Lesson 1-4: Measuring Segments and Angles. Consider the following section of a ruler showing 1 and 2 :

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1 Lesson -4: Measuring Segments and ngles onsider the following section of a ruler showing and : How many points are there between the and the marks? Did you say three? Don t be fooled by the fact that only the ½ and ¼ marks are displayed. There are really an infinite number of points between and. Just keep dividing the gaps in half. You can keep going forever! Now, what would you say the distance is between the following two points? ssume it is in proportion with the ruler above Did you estimate? That would be correct. Did you subtract point s location on the ruler ( ) from point s location ( )? = ould you do? Sure! Remember, distance is an absolute value of the difference between two points on a number line. So - = - =. Excellent, now a quick digression hang with me here What does one-to-one correspondence mean? It means you can pair every item in one set up with one and exactly one item in another set with none left over in either set you use them all up. -to- -to- Not -to Page of 6

2 Lesson -4: Measuring Segments and ngles Postulate -5 (the Ruler Postulate) Points on a line can be put into a one-to-one correspondence with the real number line so that the distance between any two points is the absolute value of the distance of the corresponding numbers. Simply put you can use a ruler (or any other straight measuring device) to find distance between points. Seem like a rather obvious thing to state? Remember, we are building the system of geometry here if we are going to assume something, we need to state it clearly and make sure it is true. Go back to lesson - and review what an axiom/postulate is. Segment length We represent the length of a segment as (no bar above the letters). Given segment and the points a, b on the real number line corresponding to and, the length of = = a - b ongruent segments: Two segments with the same length. Represented by the symbol Here is the relationship stated in math : o If = FE then FE Here it is in English o If the lengths of segments and FE are equal, the segments are congruent. Example: Pg 9, problem # D E Is D E? D = 9 E = No, they are not congruent. Page of 6

3 Lesson -4: Measuring Segments and ngles Now, look at, and how do they relate? D E In math: o + = In English: o The sum of the lengths of segments and equals the length of segment. Postulate -6 (the Segment ddition Postulate) If point is on and between points and, then + =. This is a powerful postulate as you will see in the following example Example: Pg 9, problem #0 R S T If RS = 3x +, ST = x, RT = 64 What is x? Using postulate -6 we know that RS + ST = RT, so What is RS? (3x + ) + (x ) = 64 (by substitution) 3x + x + = 64 5x = 64 5x = 65 x = 3 RS = 3x + = 3(3) + = 40 What is RT? o ST = x = (3) = 4 Page 3 of 6

4 Lesson -4: Measuring Segments and ngles onsider the following: If =, what would you call point? It is the midpoint of the segment. Midpoint of a segment: The point that divides the segment into two congruent segments. What would we have if two rays shared the same endpoint but were not collinear? The two rays would form an angle. ngle: Two rays that share a common endpoint provided the two rays do not lie on the same line. The rays are the sides of the angle. The endpoint is the vertex of the angle. Represented by the symbol Named by (see above diagram): o The three points defining the rays: (vertex point always in the middle). o y the vertex point (if there is no ambiguity): o You can also number the angles and refer to them by the number: In the following, does it make sense to name any of the angles? D No, there are 3 angles shown ( D, & D ) which one is it? Page 4 of 6

5 Lesson -4: Measuring Segments and ngles Remember the Ruler Postulate? We found a way of measuring segments by recognizing a -to- correspondence with a ruler. an you think of something we could do the same with so we can measure angles? Postulate -7 (the Protractor Postulate) Let rays Oand O be opposite rays in a plane. Rays O, O and all rays with endpoint O can be drawn on one side of can be pared with the real numbers from 0 to 80 so that:. Ois pared with 0 and O is pared with 80. if O is pared with x and OD is pared with y, then m OD = x - y We note the measure of OD as m OD. gain, this simply means we can measure angles in degrees. Now that we can measure angles, we can classify special types of angles. Types of angles cute: 0 < x < 90 Right: x = 90 Obtuse: 90 < x < 80 Straight: x = 80 Remember the Segment ddition Postulate? Given that we re talking about angles, what do you think our next postulate will be? Postulate -8 (the ngle ddition Postulate) If point D lies in the interior of, then m D + m D = m D Page 5 of 6

6 Lesson -4: Measuring Segments and ngles Now, what do you imagine DEF means? ongruent angles: ngles with the same measure. If m OD = m FGH, then OD FGH ssign homework p. 9-3 odd odd odd Page 6 of 6

Basics of Geometry Unit 1 - Notes. Objective- the students will be able to use undefined terms and definitions to work with points, lines and planes.

Basics of Geometry Unit 1 - Notes. Objective- the students will be able to use undefined terms and definitions to work with points, lines and planes. asics of Geometry Unit 1 - Notes Objective- the students will be able to use undefined terms and definitions to work with points, lines and planes. Undefined Terms 1. Point has no dimension, geometrically