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Susan Chaffee Geometry Chapter 1 Spring 2009 Lesson 3 – Sections 1.5-1.6 Unit: Chapter 1 Tools of Geometry Lesson: Section 1.5 Section 1.6 Measuring Segments Measuring Angles Lesson Goals: Students will understand how to find segment and angle measures. Lesson Objectives: Materials and/or Special Notes: Angle Names worksheet (optional) Video Introduction: Congruent – Two figures are both equal in size and similar in shape Motivation: Recall the meaning of the word geometry, comes from the Greek geo: earth and metrein: to measure. Geometry originated as the science of measuring land. One of the wonders of the world, the pyramids, depended on geometry to assure the sides of the pyramid all meet at the same point. Babylonian tablets dating back to 2500 B.C. demonstrate their understanding of what later became known as Pythagorean theorem (ask if anyone remembers what it’s used for). Section 1-5 Measuring Segments Today we look at measuring segments and angles. What seems like simple and intuitive properties, measuring, are included in Euclid’s Postulates. Ask if students remember what a postulate is. How is it different than a theorem? Recall a segment. Do you think a line or ray has length? TEACHER and STUDENT ACTIVITIES : The measurement of a segment is defined by the following postulate (fancy words for what we already know to be true). SB-1: Show line with arrows. SB-2: Ruler Postulate: The points on a line can be assigned the real numbers, called coordinates, such that the distance is defined as follows: If A has coordinate a and B has coordinate b on a line. Then the distance between A and B is the number given by: |𝑎 − 𝑏| |9-4|=5 SB-3: Def. Length of a segment – Defined by the ruler postulate as the absolute value of the distance between coordinates. Notation: 𝐴𝐵 is the notation to represent of the length of segment ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ with coordinates a and b has length |𝑎 − 𝑏| Example: A segment 𝐴𝐵 SB-4: GD: Congruent – Two figures are both equal in size and similar in shape. ̅̅̅̅ ≅ 𝐶𝐷 ̅̅̅̅ implies that the segments have the same length, i.e. AB=CD Notation: 𝐴𝐵 Susan Chaffee Geometry Chapter 1 Spring 2009 Example: Geometric shapes can be congruent, such as triangles, segments, angles, etc… NonExample: The equal sign “=” not used to compare geometric figures. Important: AB represents a real number, ̅̅̅̅ 𝐴𝐵 is a geometric shape, a segment 4 equivalent statements of congruence, show on board. GD: Postulate 1-6: Segment Addition Postulate. If 3 points A, B, and C are collinear and B is between A and C then AB+BC=AC Example 1: Find XY and YX. Example 2: XY=4 and YZ=5, what is XZ? Example 3: AB=2 and AC=10, what is BC? Example 4: Line AB=25, Find x 2x-6 x+7 GD: Midpoint – A midpoint divides a segment into two congruent segments. Example: If M is midpoint of ̅̅̅̅ 𝐴𝐵, then ̅̅̅̅̅ 𝐴𝑀 ≅ ̅̅̅̅̅ 𝑀𝐵 NonExample: A ray or line can not have a midpoint. Example 5: In section 1-7 we will look at distance in 2 dimensions. Brain Break or video http://studio4learning.tv/sub_subject.php?pl=182&id=630 Section 1-6 Measuring Angles Angle – formed by two rays with same endpoint ⃗⃗⃗⃗⃗ 𝑎𝑛𝑑 𝐵𝐶 ⃗⃗⃗⃗⃗ . Notation: ∠𝐴𝐵𝐶 is an angle with vertex B and rays, 𝐵𝐴 Example: 4 ways to name the angle as shown. Postulate 1-7: Protractor Postulate. Let OA and OB be opposite rays in a plane. All rays with endpoint ⃡⃗⃗⃗⃗ are assigned unique real numbers between 0 and 180. Assign 𝑂𝐴 ⃗⃗⃗⃗⃗ to O, lying on one side of 𝐴𝐵 coordinate 0 and ⃗⃗⃗⃗⃗ 𝑂𝐵 coordinate to 180. ⃗⃗⃗⃗⃗ If a ray 𝑂𝐶 has coordinate c and a second ray ⃗⃗⃗⃗⃗⃗ 𝑂𝐷 has coordinate d, then 𝑚∠𝐶𝑂𝐷 = |𝑐 − 𝑑| Example : 𝑚∠𝐶𝑂𝐷 = 80. Must put “m” in front of angle. Demonstrate how to measure with protractor – line up origin of protractor at end of the ray. When comparing angles, we call them congruent if they have the same measure (just as with segments) For example : 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐷𝐸𝐹 same measure, implies ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹. Tik marks are also used. Worksheet. Susan Chaffee Classify angles Angle Measure Geometry Chapter 1 Angle name Between 0 and 90 Acute 90 Right Between 90 and 180 Obtuse 180 Straight Spring 2009 Example Worksheet: Use protractor to measure and classify angles. Given a right angle, find the missing: Only makes sense if ray is between the other 2 rays. ⃗⃗⃗⃗⃗ is in between 𝑂𝐴 ⃗⃗⃗⃗⃗ and 𝑂𝐶 ⃗⃗⃗⃗⃗ , then 𝑚∠𝐴𝑂𝐶 = 𝑚∠𝐴𝑂𝐵 + Postulate 1-8: Angle Addition Postulate . If 𝑂𝐵 𝑚∠𝐵𝑂𝐶 Example: Suppose m∠ABC=88 and 𝑚∠1=42, what is measure of angle 2? Angle Pairs – have special names. 2 angles with special relationships to each other. NAME Description Congruent 2 angles with same measure Vertical angles 2 angles whose sides are opposite rays Adjacent 2 coplanar angles with common side, vertex and no common interior points Complementary 2 angles whose measurements sum to 90 Supplementary 2 angles whose measures sum to 180 We will revisit these names often, so know them well! Example In the following diagram identify 1. vertical pairs, 2. Supplementary pairs 3. complementary pairs 4. Adjacent pairs 5. Congruent Learning Log: Measuring Angles – Write a step by step description of how you use the protractor to measure an angle. Draw a diagram and label with points, refer to this in your description. Susan Chaffee Geometry Chapter 1 Spring 2009 Lab exercise: If time allows, we will use the computer lab for experimenting with angle constructions, bisectors, and perpendicular bisectors. At end of lesson, students will write learning log describing what they have observed. Lesson Closure: We learned about congruence, a word used in geometry to describe equivalence between angles and segments. Two angles are not equal, they are congruent. In our next lesson we will learn how to construct segments and angles using compass and straightedge! Assignments: p.33,#1-15(odd),16-19,29-32, p. 40,#1,9-13,16-21,24,25,27,28,31,42 Assessment: Formative: Summative: Lesson Accommodations: