Focus Graph and label the following points on a coordinate grid. P(-1, -1), Q(0, 4), R(-3, 5), S(2, 5), and T(3, -4) Name three noncollinear points. Name three collinear points. Name two intersecting lines Lesson 4: Measuring Segments and Angles

Measuring Segments and Angles Lesson 1-4 Measuring Segments and Angles Lesson 4: Measuring Segments and Angles

The Ruler Postulate (1-5) The Ruler Postulate: Points on a line can be paired with the real numbers in such a way that: Any two chosen points can be paired with 0 and 1. The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points. Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │ Lesson 4: Measuring Segments and Angles

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 Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Between Definition: X is between A and B if AX + XB = AB. AX + XB = AB AX + XB > AB Lesson 4: Measuring Segments and Angles

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

Lesson 4: Measuring Segments and Angles 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: Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Midpoint Definition: A point that divides a segment into two congruent segments Formulas: On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is . In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and is . Lesson 4: Measuring Segments and Angles

Midpoint on Number Line - Example Find the coordinate of the midpoint of the segment PK. Now find the midpoint on the number line. Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Segment Bisector Definition: Any segment, line or plane that divides a segment into two congruent parts is called segment bisector. Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Angle and Points ray vertex ray Angles can have points in the interior, in the exterior or on the angle. A E D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex. Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points: vertex must be the middle letter This angle can be named as Using 1 point: using only vertex letter * Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called . A C B Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Naming an Angle - continued Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as . A B 2 C * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present. Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Example K is the vertex of more than one angle. Therefore, there is NO in this diagram. There is Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles 4 Types of Angles Acute Angle: an angle whose measure is less than 90. Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is between 90 and 180. Straight Angle: an angle that is exactly 180 . Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Measuring Angles Just as we can measure segments, we can also measure angles. We use units called degrees to measure angles. A circle measures _____ A (semi) half-circle measures _____ A quarter-circle measures _____ One degree is the angle measure of 1/360th of a circle. 360º ? 180º ? ? 90º Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Adding Angles m1 + m2 = mADC also. Therefore, mADC = 58. Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Angle Addition Postulate Postulate: The sum of the two smaller angles will always equal the measure of the larger angle. MRK KRW MRW Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Example: Angle Addition K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK. First, draw it! 3x + x + 6 = 90 4x + 6 = 90 – 6 = –6 4x = 84 x = 21 3x x+6 Are we done? mMRK = 3x = 3•21 = 63º Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Angle Bisector An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles. Example: Since 4   6, is an angle bisector. 5 3 Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. The symbol for congruence is  3 5 Example: 3   5. Lesson 4: Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles Example Draw your own diagram and answer this question: If is the angle bisector of PMY and mPML = 87, then find: mPMY = _______ mLMY = _______ Lesson 4: Measuring Segments and Angles