Chapter 1: Basics of Geometry

1-1: Patterns and Inductive Reasoning

conjecture an unproven statement that is based on observations

a process that includes looking for patterns and making conjectures inductive reasoning a process that includes looking for patterns and making conjectures

Conjecture Information To prove a conjecture is true, you need to prove it is true in ALL cases. To prove a conjecture is false, provide a single counterexample. Conjectures that are not known to be true or false are called unproven or undecided.

counterexample an example that shows a conjecture is false

Let’s Go on To the Next Section! 

1-2: Points, Lines, and Planes

Today’s Objectives name points, lines, segments, rays and planes name a point that is collinear with given points name a point that is coplanar with given points

definition uses known words to describe a new word

undefined terms a word that is not formally defined (point, line, plane) although there is general agreement about what the word means

point a point has no dimension and is represented by a small dot ●A

line a line extends in one dimension and is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions ● ● A B AB or BA

Facts About Lines is designated by 2 capital letters which represent two points on the line may also be designated by a small letter that labels the whole line is understood to be straight unless stated otherwise is the shortest distance between two points

plane a collection of points that forms a flat surface infinitely wide and infinitely long plane M

collinear points points that lie on the same line segments and rays are collinear if they lie on the same line

coplanar points points that lie on the same plane segments, rays, and lines are coplanar if they lie on the same plane

line segment/segment part of a line that consists of two endpoints and all points on the line that are between the endpoints ● ● A B AB or BA

endpoints points at either end of a segment name segments using the endpoints

ray part of a line that consists of an initial point and all points on the line that extend in one direction ● ● B A BA only

initial point the point at the beginning of a ray when naming rays, start with the initial point

Question Describe what each of these symbols means: PQ QP

opposite rays if C is between A and B, then ray CA and ray CB are opposite rays ● ● ● A C B

Question Name two pairs of opposite rays in the figure. ● ● ● ● E F G H

intersect to have one or more points in common ● A

intersection the set of points that two or more geometric figures have in common

Question How can lines intersect at more than one point?

Question When you see a dashed line in a diagram, what does that usually imply?

Skill 1-2a I will name points, lines, segments, rays and planes.

Number 1a ●B

Number 2a

Number 3a

Number 4a

Number 5a

Skill 1-2b I will name a point that is collinear with given points.

Number 1b point D

Number 2b point H

Number 3b point G

Number 4b Points C and Q

Number 5b Points P and R

Number 6b Points A and Q

Skill 1-2c I will name a point that is coplanar with given points.

Number 1c point D

Number 2c point H

Number 3c Points D, E, and H

Number 4c Points E, F, and G

Number 5c Points G, H, and A

Let’s Do Some Homework! 

1-3: Segments and Their Measures

Today’s Objectives use the segment addition postulate correctly use the distance formula to measure distances

postulates/axioms rules that are accepted without proof

Postulate 1— Ruler Postulate The points on a line can be matched one to one with the real numbers (coordinate).

length of a segment the distance between the endpoints of a segment

Postulate 2—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.

Question Draw a sketch of three collinear points. Label them. Then write the Segment Addition Postulate for the points.

Question A car with a trailer has a total length of 27 feet. If the trailer has a total length of 13 feet, how long is the car?

Distance Formula If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is

The Distance Formula “On Top of Old Smokey” When finding the distance between two points subtract the two x’s. Do the same for the y’s. Now square both these numbers, and find out their sum. When you take the square root, then you are all done!

congruent segments segments that have the same length

Question Explain the difference between the terms equal and congruent.

Skill 1-3a I will use the segment addition postulate correctly.

Number 1a Find XY

Number 2a Find MN

Number 3a Find MO

Number 4a Points A, D, F, and X are on a segment in this order. AD = 15, AF = 22, and AX = 30. Find DF and FX.

Skill 1-3b I will use the distance formula to measure distances.

Number 1b C(0, 0) and D(5, 2)

Number 2b G(3, 0) and H(8, 10)

Number 3b V(-2, -6) and W(1, -2)

Let’s Do Some Homework! 

1-4: Angles and Their Measures

Today’s Objectives measure angles using a protractor use angle addition postulates correctly classify angles as acute, right, obtuse, or straight find the intersection and union of geometric figures

angle consists of two different rays that have the same initial point vertex ● side

sides of an angle the rays that make up an angle

vertex of an angle the initial point of the angle or where B is the vertex and are the sides

Naming an Angle Use three letters. The center letter corresponds to the vertex. Place a number or letter at the vertex in the interior of the angle. This may be used only if there is one angle at the vertex.

Question Name the vertex of angle XYZ.

congruent angles angles that have the same measure

measure of an angle the size of the opening of the angle measured in degrees

Postulate 3—Protractor Postulate Consider a point A on one side of . The rays of the form can be matched one to one with the real numbers from 0 to 180. ●A ● ● O B

interior of an angle all points between the points that lie on each side of the angle interior

exterior of an angle all points not on the angle or in its interior exterior exterior

Postulate 4—Angle Addition Postulate If P is in the interior of , then R● S● P● T●

Question How are the Segment Addition Postulate and the Angle Addition Postulate similar?

Question How are the Segment Addition Postulate and the Angle Addition Postulate different?

acute angle an angle with measure between 0° and 90°

right angle an angle with measure equal to 90°

obtuse angle an angle with measure between 90° and 180°

straight angle an angle with measure equal to 180°

reflex angle an angle with measure greater than 180° and less than 360°

adjacent angles two angles with a common vertex and side but no common interior points 1 2

Question Name the angles in the figure. M N O ● ● ● ●P

Intersection and Union An intersection of two sets A and B is the set consisting of all the members that belong to both sets A and B. Look to see what is in common A union of two sets A and B is the set consisting of all members belonging to at least one of the sets A and B. Put the sets together

Skill 1-4a I will measure angles using a protractor.

Number 1a

Number 2a

Number 3a

Number 4a

Skill 1-4b I will use angle addition postulates correctly.

Number 1b

Number 2b mLAOB = 37° and mLBOC = 24° Find mLAOC

Number 3b mLAOB = 46° and mLAOC = 84° Find mLBOC

Skill 1-4c I will classify angles as acute, right, obtuse, or straight.

Number 1c

Number 2c

Number 3c

Skill 1-4d I will find the intersection and union of geometric figures.

Number 1d

Number 2d

Number 3d

Number 4d

Let’s Do Some Homework! 

1-5: Segment and Angle Bisectors

Today’s Objectives use construction tools to bisect a segment find the midpoint or endpoint of a segment using the midpoint formula use construction tools to bisect an angle find angle measures after an angle has been bisected

midpoint the point that divides, or bisects, a segment into two congruent segments

bisects to divide into two congruent parts ● l ● l ●

Question How do you indicate congruent segments in a diagram?

segment bisector a segment, ray, line, or plane that intersects a segment at its midpoint ● l ● l ●

bisecting a segment do a compass and straightedge construction use paper folding use the midpoint formula

compass a construction tool used to draw arcs

straightedge a construction tool used to draw segments usually a ruler without marks

construction a geometric drawing that uses a limited set of tools, usually a compass and a straightedge

Constructing a Segment Bisector You are given a segment with endpoints AB. Place the compass point at point A. Use a compass setting greater than half the length of segment AB. Draw an arc.

Keep the same compass setting. Place the compass point at B. Draw an arc that should intersect the other arc in two places above and below the segment. Use a straightedge to draw a segment through the points of intersection.

Midpoint Formula If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of has coordinates

The Midpoint Formula “The Itsy Bitsy Spider”  When finding the midpoint of two points on a graph, Add the two x’s and cut their sum in half. Add up the y’s and divide ‘em by a two. Now write ‘em as an ordered pair. You’ve got the middle of the two.

angle bisector a ray that divides an angle into two adjacent angles that are congruent

Question How do you indicate congruent angles in a diagram?

Constructing an Angle Bisector You are given angle C. Place the compass point at C. Draw an arc that intersects both sides of the angle. Label the intersections A and B. Place the compass point at A. Draw an arc.

Then place the compass point at B. Using the same compass setting, draw another arc. Label the intersection D. Use a straightedge to draw a ray through C and D.

Skill 1-5a I will use construction tools to bisect a segment.

Number 1a

Number 2a

Skill 1-5b I will find the midpoint or endpoint of a segment using the midpoint formula.

Number 1b endpoints D(3, 5) and E(-4, 0)

Number 2b Endpoints A(-2, 3) and B(5, -2)

Number 3b M(3, -4) one endpoint of XY is Y(-3, -1)

Number 4b M(2, 4) one endpoint of RP is R(-1, 7)

Skill 1-5c I will use construction tools to bisect an angle.

Number 1c

Number 2c

Skill 1-5d I will find angle measures after an angle has been bisected.

Number 1d  

Number 2d  

Number 3d  

Number 4d  

Let’s Do Some Homework! 

1-6: Angle Pair Relationships

Today’s Objectives identify and find values for vertical angles and linear pairs identify and find values for complementary and supplementary angles

vertical angles two angles whose sides form two pairs of opposite rays 1 angles 1 & 3 4 2 angles 2 & 4 3

linear pair two adjacent angles whose non-common sides are opposite rays 5 6 angles 5 & 6

two angles whose measures have the sum 90° complementary angles two angles whose measures have the sum 90° can be adjacent or non-adjacent

complement the sum of the measures of an angle and its complement is 90°

supplementary angles two angles whose measures have the sum 180° can be adjacent or non-adjacent

supplement the sum of the measures of an angle and its supplement is 180°

Question The angles in a linear pair are always _____.

Question Explain the difference between complementary angles and supplementary angles.

Skill 1-6a I will identify and find values for vertical angles and linear pairs.

Number 1a Name all of the vertical angles.

Number 2a Name all of the linear pairs.

Number 3a Solve for x.

Number 4a Solve for each angle.

Skill 1-6b I will identify and find values for complementary and supplementary angles.

Number 1b Two supplementary angles have measures of (4x + 15)° and (5x + 30).° Find x.

Number 2b Given that angle U is a complement of angle V, and the measure of angle U is 73°, find .

Number 3b Solve for each angle.

Let’s Do Some Homework! 

1-7: Introduction to Perimeter, Circumference, and Area

Today’s Objectives find the perimeter and area of common plane figures

Perimeter abbreviated P the distance around a figure measured in units

Circumference abbreviated C the perimeter of a circle measured in units

Area abbreviated A the space inside a figure measured in units2

Formulas for a Square side length s P = 4s A = s2

Formulas for a Rectangle length l and width w P = 2l + 2w OR P = 2(l + w) A = lw

Formulas for a Triangle side lengths a, b, and c P = a + b + c base b height h (forms a right angle with the base) A = ½bh

Formulas for a Circle radius r = ½d diameter d = 2r C = 2 r or d A = r2 is approximately 3.14

Pythagorean Theorem a2 + b2 = c2 only works for right triangles a and b are legs c is the hypotenuse this is the longest side located across from the right angle

Question What do you call the perimeter of a circle?

Question Explain how to find the circumference and area of a circle if you know its diameter.

Question What is the difference between the area and perimeter of a figure?

Skill 1-7 I will find the perimeter and area of common plane figures.

Number 1 The perimeter of a square is 12 meters. What is the length of a side of the square?

Number 2 You are putting a fence around a rectangular garden with length 15 feet and width 8 feet. What is the length of the fence that you will need?

Number 3 Find the area and circumference.

Number 4 Find the area and perimeter.

Number 5 Find the area and perimeter.

Let’s Do Some Homework! 