Download - Geometry 1 Unit 1: Basics of Geometry
![Page 1: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/1.jpg)
Geometry 1 Unit 1: Basics of Geometry
![Page 2: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/2.jpg)
Geometry 1 Unit 1
1.1 Patterns and Inductive Reasoning
![Page 3: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/3.jpg)
EXAMPLE 1 Describe a visual pattern
Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure.
SOLUTIONEach circle is divided into twice as many equal regions as the figure number.
Sketch the fourth figure by dividing a circle into eighths.
Shade the section just above the horizontal segment at the left.
![Page 4: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/4.jpg)
GUIDED PRACTICE for Examples 1 and 2
Sketch the fifth figure in the pattern in example 1.
ANSWER
![Page 5: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/5.jpg)
EXAMPLE 2 Describe a number pattern
Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern.Notice that each number in the pattern is three times the previous number.
Continue the pattern. The next three numbers are –567, –1701, and –5103.
ANSWER
![Page 6: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/6.jpg)
GUIDED PRACTICE for Examples 1 and 2
Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern.
2.
5.13
Notice that each number in the pattern is increasing by 0.02.
5.11
+0.02
5.09
+0.02
5.07
+0.02
5.05
+0.02
5.03
+0.02
5.01
+0.02
Continue the pattern. The next three numbers are 5.09, 5.11 and 5.13
ANSWER
![Page 7: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/7.jpg)
Patterns and Inductive Reasoning Conjecture
An unproven statement that is based on observations.
Inductive Reasoning The process of looking for patterns and
making conjectures.
![Page 8: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/8.jpg)
EXAMPLE 3 Make a conjecture
Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points.
SOLUTION
Make a table and look for a pattern. Notice the pattern in how the number of connections increases. You can use the pattern to make a conjecture.
![Page 9: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/9.jpg)
EXAMPLE 3 Make a conjecture
Conjecture: You can connect five collinear points 6 + 4, or 10 different ways.
ANSWER
![Page 10: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/10.jpg)
EXAMPLE 4 Make and test a conjecture
Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers.
SOLUTION
STEP 1Find a pattern using a few groups of small numbers.
3 + 4 + 5 7 + 8 + 910 + 11 + 12 16 + 17 + 18
= 12= 4 3 = 12= 8 3= 33= 11 3 = 51= 17 3
Conjecture: The sum of any three consecutive integers is three times the second number.
ANSWER
![Page 11: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/11.jpg)
EXAMPLE 4 Make and test a conjecture
STEP 1Test your conjecture using other numbers. For example, test that it works with the groups –1, 0, 1 and 100, 101, 102.
–1 + 0 + 1 100 + 101 + 102= 0= 0 3 = 303= 101 3
![Page 12: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/12.jpg)
GUIDED PRACTICE for Examples 3 and 4
3. Make and test a conjecture about the sign of the product of any three negative integers.
Test: Test conjecture using the negative integers –2, –5 and –4
–2 –5 –4 = –40
Conjecture: The result of the product of three negative numbers is a negative number.
ANSWER
![Page 13: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/13.jpg)
Patterns and Inductive Reasoning Counterexample
An example that shows a conjecture is false.
![Page 14: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/14.jpg)
EXAMPLE 5 Find a counterexample
A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture.
Conjecture: The sum of two numbers is always greater than the larger number.
SOLUTION
To find a counterexample, you need to find a sum that is less than the larger number.
![Page 15: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/15.jpg)
EXAMPLE 5 Find a counterexample
–2 + –3
–5 > –2
= –5
Because a counterexample exists, the conjecture is false.
ANSWER
![Page 16: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/16.jpg)
GUIDED PRACTICE for Examples 5 and 6
5. Find a counterexample to show that the following conjecture is false.
Conjecture: The value of x2 is always greater than the value of x.
12( )
2=
14
14
> 12
Because a counterexample exist, the conjecture is false
ANSWER
![Page 17: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/17.jpg)
Unit 1-Basics of Geometry
1.2: Points, Lines and Planes
![Page 18: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/18.jpg)
Points, Lines, and Planes Definition
Uses known words to describe a new word.
Undefined terms Words that lack a formal definition. In Geometry it is important to have a
general agreement about these words. The building blocks of Geometry are
undefined terms.
![Page 19: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/19.jpg)
Points, Lines, and Planes The 3 Building Blocks of Geometry:
Point Line Plane
These are called the “building blocks of geometry” because these terms lay the foundation for Geometry.
![Page 20: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/20.jpg)
Points, Lines, and Planes
Point The most basic building block of
Geometry Has no size A location in space Represented with a dot Named with a Capital Letter
![Page 21: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/21.jpg)
Points, Lines, and Planes
Example: point P
P
![Page 22: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/22.jpg)
Points, Lines, and Planes
Line Set of infinitely many points One dimensional, has no thickness Goes on forever in both directions Named using any two points on the
line with the line symbol over them, or a lowercase script letter
![Page 23: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/23.jpg)
Points, Lines, and Planes
Example: line AB, AB, BA or l
B
A
**2 points determine a line
l
![Page 24: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/24.jpg)
Points, Lines, and Planes
Plane Has length and width, but no thickness A flat surface that extends infinitely in 2-
dimensions (length and width) Represented with a four-sided figure like
a tilted piece of paper, drawn in perspective
Named with a script capital letter or 3 points in the plane
![Page 25: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/25.jpg)
Points, Lines, and Planes
Example: Plane P or plane ABC
A C
B P
**3 noncollinear points determine a plane
![Page 26: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/26.jpg)
Points, Lines, and Planes Collinear
Points that lie on the same line
Points A, B, and C are Collinear
A
B
C
![Page 27: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/27.jpg)
Points, Lines, and Planes Coplanar
Points that lie on the same plane
Points D, E, and F are Coplanar
D
E
F
![Page 28: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/28.jpg)
Points, Lines, and Planes Line Segment
Two points (called the endpoints) and all the points between them that are collinear with those two points
Named line segment AB, AB, or BA
line AB segment AB
A B A B
![Page 29: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/29.jpg)
Points, Lines, and Planes Ray
Part of a line that starts at a point and extends infinitely in one direction.
Initial Point Starting point for a ray.
Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D “It begins at C and goes through D and on
forever”
![Page 30: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/30.jpg)
Segments and Their Measures Between
When three points are collinear, you can say that one point is between the other two.
A
BC
DE
F
Point B is between A and C
Point E is NOT between D and F
![Page 31: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/31.jpg)
Points, Lines, and Planes Opposite Rays
If C is between A and B, then CA and CB are opposite rays.
Together they make a line.
A BC
![Page 32: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/32.jpg)
Points, Lines, and Planes
C Y D C Y D C Y D
Line CD Ray DC Ray CD CD and CY represent the same ray. Notice CD is not the same as DC.ray CD is not opposite to ray DC
![Page 33: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/33.jpg)
Points, Lines, and Planes The intersection of two lines is a
point.
The intersection of two planes is a line.
![Page 34: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/34.jpg)
Unit 1-Basics of Geometry
1.3: Segments and Their Measures
![Page 35: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/35.jpg)
Segments and Their Measures Postulates
Rules that are accepted without proof. Also called axioms
![Page 36: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/36.jpg)
Segments and Their Measures Ruler Postulate
The points on a line can be matched one to one with the real numbers.
The real number that corresponds to a point is called the coordinate of the point.
The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B.
AB is also called the length of AB.
![Page 37: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/37.jpg)
Segments and Their Measures Segment length can be given in
several different ways. The following all mean the same thing. A to B equals 2 inches AB = 2 in. mAB = 2 inches
![Page 38: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/38.jpg)
Segments and Their Measures Example 1
Measure the length of the segment to the nearest millimeter.
D E
![Page 39: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/39.jpg)
Segments and Their Measures Between
When three points are collinear, you can say that one point is between the other two.
A
BC
DE
F
Point B is between A and C
Point E is NOT between D and F
![Page 40: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/40.jpg)
Segments and Their Measures Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
If AB + BC = AC, then B is between A and C. AC
AB BC
A B C
![Page 41: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/41.jpg)
Segments and Their Measures Example 2
Two friends leave their homes and walk in a straight line toward the others home. When they meet, one has walked 425 yards and the other has walked 267 yards. How far apart are their homes?
![Page 42: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/42.jpg)
Segments and Their Measures The Distance Formula
A formula for computing the distance between two points in a coordinate plane.
If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the distance between A and B is
2 2
2 1 2 1( ) ( )AB x x y y
![Page 43: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/43.jpg)
Segments and Their Measures Example 3
Find the lengths of the segments. Tell whether any of the segments have the same length.
![Page 44: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/44.jpg)
Segments and Their Measures Congruent
Two segments are congruent if and only if they have the same measure.
The symbol for congruence is .
We use = between equal numbers and between congruent figures.
![Page 45: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/45.jpg)
Segments and Their Measures
Markings on figures are used to show congruence. Use identical markings for each pair of congruent parts.
A 2.5 B
AB = DC = 2.5
AB DC
D 2.5 CAD BC
![Page 46: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/46.jpg)
Segments and Their Measures Distance Formula and Pythagorean
Theorem
|x2 – x1|
|y2 – y1|
A(x1, y1)
B(x2, y2)
C(x2, y1)
(AB)2 = (x2 – x1)2 + (y2 – y1)2 c2 = a2 + b2
b
ca
![Page 47: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/47.jpg)
Segments and Their Measures Example 4
On the map, the city blocks are 410 feet apart east-west and 370 feet apart north south.• Find the walking distance between C and D.• What would the distance be if a diagonal
street existed between the two points?
![Page 48: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/48.jpg)
Unit 1-Basics of Geometry
1.4: Angles and Their Measures
![Page 49: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/49.jpg)
Angles and Their Measures Angle
Formed by two rays that share a common endpoint.
Sides The rays that make the angle.
Vertex The initial point of the rays.
![Page 50: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/50.jpg)
Angles and Their Measures When naming an angle, the vertex
must be the middle letter.
angle CAT, angle TAC, CAT or TAC
C
A T
![Page 51: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/51.jpg)
Angles and Their Measures If a vertex has only one angle then
you can name it with that letter alone.
TAC could also be called A.
C
A T
![Page 52: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/52.jpg)
Angles and Their Measures Example 1
Name all the angles in the following drawing
B C
A D
1
![Page 53: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/53.jpg)
Angles and Their Measures Protractor
Geometry tool used to measure angles. Angles are measured in Degrees.
Things to know A full circle is 360 degrees, or 360º. A line is 180º.
![Page 54: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/54.jpg)
Angles and Their Measures Measure of an Angle
The smallest rotation between the two sides of the angle.
Congruent angles Angles that have the same measure.
![Page 55: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/55.jpg)
Angles and Their Measures Angle measure notation
Use an m before the angle symbol to show the measure:
mA = 34º or measure of A = 34º
![Page 56: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/56.jpg)
Angles and Their Measures Protractor Postulate
Consider a point A not on OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180.
The measure of an angle is equal to the number on the protractor which one side of the angle passes through when the other side goes through the number zero on the same scale.
A
O B
![Page 57: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/57.jpg)
Angles and Their Measures
Step 1: Place the center mark of the protractor on the vertex.
Step 2: Line up the 0-mark with one side of the angle.
Step 3: Read the measure on the protractor scale.
**Be sure you are reading the scale with the 0-mark you are using.
![Page 58: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/58.jpg)
Angles and Their Measures Interior
A point is in the interior if it is between points that lie on each side of the angle.
Exterior A point is in the exterior of an angle if it
is not on the angle or in its interior.
exteriorinterior
D
E
![Page 59: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/59.jpg)
Angles and Their Measures Angle Addition Postulate
If P is in the interior of RST, then mRSP + mPST = mRST
S
T
R
P
m RSTm RSP
m PST
![Page 60: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/60.jpg)
Angles and Their Measures Example 2
The backyard of a house is illuminated by a light fixture that has two bulbs.
Each bulb illuminates an angle of 120°. If the angle illuminated only by the right bulb
is 35°, what is the angle illuminated by both bulbs?
Left only Right only
Both bulbs
![Page 61: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/61.jpg)
Angles and Their Measures Acute Angle
An angle whose measure is greater than 0° and less than 90º.
![Page 62: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/62.jpg)
Angles and Their Measures Right Angle
An angle whose measure is 90º
![Page 63: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/63.jpg)
Angles and Their Measures Obtuse Angle
An angle whose measure is greater than 90º and less than 180º.
![Page 64: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/64.jpg)
Angles and Their Measures Straight Angle
An angle whose measure is 180°.
A
![Page 65: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/65.jpg)
Angles and Their Measures Example 3
Plot the following points.• A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2)
Measure and classify the following angles as acute, right, obtuse or straight.a. DBEb. EBCc. ABCd. ABD
![Page 66: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/66.jpg)
Angles and Their Measures
![Page 67: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/67.jpg)
Angles and Their Measures Adjacent Angles
Angles that share a common vertex and side, but have no common interior points.
A
B
C
D
![Page 68: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/68.jpg)
Angles and Their Measures Example 4
Use a protractor to draw two adjacent angles LMN and NMO so that LMN is acute and LMO is straight.
![Page 69: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/69.jpg)
Unit 1-Basics of Geometry
1.5: Segment and Angle Bisectors
![Page 70: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/70.jpg)
Segment and Angle Bisectors Midpoint
The point on the segment that is the same distance from both endpoints.
This point bisects the segment.
Bisect To cut in half (two equal pieces).
![Page 71: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/71.jpg)
Segment and Angle Bisectors
M is the midpoint of LN
L M N
LM MN
![Page 72: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/72.jpg)
Segment and Angle Bisectors Segment bisector
A segment, ray, line, or plane that intersects a segment at its midpoint.
![Page 73: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/73.jpg)
Segment and Angle Bisectors Compass
Geometric tool that is used to construct circles and arcs.
Straightedge Ruler without marks.
Construction Geometric drawing that uses a compass
and straightedge.
![Page 74: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/74.jpg)
Segment and Angle Bisectors Construct a Segment Bisector and Midpoint
Use the following steps to construct a bisector of AB and find the midpoint M of AB. 1. Place the compass point at A. Use a compass
setting greater than half of AB. Draw an arc.2. Keep the same compass setting. Place the
compass point at B. Draw an arc. It should intersect the other arc in two places.
3. Use a straightedge to draw a segment through the points of intersection. This segment bisects AB at M, the midpoint of AB.
![Page 75: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/75.jpg)
Segment and Angle Bisectors Midpoint Formula
Given two points (x1, y1) and (x2, y2) the coordinates of the midpoint are:
x1 + x2 , y1 + y2
2 2
![Page 76: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/76.jpg)
Segment and Angle Bisectors Example 1
Find the coordinates of the midpoint of the segment with endpoints at (12, -8) and (-3, 15).
![Page 77: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/77.jpg)
Segment and Angle Bisectors Example 2
Find the coordinates of the midpoint of the segment with endpoints at (5, 8) and (7, -2).
![Page 78: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/78.jpg)
Segment and Angle Bisectors Example 3
One endpoint is (17,-3) and the midpoint is (8,2). Find the coordinates of the other endpoint.
![Page 79: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/79.jpg)
Segment and Angle Bisectors Example 4
One endpoint is (-5,8) and the midpoint is (6,3). Find the coordinates of the other endpoint.
![Page 80: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/80.jpg)
Segment and Angle Bisectors Angle bisector
A ray that divides an angle into two adjacent angles that are congruent.
D
C
A
BmACD = mBCD
![Page 81: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/81.jpg)
Segment and Angle Bisectors Construct an Angle Bisector
1. Place the compass point at C. Draw an arc that intersects both sides of the angle. Label the intersections A and B.
2. Place the compass point at A. Draw another arc. Then place the compass point at B. Using the same compass setting, draw a third arc to intersect the second one.
3. Label the intersection D. Use a straightedge to draw a ray from C through D. This is the angle bisector.
![Page 82: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/82.jpg)
Segment and Angle Bisectors Example 5
JK bisects HJL. Given that mHJL = 42°, what are the measures of HJK and KJL?
![Page 83: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/83.jpg)
Segment and Angle Bisectors Example 6
A cellular phone tower bisects the angle formed by the two wires that support it. Find the measure of the angle formed by the two wires.
wire wire47°
Cellular phone tower
![Page 84: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/84.jpg)
Segment and Angle Bisectors Example 7
MO bisects LMN. The measures of the two congruent angles are (3x – 20)° and (x + 10) °. Solve for x.
![Page 85: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/85.jpg)
Unit 1-Basics of Geometry
1.6 Angle Pair Relationships
![Page 86: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/86.jpg)
Angle Pair Relationships Vertical Angles
Angles whose sides form opposite rays.
4
3
21
1 and 3 are vertical angles.
2 and 4 are vertical angles.
![Page 87: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/87.jpg)
Angle Pair Relationships Linear Pair of Angles
Angles that share a common vertex and a common side. Their non-common sides form a line.
5 and 6 are a linear pair of angles.
5 6
![Page 88: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/88.jpg)
Angle Pair Relationships Example 1
a. Are 1 and 2 a linear pair?b. Are 4 and 5 a linear pair?c. Are 5 and 3 vertical angles?d. Are 1 and 3 vertical angles?
1
2 3
45
![Page 89: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/89.jpg)
Angle Pair Relationships Example 2
![Page 90: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/90.jpg)
Angle Pair Relationships Example 3
Solve for x and y. Then find the angle measures.
(4x + 15)° (5x + 30)°
(3y – 15)°(3y + 15)°
L
M
O
NP
![Page 91: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/91.jpg)
Angle Pair Relationships Complementary Angles
Two angles that have a sum of 90º Each angle is a complement of the
other.
![Page 92: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/92.jpg)
Angle Pair Relationships Supplementary Angles
Two angles that have a sum of 180º Each angle is a supplement of the
other.
![Page 93: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/93.jpg)
Angle Pair Relationships Example 4
State whether the two angles are complementary, supplementary or neither.• The angles formed by the hands of a clock
at 11:00 and 1:00.
![Page 94: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/94.jpg)
Angle Pair Relationships Example 5
Given that G is a supplement of H and mG is 82°, find mH.
Given that U is a complement of V, and mU is 73°, find mV.
![Page 95: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/95.jpg)
Angle Pair Relationships Example 6
T and S are supplementary. The measure of T is half the measure of S. Find mS.
![Page 96: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/96.jpg)
Angle Pair Relationships Example 7
D and E are complements and D and F are supplements. If mE is four times mD, find the measure of each of the three angles.
![Page 97: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/97.jpg)
Unit 1-Basics of Geometry
1.7: Introduction to Perimeter, Circumference, and Area
![Page 98: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/98.jpg)
Introduction to Perimeter, Circumference, and Area Square
Side length s P = 4s A = s2
s
![Page 99: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/99.jpg)
Introduction to Perimeter, Circumference, and Area
Rectangle Length l and width w P = 2l + 2w A = lw
w
l
![Page 100: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/100.jpg)
Introduction to Perimeter, Circumference, and Area Triangle
Side lengths a, b, and c, Base b, and height h P = a + b + c A = ½bh
a c
b
h
![Page 101: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/101.jpg)
Introduction to Perimeter, Circumference, and Area Circle
Radius r C = 2π r A = π r2
Pi (π) is the ratio of the circle’s circumference to its diameter. π ≈ 3.14
r
![Page 102: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/102.jpg)
Introduction to Perimeter, Circumference, and Area Example 1
Find the perimeter and area of a rectangle of length 4.5m and width 0.5m.
![Page 103: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/103.jpg)
Introduction to Perimeter, Circumference, and Area Example 2
A road sign consists of a pole with a circular sign on top. The top of the circle is 10 feet high and the bottom of the circle is 8 feet high.
Find the diameter, radius, circumference and area of the circle. Use π ≈ 3.14.
![Page 104: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/104.jpg)
Introduction to Perimeter, Circumference, and Area Example 3
Find the area and perimeter of the triangle defined by H(-2, 2), J(3, -1), and K(-2, -4).
![Page 105: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/105.jpg)
Introduction to Perimeter, Circumference, and Area Example 4
A maintenance worker needs to fertilize a 9-hole golf course. The entire course covers a rectangular area that is approximately 1800 feet by 2700 feet. Each bag of fertilizer covers 20,000 square feet. How many bags will the worker need?
![Page 106: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/106.jpg)
Introduction to Perimeter, Circumference, and Area Example 5
You are designing a mat for a picture. The picture is 8 inches wide and 10 inches tall. The mat is to be 2 inches wide. What is the area of the mat?
![Page 107: Geometry 1 Unit 1: Basics of Geometry](https://reader036.vdocuments.us/reader036/viewer/2022062422/56812d5c550346895d9266b7/html5/thumbnails/107.jpg)
Introduction to Perimeter, Circumference, and Area Example 6
You are making a triangular window. The height of the window is 18 inches and the area should be 297 square inches. What should the base of the window be?