Slide 1 / 210 Slide 2 / 210 Table of Contents Click on the topic to go to that section · Triangles · Triangle Sum Theorem · Exterior Angle Theorem · Inequalities in Triangles · Similar Triangles · Questions from Released PARCC Test Slide 3 / 210 Slide 4 / 210 Geometry Triangles Triangles Return to Table of Contents www.njctl.org Slide 5 / 210 Slide 6 / 210 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 7 / 210 Slide 8 / 210 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 / 210 Slide 10 / 210 Parts of a Triangle 2 The letter on this triangle that represents a vertex is: C Each vertex is named with a letter. 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 followed by the three letters of ______ ______ ______ its vertices. Slide 11 / 210 Slide 12 / 210 3 What is the name of the side shown in red? 4 What is the name of the side shown in red? C A AB A AB C B BC B BC C AC C AC A B A B
Slide 13 / 210 Slide 14 / 210 Parts of a Triangle 5 Which of the following are names of this triangle? A ΔABC D ΔCAB C B ΔBCA E all of these C ΔACB C A side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle. A B In the above, the red side is ________________ A, A B while the green sides are ________________ to A. Slide 15 / 210 Slide 16 / 210 6 Which side is opposite angle B? 7 Which side is opposite angle A? A AB A AB C C CA CA B B C BC C BC D None D None B A A B Slide 17 / 210 Slide 18 / 210 8 Which sides are adjacent to angle C? 9 Which sides are adjacent to angle B? C A AB & BC A AB & BC B CA & BA C B CA & BA C BC & CA C BC & CA D None D None B A B A
Slide 19 / 210 Slide 20 / 210 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 angles. Euclid defined the names for a number of these in his definitions. In this definition, Euclid used the sides. In his next definition, Euclid uses the angles. Slide 21 / 210 Slide 22 / 210 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 23 / 210 Slide 24 / 210 Classifying Triangles Classifying Triangles Right Triangles Isosceles Triangles A right triangle has one right angle and two acute angles. An isosceles triangle has 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 x
Slide 25 / 210 Slide 26 / 210 Classifying Triangles Classifying Triangles Isosceles Triangles Obtuse Triangles An obtuse triangle has one angle which is greater than 90 º and The equal angles, of measure x in this two acute angles. diagram, are called the base angles. The side between them is called the base. Definition 21: "... an obtuse-angled The other two sides, opposite the base triangle is that which has an obtuse angles and congruent to each other are angle..." called the legs. This is a special case of an acute triangle. Notice that one angle is greater than 90º, which means that the other two sum to less than 90º; and they are acute. . x x Slide 27 / 210 Slide 28 / 210 Classifying Triangles Classifying Triangles Equiangular / Equilateral Triangles Scalene Triangles An equiangular, or equilateral, triangle has angles of equal None of the sides or angles of a scalene triangle are congruent measure and sides of equal length. with one another. Definition 20: "... an equilateral triangle is x Definition 20: "...a scalene triangle is that which has its three sides equal..." that which has its three sides unequal..." All the angles are of equal measure and all the sides are of equal length. Note that in this triangle none of the sides or angles are equal. Each angle measures 60º. x x This is a special acute triangle. Slide 29 / 210 Slide 30 / 210 10 An isosceles triangle is _______________ an 11 An obtuse triangle is _______________ an isosceles equilateral triangle. triangle. A Sometimes A Sometimes B Always B Always C Never C Never
Slide 31 / 210 Slide 32 / 210 12 A triangle can have more than one obtuse angle. 13 A triangle can have more than one right angle. True True False False Slide 33 / 210 Slide 34 / 210 14 Each angle in an equiangular triangle measures 60° 15 An equilateral triangle is also an isosceles triangle True True False False Slide 35 / 210 Slide 36 / 210 16 This triangle is classified as _____. 17 This triangle is classified as _____. (Choose all that apply.) (Choose all that apply.) A acute A acute B right B right 57 º 60 º 8.6 8.6 C isosceles C isosceles 6.1 8.7 D obtuse D obtuse 60 º 60 º 79 º 44 º 8.6 E equilateral E equilateral 7.4 F equiangular F equiangular G scalene G scalene
Slide 37 / 210 Slide 38 / 210 18 This triangle is classified as _____. 19 This triangle is classified as _____. (Choose all that apply.) Choose all that apply. A acute A acute B right B right 26° 4.8 4.8 4.5 C isosceles C isosceles 2.5 45° 45° D obtuse D obtuse 128° 26° 6.8 2.5 E equilateral E equilateral F equiangular F equiangular G scalene G scalene Slide 39 / 210 Slide 40 / 210 Example Example Measure and Classify the triangle by sides and angles Measure and Classify the triangle by sides and angles scalene, obtuse Click for Answer Click for Answer isosceles, acute Click for Answer Click for Answer Slide 41 / 210 Slide 42 / 210 Example 20 Classify the triangle with the given information: Measure and Classify the triangle by sides and angles Side lengths: 3 cm, 4 cm, 5 cm A Equilateral D Acute B Isosceles E Equiangular C Scalene F Right G Obtuse Click for Answer Click for Answer scalene, acute
Slide 43 / 210 Slide 44 / 210 21 Classify the triangle with the given information: 22 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm Side lengths: 5 cm, 5 cm, 5 cm D Acute D Acute A Equilateral A Equilateral E Equiangular E Equiangular B Isosceles B Isosceles F Right F Right C Scalene C Scalene G Obtuse G Obtuse Slide 45 / 210 Slide 46 / 210 24 Classify the triangle with the given information: 23 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90° Angle Measures: 25°, 120°, 35° A Equilateral D Acute A Equilateral D Acute B Isosceles E Equiangular B Isosceles E Equiangular F Right F Right C Scalene C Scalene G Obtuse G Obtuse Slide 47 / 210 Slide 48 / 210 26 Classify the triangle by sides and angles 25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm A Equilateral B Angle measures: 37°, 53°, 90° A B Isosceles 120° D Acute A Equilateral C Scalene E Equiangular B Isosceles D Acute F Right C Scalene E Equiangular G Obtuse F Right C G Obtuse
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