Draw six segments that pass through every dot in the figure without taking your pencil off the paper. Session 2

Counterexample It only takes 1 false example to show that a conjecture is not true. Example 4: Find a counterexample for these statements… All dogs have spots. All prime numbers are odd.

Counterexample Example 5: Find a counterexample: For all real numbers x, the expression x 2 is greater than or equal to x = 0.25 and 0.25  0.5

Point Has no size, no dimension Is represented by a dot Named by using a capital letter We would call this one “point E.”

Has one dimension Is made up of infinite number of points and is straight Arrows show that the line extends without end in both directions Can be named with a single lowercase cursive letter OR by any 2 points on the line Symbol Line Names of these lines:

COLLINEAR Points lie on the same line NONCOLLINEAR Points do NOT lie on the same line

Example Points D, B, & C are in a straight line so they are _______________ Points A, B, & C are ________________ A B C D E

2 dimensions Extends without end in all directions Takes at least 3 noncollinear pts. to make a plane Named with a single uppercase script letter or by 3 noncollinear pts. Plane Names of these planes: M

COPLANAR Points lie in the same plane NONCOPLANAR Points do NOT lie in the same plane

Is straight and made up of points Has a definite beginning and definite end Name a line segment by using the endpoints only You will always use two letters to name a segment Symbol Line Segment Name of these segments: A B C D E F G H Name of segment from 3 to 0.

Is straight and made up of points Has a beginning but no end Starting pt. of a ray is called the endpoint Name a ray by using the endpt. 1st and another point on the ray You will always use two letters to name a ray Symbol Ray Names of these rays:

Made up of two rays with a common endpoint The point is called the vertex of the angle The rays are called the sides of the angle Symbol  Several ways to name an angle Angles Names:

Postulates Postulates are facts about geometry that are accepted as true.

Ruler Postulate Every point on a line can be matched with a coordinate on the number line. The distance between two points is the absolute value of the difference of the coordinates. PQPQ Do not copy all of this

Ruler Postulate (in other words) If P is at 15 and Q is at 18, the distance from P to Q is P Q The distance from P to Q is written: PQ How is this different from PQ?

Ex. 1 Use the number line to find each measure. 01 F EDBIH a) DHb) EIc) FB

Segment Addition Postulate If B is between A and C, then AB + BC = AC. ABC AB + BC = AC

Example 2 Points X, Y, and Z are collinear. If XY = 12, YZ = 47, and XZ = 35, determine which point is between the other two. XY 12 XZ 35 YZ 47

Example 3 If QS = 29 and QT = 52, find ST. P Q RST QS + ST = QT 29 + ST = 52 ST = 23

Example 4 If FG = 12 and FJ = 47, find GJ. F G HJ FG + GJ = FJ 12 + GJ = 47

If A(x 1, y 1 ) and B(x 2, y 2 ) are points on the coordinate plane, then:

5. Find the distance between the points. Round to the nearest tenth.

6. Find the distance between the points. Round to the nearest tenth.

7. Find the distance between the points. Round to the nearest tenth.

Congruent Segments Two segments are congruent if and only if they have the same length. YXAB

TWO DIFFERENT RAYS THE SAME INITIAL POINT SIDES VERTEX

A B C 1

OR M A T H 1 2

A B C

How To Measure An Angle

R S T P Why can’t you name any of the angles  S?

R S T P 1 Example 1

Y 2 Example 2 N M K J

A L U Y Example 3

ACUTE ANGLES = Greater than 0 and less than 90 RIGHT ANGLES = Measure exactly 90 OBTUSE ANGLES = Greater than 90 and less than 180 STRAIGHT ANGLES = Measure exactly 180