Honors Geometry Fundamentals Prerequisite Skills Copy and complete

Honors Geometry Fundamentals

Prerequisite Skills Copy and complete the statement. 1) The distance around a rectangle is called its perimeter _____, and the distance around a circle is circumference called its _______. 2) The number of square units covered by a figure is called its area ______.

Prerequisite Skills Evaluate the expression) 3) |4 – 6| 2 4) |3 – 11| 8 5) |– 4 + 5| 1 6) |– 8 – 10| 18

What do you think? A) Where would you live, if your house was at the NW corner of B St and 3 rd Ave ? B) Your friend lives at A Ave and 1 st St. How many blocks is that from your house? 7 Blocks!

What do you know? (5, 7) (– 5, – 2) I III (0, 6) none (6, – 1) IV (– 8, 9) II

What do you know? 2 Find the coordinates of three points that lie on the graph of y = -2 x + 5 plot the points, then graph the line. (1, 3) (4, -3) (-2, 9) Are these the only points possible?

Relationships How many ways are there to show a relationship between two variables? Table (Coordinates) Graph Equation (Inequality)

What do you think? What is a point? What is a line? What is a plane?

Points, Lines and point �a position in space Planes A named by capital letters Point A line �an infinite set of points stretching in opposite directions named by single lower case cursive letters or by 2 points on the line AB

Definition of Collinear Points – 3 or more points that lie on the same line.

Collinear? A B C D E F H G I Name three sets of collinear points.

Points, Lines and Planes (cont) plane → an infinite flat surface (drawn as a parallelogram) named by capital cursive letters or by 3 noncollinear points

Definition Coplanar Points → 4 or more noncollinear points that lie in the same plane.

Discuss with your Neighbor 1) How do we name lines? 2) How do we name planes?

Practice 1 List all the possible names for the following: a) b)

Practice 2 For each of the following determine what information that you can say about each figure: a) b) c) d)

Between For a point to be between two points (say A & B) in a plane it must be collinear with the points (A & B) A C D B C is between A & B D is NOT between A & B

Naming Segments and Rays Definition of a Segment A segment is two given points in a plane and all of the points between the given points. A segment is named by its endpoints with a bar over them in no particular order. R T RT or TR

Naming Segments and Rays Informal definition of a ray A ray can be considered a half-line Formal definition of a ray A ray is a segment (call it AB) and all of the possible points C such that B is between A and C A B C

Naming Segments and Rays are named by their endpoint, then any other point on the ray with a one sided arrow pointing from left to right A B AB or AC C

EXAMPLE 1 Name points, lines, and planes a) Give two other names for PQ and for plane R. b) Name three points that are collinear. c) Name four points that are coplanar. SOLUTION a) Other names for PQ are QP and line n. Other names for plane R are plane SVT and plane PTV. b) Points S, P, and T lie on the same line, so they are collinear. c) Points S, P, T, and V lie in the same plane, so they are coplanar. S, P, T and Q are also coplanar, even though the plane that contains them is not drawn.

GUIDED PRACTICE 1) for Example 1 Give two other names for ST. Name a point that is not coplanar with points Q, S, and T. Other names for ST are TS and PT. Point V is not coplanar with points Q, S and T

EXAMPLE 2 Name segments, rays, and opposite rays a) Give another name for GH. b) Name all rays with endpoint J. Which of these rays are opposite rays? a) Another name for GH is HG. b) The rays with endpoint J are JE , JG , JF , and JH. The pairs of opposite rays with endpoint J are JE and JF , and JG and JH.

GUIDED PRACTICE 2) for Example 2 Give another name for EF Another name for EF is FE 3) Are HJ and JH the same ray ? Are HJ and HG the same ray? Explain. No; HJ and JH have different endpoints, so they are not the same ray Yes; points J and G lie on the same side of H, so they are the same ray

ASSIGNMENT #1 • P. 5 -8 1 -6, 8 -13, 17 -22, 27 -37 odd, 39, 44, 46

Daily Homework Quiz For use after Lesson 1 -1 Use the figure for exercises 1 -2 1) Give two other names for AE. ANSWER 2) EC , AC Give another name for plane S. ANSWER plane DEF or EBF

Daily Homework Quiz For use after Lesson 1 -1 Use the figure for exercises 3 -4 3) Name three collinear points. A, E, C or D, E, B 4) Name the intersection of AC and plane S. E

THE RULER POSTULATE • The points on a line can be matched one to one with the real numbers. The real number that corresponds to a particular point is the coordinate of that point. • The distance between points A and B (written as AB), is the absolute value of the difference of the coordinates of A and B. • The importance of the ruler postulate is that it allows use to use a ruler. (To assign a length in some unit to an object) AB = | x 2 – x 1 |

EXAMPLE 1 Apply the Ruler Postulate Measure the length of ST to the nearest tenth of a centimeter. SOLUTION Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align S with 2, T appears to align with 5. 4 ST = |5. 4 – 2| = 3. 4 The length of ST is about 3. 4 centimeters

What do you think? In which of the following, would you say that point S is between points R and T? A B In order for a point to be BETWEEN two other points, it must be collinear with the other two.

More on Between Compare RS + ST to RT, is: a) RS + ST > RT b) RS + ST = RT c) RS + ST < RT We decided that in order to consider S to be between R and T, the points needed to be collinear.

Even more on Between Compare RS + ST to RT, is: a) RS + ST > RT b) RS + ST = RT c) RS + ST < RT

The 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. Start ST O P ! This means that if a segment that is divided into two parts by a point, then the length of the segment is equal to the sum of the segments on either sides of that point) or This means that if the length of a segment that is divided into two parts by a point is equal to the sum of the segments on either sides of that point, then the point is between the endpoints of the segment)

EXAMPLE 2 Maps Apply the Segment Addition Postulate The cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri. SOLUTION Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate. LS = LT + TS = 380 + 360 = 740 The distance from Lubbock to St. Louis is about 740 miles.

GUIDED PRACTICE for Examples 1 and 2 In Examples 1 and 2, use the diagram shown. 1) Use the Segment Addition Postulate to find XZ. XZ = XY + YZ = 23 + 50 = 73 2) In the diagram, WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z? NO; Because W is not between X and Z.

EXAMPLE 3 Find a length Use the diagram to find GH. SOLUTION Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. FH = FG + GH Segment Addition Postulate. 36 = 21 + GH Substitute 36 for FH and 21 for FG. 15 = GH Subtract 21 from both sides.

EXAMPLE 4 Compare segments for congruence Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JK and LM are congruent. SOLUTION To find the length of a horizontal segment, find the absolute value of the difference of the xcoordinates of the endpoints. Use Ruler Postulate. JK = |2 – (– 3)| = 5 To find the length of a vertical segment, find the absolute value of the difference of the ycoordinates of the endpoints. Use Ruler Postulate. LM = |– 2 – 3| = 5 Since JK = 5 and LM = 5, then JK = LM and therefore JK ≅ LM

GUIDED PRACTICE for Examples 3 and 4 5) Use the diagram below to find WX. Use the segment addition postulate to write an equation. Then solve the equation to find WX SOLUTION VX = VW + WX 144 = 37 + WX 107 = WX

GUIDED PRACTICE for Examples 3 and 4 6) Plot the points A(– 2, 4), B(3, 4), C(0, 2), and D(0, – 2) in a coordinate plane. Then determine whether AB and CD are congruent. ANSWER Length of AB is not equal to the length of CD, so they are not congruent

Assignment #2 p. 12 -14 6 -30 even, 31, 33, 36