Use Segments and Congruence Lesson 1-2 Use Segments and Congruence

Warm Up Problems Solve for x.

Theorem Definition: A true statement that follows as a result of other true statements. A statement that can be proven. We will investigate theorems in greater detail later. Your book lists all the Theorems (and Postulates ) we will use this year. Page 926

Postulate or axiom Definition: A rule that is accepted without proof. Examples: Through any two points there is exactly one line. A line contains at least two points. Through any three points, there is exactly one plane. A plane contains at least three points.

Postulates Examples: If two planes intersect, then the intersecting is a line. If two points lie in a plane, then the line containing the two points lie in the same plane.

POSTULATE 1 The Ruler Postulate The Ruler Postulate: Points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written AB, is the absolute value of the difference of the coordinates of A and B. x1 –small numbers are called subscripts (read “x sub one”)

Ruler Postulate : Example Find the distance between P and K. Note: The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points. Therefore, the coordinates of points P and K are 3 and -2 respectively. Substituting the coordinates in the formula │a – b │ PK = | 3 - -2 | = 5 Remember : Distance is always positive

Between AB + BC = AC AX + XB > AB Definition: B is between A and C if AB + BC = AC. AB + BC = AC AX + XB > AB

POSTULATE 2 The Segment Addition Postulate If B is between A and C, then AB + BC = AC. Example: If AB = x , BC = 2x and AC = 12, then, find x, AB and BC. A C B 2x x 12 Step 1: Draw a figure Step 2: Label fig. with given info. AB + BC = AC x + 2x = 12 3x = 12 x = 4 Step 3: Write an equation x = 4 AB = 4 BC = 8 Step 4: Solve and find all the answers

Apply the Segment Addition Postulate Maps 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. ANSWER

Congruent Segments Definition: Segments with equal lengths. (congruent symbol: ) Congruent segments can be marked with dashes. If numbers are equal the objects are congruent. AB: the segment AB ( an object ) AB: the distance from A to B ( a number ) Correct notation: Incorrect notation:

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 x-coordinates of the endpoints. JK = 2 – (– 3) = 5 Use Ruler Postulate.

Compare segments for congruence (continued) To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. LM = – 2 – 3 = 5 Use Ruler Postulate. JK and LM have the same length. So, JK LM. = ~ ANSWER

Homework #2 p. 878: 7-12 p. 12: 1-8, 12-23, 32-34, 37-45