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Understanding Triangles and Angles: Definitions and Classifications

This comprehensive guide covers the definition of triangles, formed by three segments joining three noncollinear points. Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, equiangular, right, obtuse). Key concepts such as the Triangle Sum Theorem, relationships between angles and sides, and how to classify triangles based on their properties are discussed. Examples illustrate how to identify triangle types, including the significance of interior and exterior angles.

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Understanding Triangles and Angles: Definitions and Classifications

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  1. 4.1 Triangles and Angles

  2. Definition: Triangle A triangle is a figure formed by three segments joining three noncollinear points. Triangles are always classified in two ways: • By Sides • By Angles

  3. Subtending Angles & Sides • When two sides are equal then their subtending angles must be equal. • Likewise, when two angles are equal then their subtending sides must be equal. B Angle A subtends Side BC Angle B subtends Side AC Angle C subtends Side AB If AB = BC, then which two angles must be equal? If <B = <C, then which two sides must be equal? A C

  4. Classification by Sides Equilateral Triangle = 3 congruent sides Isosceles Triangle = At least 2 congruent sides Scalene Triangle = No congruent sides

  5. Classification by Angles Acute Triangle: Three Acute Angles (all measure less than 90o) Equiangular Triangle: Threecongruent angles (each measures 60o) Right Triangle; One Right Angle (90o) Obtuse Triangle: One obtuse Angle (more than 90o) and two acute angles (less than 90o)

  6. Example 1: Classifying Triangles When you classify a triangle, you must give TWO classifications (sides & angles.) How would you classify triangle ABC? Sides: _____________ Angles: ____________

  7. Example 2: Classifying Triangles How would you classify this triangle? Sides: _____________ Angles: ____________

  8. Example 3: Classifying Triangles How would you classify this triangle? Sides: Angles:

  9. Special Labels for Right and Isosceles Triangles Leg Leg Hypotenuse Leg Base Leg

  10. By extending the sides we create Interior and Exterior Angles Interior Angles are “inside” the triangle. Exterior Angles are “outside the triangle. Label each interior angle with an “I” and each exterior angle with an “E.” When an interior angle and an exterior angle are together they make a straight line and sum to 180o.

  11. Triangle Sum Theorem • The three angles inside of a triangle must always add to 180o. If m<A = 75o and m<B = 85o, what must the measure of angle C be?

  12. A special and useful property! • The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent interior angles If m<A = 90o and m<B = 45o, then what is m<1? If m<1 = 102o and m<B = 55o, then what is m<A? B 1 A C

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