Angles MP4: Model with mathematics. MP5: Use appropriate tools - PDF document

Slide 1 / 190 Slide 2 / 190 Geometry Angles 2015-10-21 www.njctl.org Slide 3 / 190 Slide 4 / 190 Table of Contents for Videos Table of Contents click on the topic to go to that section Demonstrating Constructions click on the topic to go Angles to that video Congruent Angles Congruent Angles Angles & Angle Addition Postulate Angle Bisectors Protractors Special Angle Pairs Proofs Special Angles Angle Bisectors Locus & Angle Constructions Angle Bisectors & Constructions PARCC Released Questions Slide 5 / 190 Slide 6 / 190 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. Angles MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that Return to Table the questions address are listed in the Pull-tab. of Contents

Slide 7 / 190 Slide 8 / 190 Angles Angles Definition 8: A plane angle is the inclination to one another of The measure of angle is the amount that one line, one ray or two lines in a plane which meet one another and do not lie in a segment would need to rotate in order to overlap the other. straight line. A Whenever lines, rays or A In this case, Ray BA would segments in a plane have to rotate through an intersect, they do so at angle of x degrees in order an angle. to overlap Ray BC. xº xº B C B C Slide 9 / 190 Slide 10 / 190 Angles Measuring angles in degrees In this course, angles will be measured with degrees, The use of 360 degrees to represent a full rotation back to the which have the symbol º. original position is arbitrary. A For a ray to rotate all the Any number could have been way around from ray BC, as used, but 360 degrees for a full shown, back to ray BC would rotation has become a represent a 360º angle. standard. 360º xº B C Slide 11 / 190 Slide 12 / 190 Measuring angles in degrees Right Angles Definition 10: When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the The use of 360 for a full rotation is thought that it come from ancient equal angles is right, and the straight line standing on the Babylonia, which used a number system based on 60. other is called a perpendicular to that on which it stands. Their number system may also be linked to the fact that there are The only way that two 365 days in a year, which is pretty close to 360. lines can intersect as shown and form equal 360 is a much easier number to work with than 365 since it is A adjacent angles, such as divided evenly by many numbers. the angles shown here These include 2, 3, 4, 5, 6, 8, 9, 10 and 12. where m ∠ ABC = m ∠ ABD, is if they are right angles, xº xº 90º. D B C

Slide 13 / 190 Slide 14 / 190 Right Angles Right Angles This definition is unchanged today and should be familiar to you. Fourth Postulate: That all right angles are equal to one another. Perpendicular lines, segments or rays form right angles. Not only are adjacent right angles equal to each other as shown below, all right angles are equal, even if they are not adjacent, for instance, all three of the below right angles are equal to one another. If lines intersect to form adjacent A equal angles, then they are perpendicular and the measure of A A those angles is 90º. 90º B C xº xº 90º D B C B When perpendicular lines meet, they form equal adjacent C angles and their measure is 90º. Slide 15 / 190 Slide 16 / 190 Right Angles Obtuse Angles Definition 11: An obtuse angle is an angle greater than a There is a special indicator of a right angle. right angle. A A It is shown in red in this case to make it easy to recognize. 135º B C B C Slide 17 / 190 Slide 18 / 190 Acute Angles Straight Angle A definition that we need that was not used in The Elements is that Definition 12: An acute angle is an angle less than a right angle. of a "straight angle." That is the angle of a straight line. A B C A 45º 2 questions to discuss with a partner: B C Is this an acute or obtuse angle? Explain why. What is the degree measurement of the angle?

Slide 19 / 190 Slide 20 / 190 Angles Reflex Angle Another modern definition that was not used in The Elements is that of a "reflex angle." That is an angle that is greater than 180º. In the next few slides we'll use our responders to review the 235º names of angles by showing angles from 0º to 360º in 45º increments. B C Angles can be of any size, not just increments of 45º, but this is just to give an idea for what a full rotation looks like. This is also a type A of obtuse angle. Slide 21 / 190 Slide 22 / 190 1 This is an example of a (an) ________ angle. 2 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute A B obtuse 0º A B obtuse B C C right C right 45º B C D reflex D reflex E straight E straight Slide 23 / 190 Slide 24 / 190 3 This is an example of a (an) ________ angle. 4 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute A B obtuse B obtuse A C right C right 135º 90º D reflex D reflex B C B C E straight E straight

Slide 25 / 190 Slide 26 / 190 5 This is an example of a (an) ________ angle. 6 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute 235º B obtuse B obtuse B C 180º C right C right B C A D reflex D reflex A E straight E straight Slide 27 / 190 Slide 28 / 190 7 This is an example of a (an) ________ angle. 8 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute 270º C B obtuse B obtuse B 315º C B C right C right D reflex D reflex A E straight E straight A Slide 29 / 190 Slide 30 / 190 Naming Angles 9 This is an example of a (an) ________ angle. Choose all that apply. An angle has three parts, it has two sides and one vertex, where the sides meet. A acute B obtuse A 360º In this example, the sides C right are the rays BA and BC B side A C and the vertex is B. D reflex θ E straight vertex side B C

Slide 31 / 190 Slide 32 / 190 Interior of Angles Naming Angles Any angle with a measure of less than 180º has an interior An angle can be named in three different ways: and exterior, as shown below. By its vertex (B in the below example) · leg A By a point on one leg, its A · vertex and a point on the θ other leg (either ABC or vertex CBA in the below example) leg B C Exterior Interior θ Or by a number or a symbol placed inside the angle (e.g. · Greek letter, θ, in the figure) B C Slide 33 / 190 Slide 34 / 190 Naming Angles Naming Angles Using the vertex to name an angle doesn't work in some The angle shown can be called ∠ ABC , ∠ CBA, or ∠B . cases. Why would it be unclear to use the vertex to name the angle in the image below? When there is no chance C of confusion, the angle may also be identified by its vertex B. D How many angles do A The sides of ∠ABC 32° you count in the B are rays BC and BA image? A θ α The measure of ∠ ABC is 32 degrees, which can be rewritten as B C m ∠ABC = 32 º. Slide 35 / 190 Slide 36 / 190 Intersecting Lines Form Angles Naming Angles When an angle is formed by either two rays or segments with a What other ways could you name ∠ ABC, ∠ ABD and ∠D BC in the shared vertex, one included angle is formed. case below? (using the side - vertex - side method) Shown as θ in the below diagram to the left. When two lines intersect, 4 angles are formed, they are numbered in the diagram below to the right. D A A 2 1 θ 3 4 α B C θ B C How could you name those 3 angles using the letters placed inside the angles?

Slide 37 / 190 Slide 38 / 190 Intersecting Lines Form Angles 10 Two lines ________________ meet at more than one point. A Always These numbers used have no special significance, but just show the 4 angles. When rays or segments intersect but do not have a common vertex, they also create 4 angles. B Sometimes C Never A 2 1 θ 3 4 B C Slide 39 / 190 Slide 40 / 190 11 An angle that measures 90 degrees is __________ a 12 An angle that is less than 90 degrees is right angle. ___________ obtuse. Always Always A A Sometimes Sometimes B B C Never C Never Slide 41 / 190 Slide 42 / 190 13 An angle that is greater than 180 degrees is _______ referred to as a reflex angle. Always A Sometimes B Congruent Never C Angles Return to Table of Contents