pairs of adjacent
complementary angles
Perpendicular and Parallel Lines
Perpendicular lines –
intersect to form right
angles
Parallel Lines – do not
intersect.
Transversal – two parallel
lines can be intersected by
a third line called a
transversal.
Alternate Interior Angles
If two lines are intersected by a transversal,
the lines are parallel if and only if the
alternate interior angles created by the
transversal have the same measure.
12) If l and m are parallel lines, explain why
the angles in each pair in parts a
through d have the same measure.
a.
b.
c.
d.
e.
Angle 2 and Angle 8
Angle 2 and Angle 4
Angle 4 and Angle 8
Angle 1 and Angle 7
Explain why Angle 3
and Angle 8 are
supplementary angles.
14) If s and t are parallel lines and the
measure of Angle f is 32o and the
measure of Angle a is 40o, determine
the measure of each of the following
angles.
a.Angle b
b.Angle c
c.Angle d
d.Angle e
e.Angle g
Curves
Simple Curve – starts and
stops without intersecting
itself.
Simple Closed Curve –
starts and stops at the
same point.
Closed Curve – like a
simple closed curve except
it intersects itself.
Classify each of the curves as simple,
simple closed, or none of these.
1
2
3
4
Concave vs. Convex
Concave - A set is concave if
it contains two points such
that the line segment joining
them does not completely lie
in the set. Concave sets are
also called nonconvex
Convex – not concave
Classify each region as concave or
convex.
1
2
Circles
Tangent
Chord
Diameter
Circle – Each point on a circle is
the same distance from a
fixed point called the
center.
Chord – A line segment whose
endpoints are on the
circle
Radius
Diameter – a chord that passes
through the center
Radius – a line segment from a
point on the circle to its
center
Tangent – a line that intersects a
circle in exactly one point
16) Use twelve hour clock to answer
the following.
a. How many degrees will the minute hand
of the clock move through when the time
changes from 8 o’clock to 8:25?
b. How many hours will have passed when
the hour hand has moved through 120o?
c. What is the measure of the obtuse angle
formed by the hour hand and the minute
hand if the time is 2:30?
Polygons
Polygon – a simple closed curve that is the
union of line segments called sides. The
endpoints of these line segments are
called vertices. Two sides of a polygon
are adjacent sides if they share a
common vertex, and two vertices are
adjacent vertices if they share a common
side.
Regular Polygons
Triangles
Right Triangle
(contains 1 right angle)
Equilateral Triangle
(all 3 sides of equal lengths)
Scalene Triangle
(all 3 sides of different lengths)
Isosceles Triangle
(at least two sides of equal
length)
Quadrilaterals
Rectangle
(pairs of opposite
sides parallel and of
equal length and all
right angles)
Square
Parallelogram
(all sides of equal
length and all right
angles)
(pairs of opposite
sides parallel and
of equal length)
Rhombus
Trapezoid
(opposites sides of
parallel and all sides
of equal length)
(exactly 1 pair of
opposite sides parallel)
24) Determine whether the following
statements are true or false. For
each false statement, show a
counterexample.
a. The two diagonals of a parallelogram
have the same length.
b. Any two angles in a parallelogram that
share a common side are
supplementary.
c. The two diagonals of a rectangle have
the same length.