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Transcript
Triangles and Angles Geometry Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: • Classify triangles by their sides and angles. • Find angle measures in triangles DEFINITION: A triangle is a figure formed by three segments joining three noncollinear points. 2 Names of triangles Triangles can be classified by the sides or by the angle Equilateral —3 congruent sides Isosceles Triangle—2 congruent sides Scalene— no congruent sides 3 Acute Triangle 3 acute angles m ABC = 70.26 m CAB = 41.76 m BCA = 67.97 B C A 4 Equiangular triangle • 3 congruent angles. An equiangular triangle is also acute. 5 Right Triangle Obtuse Triangle • 1 right angle 6 Parts of a triangle • Each of the three points joining the sides of a triangle is a vertex.(plural: vertices). A, B and C are vertices. • Two sides sharing a common vertext are adjacent sides. • The third is the side opposite an angle Side opposite A B C adjacent adjacent A 7 Right Triangle hypotenuse • Red represents the hypotenuse of a right triangle. The sides that form the right angle are leg the legs. leg 8 Isosceles Triangles • An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two base sides are the legs of the isosceles triangle. The third is the base. leg leg 9 Identifying the parts of an isosceles triangle • About 7 ft. 5 ft • 5 ft Explain why ∆ABC is an isosceles right triangle. In the diagram you are given that C is a right angle. By definition, then ∆ABC is a right triangle. Because AC = 5 ft and BC = 5 ft; AC BC. By definition, ∆ABC is also an isosceles triangle. 10 Identifying the parts of an isosceles triangle Hypotenuse & Base • About 7 ft. • 5 ft leg 5 ft leg Identify the legs and the hypotenuse of ∆ABC. Which side is the base of the triangle? Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse. Because AC BC, side AB is also the base. 11 Using Angle Measures of Smiley faces are Triangles interior angles and hearts represent the exterior angles B A C Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex. 12 Ex. 3 Finding an Angle Measure. Exterior Angle theorem: m1 = m A +m B x + 65 = (2x + 10) 65 = x +10 65 x 55 = x (2x+10) 13 Triangle Sum Theorem • The sum of the measures of all interior angles in a triangle is equal to 180°. B C A M<A+m<B+m<C=180° 14 Finding angle measures • Corollary to the triangle sum theorem • The acute angles of a right triangle are complementary. • m A + m B = 90 2x x 15 Finding angle measures X + 2x = 90 3x = 90 X = 30 • So m A = 30 and the m B=60 B 2x C x A 16