Slide 1 / 232 Slide 2 / 232 Geometry Triangles 2015-12-08 www.njctl.org Slide 3 / 232 Slide 4 / 232 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. Click on the topic to go to that section MP1: Making sense of problems & persevere in solving them. · Triangles MP2: Reason abstractly & quantitatively. · Triangle Sum Theorem MP3: Construct viable arguments and critique the reasoning of others. · Exterior Angle Theorem MP4: Model with mathematics. MP5: Use appropriate tools strategically. · Inequalities in Triangles MP6: Attend to precision. · Similar Triangles MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. · PARCC Sample Question and Applications 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 the questions address are listed in the Pull-tab. Slide 4 (Answer) / 232 Slide 5 / 232 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 Math Practice others. MP4: Model with mathematics. Triangles MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math [This object is a pull tab] Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. Return to Table of Contents If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
Slide 6 / 232 Slide 7 / 232 Geometric Figures Geometric Figures Definition 19. Rectilinear figures are those which are contained by Definition 13. A boundary is that which is an extremity of anything. straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those Definition 14. A figure is that which is contained by any boundary contained by more than four straight lines. or boundaries. Euclid now makes the transitions to geometric figures, which are created by a boundary which separates space His definitions from 15 to 18 relate to circles, which we will discuss into that which is within the figure and that which is not. later. In this chapter, we will be discussing triangles, which are an example of a rectilinear figure: a figure bounded by straight lines. A triangle is bounded by three lines. Slide 8 / 232 Slide 9 / 232 Parts of a Triangle 1 The letter on this triangle that corresponds to a side is: Each triangle has three sides and three vertices. A Each vertex is where two sides meet. A pair of sides and the vertex define an angle, so each triangle C includes three angles. Write "side" next to each side and circle the vertices on the triangle below. B Slide 9 (Answer) / 232 Slide 10 / 232 1 The letter on this triangle that corresponds to a side is: 2 The letter on this triangle that represents a vertex is: A C Answer C C B B A [This object is a pull tab]
Slide 10 (Answer) / 232 Slide 11 / 232 Parts of a Triangle 2 The letter on this triangle that represents a vertex is: C Each vertex is named with a letter. Answer A C The sides can then be named with the letters of the two vertices on either side of it. B A B A The triangle is named with a Name the 3 sides of this triangle triangle symbol Δ in front [This object is a pull tab] followed by the three letters of ______ ______ ______ its vertices. Slide 12 / 232 Slide 12 (Answer) / 232 3 What is the name of the side shown in red? 3 What is the name of the side shown in red? A AB A AB C C B BC B BC Answer C AC C AC C A B A B [This object is a pull tab] Slide 13 / 232 Slide 13 (Answer) / 232 4 What is the name of the side shown in red? 4 What is the name of the side shown in red? C C A AB A AB B BC B BC Answer C AC C AC A A B A B [This object is a pull tab]
Slide 14 / 232 Slide 14 (Answer) / 232 5 Which of the following are names of this triangle? 5 Which of the following are names of this triangle? A A ΔABC D ΔCAB ΔABC D ΔCAB B ΔBCA B ΔBCA E all of these E all of these C ΔACB C ΔACB Answer C C E [This object is a pull tab] A B A B Slide 15 / 232 Slide 15 (Answer) / 232 Parts of a Triangle Parts of a Triangle C C opposite adjacent Answer A side is opposite an A side is opposite an angle if it does not angle if it does not touch it. Otherwise, it touch it. Otherwise, it Questions on this slide is adjacent to the is adjacent to the A B A B angle. angle. address MP6 [This object is a pull tab] In the above, the red side is ________________ A, In the above, the red side is ________________ A, while the green sides are ________________ to A. while the green sides are ________________ to A. Slide 16 / 232 Slide 16 (Answer) / 232 6 Which side is opposite angle B? 6 Which side is opposite angle B? A AB A AB C C B CA B CA Answer C BC C BC B D None D None A B A B [This object is a pull tab]
Slide 17 / 232 Slide 17 (Answer) / 232 7 Which side is opposite angle A? 7 Which side is opposite angle A? A AB A AB C C B CA B CA Answer C BC C BC C D None D None B B A A [This object is a pull tab] Slide 18 / 232 Slide 18 (Answer) / 232 8 Which sides are adjacent to angle C? 8 Which sides are adjacent to angle C? C C A AB & BC A AB & BC B CA & BA B CA & BA C BC & CA C BC & CA Answer C D None D None B B A A [This object is a pull tab] Slide 19 / 232 Slide 19 (Answer) / 232 9 Which sides are adjacent to angle B? 9 Which sides are adjacent to angle B? A AB & BC A AB & BC B CA & BA C B CA & BA C C BC & CA C BC & CA Answer A D None D None B B A A [This object is a pull tab]
Slide 20 / 232 Slide 21 / 232 Classifying Triangles Types of Triangles In general, a triangle can have sides of all different lengths and angles Definition 20: Of trilateral figures, an equilateral triangle is that which of all different measure. has its three sides equal, an isosceles triangle is that which has two of its sides alone equal, and a scalene triangle is that which has its However, there are names given to triangles which have specific or three sides unequal special angles or some number of equal sides or angles. Triangles can be classified by their sides or by their Euclid defined the names for a number of these in his definitions. angles. In this definition, Euclid used the sides. In his next definition, Euclid uses the angles. Slide 22 / 232 Slide 23 / 232 Classifying Triangles Classifying Triangles Acute Triangles Definition 21: Further, of trilateral figures, a right-angled triangle is In an acute triangle, every angle of a triangle is acute. that which has a right angle, an obtuse-angled triangle is that which has an obtuse angle, and an acute-angled triangle is that which has Definition 21: "... an acute-angled triangle is that which has its its three angles acute. three angles acute." We will draw from both definitions, since in several cases both Notice that no angle is equal to or definitions apply to the same triangle. greater than 90º in this triangle. Slide 24 / 232 Slide 25 / 232 Classifying Triangles Classifying Triangles Right Triangles Isosceles Triangles A right triangle has one right angle and two acute angles. An isosceles triangle has at least two sides with equal length. Definition 21: "... a right-angled triangle is that which has a right angle..." The angles opposite those equal sides are of equal measure. Notice that one angle is 90º, which means that the other two sum to 90º; and they are acute. Definition 20: "...an isosceles triangle is that which has two of its sides alone equal..." The side opposite the right angle is called the hypotenuse and the other two sides are called the legs. x º x º
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