Geometry
Unit 1 – Points, Lines, Planes, and Angles
1
Geometry
Chapter 1 – Points, Lines, Planes, and Angles
***In order to get full credit for your assignments they must me done on time and you must
SHOW ALL WORK. ***
____ Mathography
____ Summer Review Packet
____ Geometry Scavenger Hunt
____ (1-1) Points, Lines, and Planes Page 9-10 # 13-20, 21, 22, 25, 26, 27, 30-46
____ (1-2) Linear Measure – Day 1 Page 17 # 12-15, 22 – 27
____ (1-2) Linear Measure – Day 2 Page 17-19 # 28-39, 58, 62
____ (1-3) Distance – Day 1 Page 25 # 13-30
____ (1-3) Midpoints – Day 2 Page 26 # 31-42
____ Take Home Test on Sections 1-1 through 1-3
____ (1-4) Angle Measure – Day 1 Page 33-34 # 4-6, 12-24
____ (1-4) Angle Measure – Day 2 Page 34 # 25-33
____ (1-4) Angle Measure – Day 3 Page 34 # 34 – 39, 52-60, 61, 63, 65
____ (1-5) Angle Relationships – Day 1 Page 42 # 11 – 25 (skip # 17)
____ (1-5) Angle Relationships – Day 2 1-5 Practice WS
____ (1-5) Angle Relationships – Day 3 Page 42 # 17, 27-30, 31-35, 37
____ (1-6) Polygons – Day 1 Page 49 – 50 # 12-25, 29-34
____ (1-6) Polygons – Day 2 Page 49 – 50 #26 – 28
____ Chapter 1 Review WS
2
3
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:
4
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.
d.) At what point do and CA intersect? DBsuur suur
5
6
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 34
inches long.
7
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!
8
9
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.
10
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
11
12
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!
13
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)
14
15
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
16
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 _______________
17
18
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 ______________________ rays that have a common
__________________.
The rays are called ____________ of the angle.
The common endpoint is the ______________.
Symbols:
19
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∠
20
21
Date: ______________________
Section 1 – 4: Angle Measure Notes – Part 2
Measure Angles
Key Concept (Classify Angles):
RIGHT ANGLE: ACUTE ANGLE: OBTUSE ANGLE:
Model: Model: Model:
Measure: Measure: Measure:
Example #1: Measure each angle, then classify as right, acute, or obtuse.
a.) b.)
c.) d.)
22
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∠
23
24
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.
25
Angle Bisector: a _________ that divides an angle into _________ congruent angles.
Ex:
If PQuuur
is the angle bisector of ___________,
then _____________________.
Example #3: In the figure, QP and QR are opposite rays, and QT bisects . RQS∠
a.) If and 56 +=∠ xRQTm 27 −=∠ xSQTm , find RQTm∠ .
b.) Find if and TQSm∠ 1122 −=∠ aRQSm 812 −=∠ aRQTm .
Example #4: In the figure, YU bisects ZYW∠ and YT bisects XYW∠ .
a.) If and , find 1051 +=∠ xm 2382 −=∠ xm 2∠m .
b.) If =82 and , find r. WYZm∠ 254 +=∠ rZYUm
26
27
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: Non-example:
Linear Pair : a pair of ________________ angles whose non-common sides are opposite
__________.
Example: Non-example:
28
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.
29
30
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.
Example #1: Find x so that . KO HM⊥suur suuur
Example #2: Find x and y so that BE and AD are perpendicular.
31
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.
PLPN ≅
QPO∠ and OPL∠ are supplementary.
PMPN ⊥
L, P, and Q are collinear.
LPM QPO ∠≅∠
POPQ ≅
PQLP ≅
LMP∠ and MNP∠ are adjacent angles.
LPN∠ and NPQ∠ are a linear pair.
LPM OPN ∠≅∠
,,, POPNPM and LQ intersect at P.
Example #4: Determine whether each statement can be assumed from the figure below. Explain.
a.) 90m VYT∠ =
b.) and are supplementary TYW∠ TYU∠
c.) and are complementary VYW∠ TYS∠
32
33
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. Example #4: Find the value of x so that AC
suur and BD
suur are perpendicular.
34
35
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:
_____________________ ________________________
36
Regular Polygon: a convex
polygon in which all the ________
are congruent and all the angles are
___________________.
Number of Sides Polygon
Ex:
3 quadrilateral 5 6 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.)
37