Download Section 1 – 1: Points, Lines, and Planes Notes A Point: is simply a

Date: ______________________
Section 1 – 1: Points, Lines, and Planes
Notes
A Point: is simply a _______________.
Example:
Drawn as a ________.
Named by a ______________ letter.
Words/Symbols:
A Line: is made up of ____________ and has no thickness or __________.
Drawn with an _________________ at each end.
Named by the _____________ representing two points on the line or a lowercase
script letter.
Points on the same _______ are said to be _____________.
Words/Symbols:
Example:
A Plane: is a _______ surface made up of ____________.
Drawn as a ____________ 4-sided figure.
Named by a _____________ script letter or by the letters naming three
___________________ points.
Points that lie on the same plane are said to be _______________.
Words/Symbols:
Example:
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Example #1: Use the figure to name each of the following.
a.) Name a line that contains point P.
b.) Name the plane that contains lines n and m.
c.) Name the intersection of lines n and m.
d.) Name a point not on a line.
e.) What is another name for line n.
f.) Does line l intersect line n or line m? Explain.
Example #2: Draw and label a figure for the following relationship.
a.) Point T lies on WR.
b.) AB intersects CD in plane Q at point P.
Example #3:
a.) How many planes appear in this figure?
b.) Name three points that are collinear.
c.) Are points A, B, C, and D coplanar? Explain.
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d.) At what point do DB and CA intersect?
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Date: ______________________
Section 1 – 2: Linear Measure
Notes – Part 1
Measure Line Segments
A line segment, or ______________, is a measurable part of a line that consists of
two points, called _________________, and all of the points between them.
A segment with endpoints A and B can be named as _______ or _______.
The length or _______________ of AB is written as ________.
Example #1: Use a metric ruler to draw each segment.
a.) Draw LM that is 42 millimeters long.
b.) Draw QR that is 5 centimeters long.
Example #2: Use a customary ruler to draw each segment.
a.) Draw DE that is 3 inches long.
b.) Draw FG that is 2
3
inches long.
4
1
Calculate Measures
Betweenness of Points: Point M is between points P and
Q if and only if P,Q, and M are ______________ and
__________________.
Example #4:
a.) Find LM.
b.) Find XZ.
c.) Find DE.
d.) Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.
e.) Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21.
Draw a picture!
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Date: ______________________
Section 1 – 2: Linear Measure
Notes – Part 2
Example: Find the value of x and LM if L is between N and M, NL = 6x – 5,
LM = 2x + 3, and NM = 30. Draw a picture!
Measure Line Segments
Key Concept (Congruent Segments):
Two __________________ having the same
Ex:
measure are __________________.
Symbol:
Example #1: Name all of the congruent segments found in the kite.
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Example #2: Find the measurement of RS.
Example #3: Use the figures to determine whether each pair of segments is congruent.
a.) AB, CD
b.) WZ , XY
c.) HO, HT
d.) MH , TH
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Date: ______________________
Section 1 – 3: Distance
Notes – Part 1
Distance Between Two Points
Key Concept (Distance Formulas):
Number Line
Coordinate Plane
The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by
d=
Pythagorean Theorem:
Example #1: Find the distance between E(-4, 1) and F(3, -1). Hint: Draw a triangle!
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Example #2: Use the number line to find QR.
Example #3: Use the number line to find CD.
Example #4: Use the number line to find AB and CD.
Example #5: Use the Distance Formula to find the distance between the following points.
a.) A(10, -2) and B(13, -7)
b.) X(-5, -7) and Y(-10, 7)
c.) G(-4, 1) and H(3, -1)
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Date: ______________________
Section 1 – 3: Midpoint
Notes – Part 2
Midpoint of a Segment
Key Concept (Midpoint):
The midpoint M of
PQ
is the point ___________________ P and Q such that
_____________________.
Number Line:
The coordinate of the
midpoint of a __________________ whose
endpoints have coordinates a and b is
Example #1: The coordinates on a number line of J and K are –12 and 16, respectively. Find the
coordinate of the midpoint of JK . Hint: Draw a number line!
Example #2: The coordinates on a number line of T and S are 5 and 8, respectively. Find the
coordinate of the midpoint of TS . Hint: Draw a number line!
Coordinate Plane: The coordinates of the
_____________________
of
a
segment
whose endpoints have coordinates (x1, y1)
and (x2, y2) are
1
Example #3: Find the coordinates of the midpoint of PQ for P(-1, 2) and Q(6, 1).
Example #4: Find the coordinates of the midpoint of GH for G(8, -6) and H(-14, 12).
Example #5: Find the coordinates of the midpoint of AB for A(4, 2) and B(8, -6).
Example #6: What is the measure of PR if Q is the midpoint of PR ?
Segment Bisector: any segment, line, or plane that interests a
segment at its _______________
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Date: ______________________
Section 1 – 4: Angle Measure
Notes – Part 1
Measure Angles
Degree: a unit of measure used in measuring
______________ and __________.
An arc of a
circle with a measure of 1° is ___________ of the
entire circle.
Ray: is a part of a ___________
It
has
one
____________________
and
extends
indefinitely in _________ direction.
Symbols:
Opposite Rays: two rays _________ and _________
such that B is between A and C
Key Concept (Angle):
An
angle
is
formed
by
two
______________________
__________________.
The rays are called ____________ of the angle.
The common endpoint is the ______________.
Symbols:
1
rays
that
have
a
common
An angle divides a plane into three distinct parts.
Points _____, _____, and _____ lie on the angle.
Points _____ and _____ lie in the interior of the
angle.
Points _____ and _____ lie in the exterior of the angle.
Example #1:
a.) Name all angles that have B as a vertex.
b.) Name the sides of ∠5 .
c.) Write another name for ∠6 .
Example #2:
a.) Name all the angles that have W as a vertex.
b.) Name the sides of ∠1 .
c.) Write another name for ∠WYZ .
d.) Name the vertex of ∠4 .
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Date: ______________________
Section 1 – 4: Angle Measure
Notes – Part 2
Measure Angles
Key Concept (Classify Angles):
RIGHT ANGLE:
Model:
Measure:
ACUTE ANGLE:
OBTUSE ANGLE:
Model:
Model:
Measure:
Measure:
Example #1: Measure each angle, then classify as right, acute, or obtuse.
a.)
b.)
c.)
d.)
1
e.)
f.)
Example #2: Measure each angle named and classify it as right, acute, or obtuse.
a.) ∠TYV
b.) ∠WYT
c.) ∠TYU
d.) ∠VYX
e.) ∠SYV
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Date: ______________________
Section 1 – 4: Angle Measure
Notes – Part 3
Congruent Angles
Key Concept (Congruent Angles):
Angles that have the same _____________________ are
congruent angles.
Arcs on the figure also indicate which angles are
___________________.
Example #1: State whether each pair of angles is congruent, and if so write a congruence statement.
a.)
b.)
Example #2: Find the value of x and the measure of one angle.
1
Angle Bisector: a _________ that divides an angle into _________ congruent angles.
Ex:
uuur
If PQ is the angle bisector of ___________,
then _____________________.
Example #3: In the figure, QP and QR are opposite rays, and QT bisects ∠RQS .
a.) If m∠RQT = 6 x + 5 and m∠SQT = 7 x − 2 , find m∠RQT .
b.) Find m∠TQS if m∠RQS = 22a − 11 and m∠RQT = 12a − 8 .
Example #4: In the figure, YU bisects ∠ZYW and YT bisects ∠XYW .
a.) If m∠1 = 5 x + 10 and m∠2 = 8 x − 23 , find m∠2 .
b.) If m∠WYZ =82 and m∠ZYU = 4r + 25 , find r.
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Date: ______________________
Section 1 – 5: Angle Relationships
Notes – Part 1
Pairs of Angles
Key Concept (Angle Pairs):
Adjacent Angles:
are two angles that lie in the same ____________, have a common
_____________, and a common ___________, but no common interior ____________
Examples:
Vertical Angles : are two non-adjacent angles formed by two __________________ lines
Examples:
Linear Pair :
Non-example:
a pair of ________________ angles whose non-common sides are opposite
__________.
Example:
Non-example:
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Example #1 : Name an angle pair that satisfies each condition.
a.) two angles that form a linear pair
b.) two acute vertical angles
c.) an angle supplementary to ∠VZX
d.) two acute adjacent angles
Key Concept (Angle Relationships):
Complementary Angles: two angles whose measures have a sum of ________
Examples:
Supplementary Angles: two angles whose measures have a sum of ________.
Examples:
Example #2: Find the measures of two supplementary angles if the measure of one angle is 6 less
than 5 times the measure of the other angle.
Example #3: Find the measures of two complementary angles if the difference in the measures of the
two angles is 12.
Example #4: The measure of an angle’s supplement is 33 less than the measure of the angle. Find the
measure of the angle and its supplement.
2
Date: ______________________
Section 1 – 5: Angle Relationships
Notes – Part 2
Perpendicular Lines
Lines that form right angles are _____________________.
Key Concept (Perpendicular Lines):
Perpendicular lines intersect to form _________ right
angles.
Perpendicular lines intersect to form _________________
_______________ angles.
________________ and _________ can be perpendicular
to lines or to other line segments and rays.
The right angle symbol in the figure indicates that the lines are ___________________.
Symbol: _______ is read is perpendicular to.
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Example #1: Find x so that KO ⊥ HM .
Example #2: Find x and y so that BE and AD are perpendicular.
1
Assumptions:
Example #3: Determine whether or not each of the following statements can be assumed or not.
All points shown are coplanar.
P is between L and Q.
PN ≅ PL
∠QPO and ∠OPL are supplementary.
PN ⊥ PM
L, P, and Q are collinear.
∠QPO ≅ ∠LPM
PQ ≅ PO
LP ≅ PQ
∠LMP and ∠MNP are adjacent angles.
∠LPN and ∠NPQ are a linear pair.
∠OPN ≅ ∠LPM
PM , PN , PO, and LQ intersect at P.
Example #4: Determine whether each statement can be assumed from the figure below. Explain.
a.) m∠VYT = 90
b.) ∠TYW and ∠TYU are supplementary
c.) ∠VYW and ∠TYS are complementary
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Date: ______________________
Section 1 – 5: Angle Relationships
Extra Examples
Example #1: Two angles are complementary. One angle measures 24° more than the
other. Find the measures of the angles.
Example #2: Find the measures of two supplementary angles if the measure of one
angle is 4 less than 3 times the measure of the other angle.
Example #3: The measure of an angle’s supplement is 22 less than the measure of the
angle. Find the measure of the angle and its supplement.
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Example #4: Find the value of x so that AC and BD are perpendicular.
1
Date: ______________________
Section 1 – 6: Polygons
Notes
Polygons
A polygon is a ______________ figure whose sides are all segments.
The sides of each angle in a polygon are called ___________ of the polygon, and the vertex of
each angle is a _____________ of the polygon.
Examples:
Polygons can be ________________ or ________________.
Examples:
_____________________
________________________
1
Number of
Sides
3
Regular
Polygon
polygon
are
quadrilateral
Polygon:
in
congruent
which
and
5
___________________.
6
Ex:
all
all
a
convex
the
________
the
angles
are
heptagon
octagon
9
decagon
12
n
Example #1: Name each polygon by the number of sides. Then classify it as convex or concave, regular
or irregular.
a.)
b.)
Perimeter
The perimeter of a polygon is the sum of the _______________ of its sides, which are
_________________.
Example #2: Find the perimeter of each polygon.
a.)
b.)
c.)
2