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Transcript
Triangle Relationships Chapter 4 Objectives: Classifying triangles and finding their angle measures. Using the Distance Formula, the Pythagorean Theorem, and its converse. Showing relationships between a triangle’s sides and angles. Sections 4.1 Classifying Triangles 4.2 Angle Measures of Triangles 4.3 Isosceles and Equilateral Triangles 4.4 The Pythagorean Theorem and the Distance Formula 4.5 The Converse of the Pythagorean Theorem 4.6 Medians of a Triangle 4.7 Triangle Inequalities Classifying Triangles Section 4.1 Objectives Identify and classify triangles by angles Identify and classify triangles by sides Key Vocabulary Acute Triangle Equiangular Triangle Obtuse Triangle Right Triangle Equilateral Triangle Isosceles Triangle Scalene Triangle Vertex Definition Triangle: a figure formed by three segments joining three noncollinear points. B A C Two methods of classifying or naming triangles: Angles Sides Triangle notation: ∆ Name triangle using three letters, therefore the above triangle is ∆ABC. Parts of a Triangle A triangle is a 3-sided polygon The sides of ∆ABC are A AB, BC, and AC A vertex of a triangle is a point that adjacent joins two sides of the triangle. The vertices of ∆ABC are B A, B, and C Two sides sharing a common vertex are adjacent sides The third side is called the opposite side All sides can be adjacent or opposite (it just depends which vertex is being used) adjacent Side opposite A C Example 1: Name the side that is opposite the angle. a. A b. B SOLUTION a. BC is the side that is opposite A. b. AC is the side that is opposite B. c. AB is the side that is opposite C. c. C Classifying Triangles by Angles Four Classifications Acute Obtuse Right Equiangular All triangles have at least two acute angles, the third angle is used to classify the triangle. Classifying Triangles by Angles One way to classify triangles is by their angles… Acute Obtuse all 3 angles are acute 1 angle is obtuse (measure < 90°) (measure > 90°) ) ) ( Right 1 angle is right (measure = 90°) Equiangular All 3 congruent acute angles (measure < 90˚ and ≅) Classifying Triangles by Angles Definition: ACUTE Triangle a triangle in which all angles are acute. E 30 D 70 80 F Classifying Triangles by Angles 40 E 30 110 F Definition: OBTUSE Triangle a triangle in which one of the angles is an obtuse angle. D Classifying Triangles by Angles A Hypotenuse B Definition: RIGHT Triangle a triangle in which one of the angles is a right angle. Leg Leg C Classifying Triangles by Angles D Definition: EQUIANGULAR Triangle a triangle in which all angles are congruent. 60 60 60 F E **EQUIANGULAR applies to any figure in which all angles are congruent** Classifying Triangles by Angles Each of the classifications (acute, obtuse, right, equiangular) is a distinct group and should not be combined. A common mistake is to place triangles into more than one of the angle classifications. Example: a right triangle cannot be classified as an acute triangle. Example 2: The triangular truss below is modeled for steel construction. Classify JMN, JKO, and OLN as acute, equiangular, obtuse, or right. Example 2: Answer: JMN has one angle with measure greater than 90, so it is an obtuse triangle. JKO has one angle with measure equal to 90, so it is a right triangle. OLN is an acute triangle with all angles congruent, so it is an equiangular triangle. Classifying Triangles by Sides Triangles can also be classified according to the number of congruent sides they have. Three classifications: Equilateral Isosceles Scalene To indicate that sides of a triangle are congruent, an equal number of hash marks is drawn on the corresponding sides. Classifying Triangles by Sides Another way to classify triangles is by their sides… Equilateral 3 congruent sides Isosceles Scalene 2 congruent sides no congruent sides Classifying Triangles by Sides D Definition: EQUILATERAL a triangle in which all sides are congruent. E F **EQUILATERAL applies to any figure in which all sides are congruent** Classifying Triangles by Sides Vertex Angle A Definition: ISOSCELES a triangle in which at least 2 sides are congruent. Leg B Leg Base Base Angles C Classifying Triangles by Sides M Definition: SCALENE a triangle in which no sides are congruent. O N Example 3: Classify the triangle by its sides. a. b. c. SOLUTION a. Because this triangle has 3 congruent sides, it is equilateral. b. Because this triangle has no congruent sides, it is scalene. c. Because this triangle has 2 congruent sides, it is isosceles. Your Turn: Classify the triangle by its sides. 1. ANSWER isosceles 2. ANSWER equilateral ANSWER scalene 3. Example 4: Name the side that is opposite the angle. a. A b. B c. C SOLUTION a. BC is the side that is opposite A. b. AC is the side that is opposite B. c. AB is the side that is opposite C. Example 5: Identify the isosceles triangles in the figure if Isosceles triangles have at least two sides congruent. Answer: UTX and UVX are isosceles. Example 6: Identify the scalene triangles in the figure if Scalene triangles have no congruent sides. Answer: VYX, ZTX, VZU, YTU, VWX, ZUX, and YXU are scalene. Your Turn: Identify the indicated triangles in the figure. a. isosceles triangles Answer: ADE, ABE b. scalene triangles Answer: ABC, EBC, DEB, DCE, ADC, ABD Example 7: ALGEBRA Find d and the measure of each side of equilateral triangle KLM if and Since KLM is equilateral, each side has the same length. So 5=d Example 7: Next, substitute to find the length of each side. KL = 7 LM = 7 KM = 7 Answer: For KLM, and the measure of each side is 7. Your Turn: ALGEBRA Find x and the measure of each side of equilateral triangle if and Answer: Review: Classifications of Triangles by Angles polygons Polygon triangles Triangle – 3 sides right acute Right One 90˚ ∠ Acute Obtuse One ∠ > 90˚ All ∠s < 90˚ Equiangular All ∠s ≅ equiangular obtuse 33 Review: Classifications of Triangles by Sides polygons Polygon Triangle – 3 sides Scalene No sides ≅ Isosceles 2 sides ≅ Equilateral 3 sides ≅ triangles scalene isosceles equilateral 34 Joke Time What did the pony say when he had a cold? I’m just a little horse! What is Beethoven doing in his grave? De-composing What do you call an arrogant household bug? A cocky roach. Assignment Sec. 4.1 Pg. 175-178: #1 – 29 odd, 30 – 36 all, 37 – 65 odd