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2.1 – What is a Triangle?

Objectives

  • Define a triangle.
  • Label the parts of a triangle: sides, angles, and vertices.
  • Use proper notation to name triangles.
  • Group triangles by their angle measurements (acute, right, and obtuse).
  • Group triangles by their side lengths (equilateral, isosceles, and scalene).
  • Use the triangle inequality theorem to build triangles.

 

Key Terms

  • Acute Triangle – A triangle in which all three interior angles are acute.
  • Angles – The corner-like spaces where the sides of a triangle meet.
  • Equilateral Triangle – A triangle with three sides of equal length.
    • The three angles of an equilateral triangle each measure 60°.
    • All equilateral triangles are isosceles.
    • All equilateral triangles are acute.
    • The word equilateral comes from the Latin prefix aequi-, meaning “equal” and the Latin root lateralis, which means “belonging to the side.”
  • Isosceles Triangle – A triangle that has at least two congruent sides.
    • The angles opposite these sides are also congruent.
    • Some isosceles triangles are equilateral, not all.
    • Isosceles triangles can be acute, obtuse, or right.
    • The word isosceles comes from the Greek isos, meaning “equal” and the Greek skelos, which means “leg.”

Geo A 2.01 - Isosceles Equilateral

  • Obtuse Triangle – A triangle that has one angle measuring more than 90°.
  • Right Triangle – A triangle that contains a right angle.
  • Scalene Triangle – A triangle in which all three sides have different lengths.
    • Scalene triangles can be acute, obtuse, or right.
  • Sides – The three line segments that form a triangle.
  • Triangle – A polygon with three sides.
    • A triangle is often shown with the symbol, followed by the three letters that label the vertices.
    • A figure formed by 3 line segments that connects 3 noncollinear points.
    • The prefix tri- means three. So the word triangle means three angles.
  • Vertices – The points at which the sides of a triangle meet.

 

Review

Angle Measures
  • Important angle measures to remember

Geo A 2.01 - Angle Measures

Notes

Triangles
  • Characteristics of a Triangle
    • Closed, plane (2 dimensional) figures
      • Closed means no gaps or holes in the figure
    • Have 3 sides
    • All sides are line segments (straight and measurable)
    • Segments connect 3 noncolliear points
  • Parts of a Triangle
    • Pay very close attention to the symbols and notation.

Geo A 2.01 - Parts of a Triangle

  • Triangles Sorted by Side Lengths
    • Equilateral triangles have all congruent sides.
      • ALL equilateral triangles are isosceles triangles (because at least 2 sides are congruent).
      • All equilateral triangles are acute (angles are 60° each).
    • Isosceles triangles have at least two congruent sides.
      • Can be obtuse, right, or acute triangles.
      • Some isosceles triangles are also equilateral (but only if they have all 3 congruent sides).
    • Scalene triangles have no congruent sides.
      • Can be obtuse, right, or acute triangles.

Geo A 2.01 - Sorting Triangles

 

Naming and Describing Triangles
Geo A 2.01 - Triangle ABC

  • Notation
    • Triangle:  \Delta ABC
    • Angle A:  \angle A
    • Angle B:  \angle B
    • Angle C:  \angle C

  • Describing a triangle
    • Sides are ALWAYS opposite angles
    • Angles are ALWAYS opposite sides
      • Side AB is included between angle A and angle B
      • Side AC is included between angle A and angle C
      • Side BC is included between angle B and angle C
      • Side AB is opposite angle C
      • Side AC is opposite angle B
      • Side BC is opposite angle A

  • Naming triangles
    • All polygons, including triangles, can be named starting with any vertex and going around clockwise or counter-clockwise.
      • \Delta ABC
      • \Delta BCA
      • \Delta CAB
      • \Delta ACB
      • \Delta CBA
      • \Delta BAC

 

Classification and Confirmation of Triangles
  • Classifying Triangles
    • The sum of interior angles of a triangle measure 180 degrees
    • Acute triangles have all little angles (the largest angle is less than 90°).
      • It’s a cute triangle.
    • Right triangles have one angle that is just right (the largest angle is exactly 90°).
      • It’s a perfect triangle.
      • You can only have ONE right angle in a triangle (as the other two angles will be less than 90°, adding up to a total of 180°).
    • Obtuse triangles have one very large angle (the largest angle is greater than 90°).
      • It’s an obese triangle.
      • You can only have ONE obtuse angle in a triangle (as the other two angles will be less than 90°, adding up to a total of 180°).

Geo A 2.01 - Angle Measures Sorting

  • Is it a Triangle?
    • Do the two shortest sides add up to more than the 3rd (longest) side?  If so, it’s a triangle!
      • ex. A triangle with sides 2, 5, and 1
        • Step 1:  Put the sides in order from smallest to greatest: 1, 2, and 5
        • Step 2:  Add up the two shorter sides (1 + 2 = 3).
        • Step 3:  Do those two shorter sides (3) add up to more than the largest side (5)?  Is 3 > 5 ?
        • No!  So, this is NOT a triangle!
      • ex. A triangle with sides 2, 5, and 4
        • Step 1:  Put the sides in order from smallest to greatest: 2, 4, and 5
        • Step 2:  Add up the two shorter sides (2 + 4 = 5).
        • Step 3:  Do those two shorter sides (6) add up to more than the largest side (5)?  Is 6 > 5 ?
        • Yes!  So, this IS a triangle!

  • Triangle Inequality Theorem
    • The sum of the lengths of any two sides of a triangle is greater than the length of the 3rd side.
    • To find the possible range of the 3rd side of a triangle, when given the other 2 sides, set up the following problem:
      • Large # (minus) Small # < x < Large # (plus) Small #
      • x represents all the possible side lengths between the small and large numbers.

 

  • Ex. Given sides 4 and 7, what is the possible range of the length of the 3rd side of this triangle?
    • 7 is the large number
    • 4 is the small number
    • 7 – 4 < x < 7 + 4
    • 3 < x < 11
      • So, the 3rd side of the triangle is a number greater than 3 (such as 3.1) but a number less than 11 (such as 10.99).
      • The integers would be 4, 5, 6, 7, 8, 9, & 10.

Examples

  • Ex 1. What are the valid names for the given triangle?

GeoA 02.01 Q01-05

Answer: Since names are denoted by vertices, M and Y are NOT part of the triangle’s name.

The names are \Delta TAX\Delta TXA\Delta AXT\Delta ATX\Delta XAT, and \Delta XTA

  • Ex 2. Classify the following triangle as acute, obtuse, or right.

GeoA 02.01 Q01-09

Answer: Since the largest angle is 70°, which is acute, then the triangle is classified as acute.

  • Ex 3. Classify the following triangle.

GeoA 02.01 Q01-02

Answer: Two sides have identical hash marks, so this is an isosceles triangle.  It also has an angle of 120°, so it is an obtuse triangle.

  • Ex 4. Classify the following triangle.

GeoA 02.01 Q01-04

Answer: The largest angle is 102, so this is an obtuse triangle.  All three sides have different lengths, so this is also a scalene triangle.

  • Ex 5. The segments shown below could form a triangle.

GeoA 02.01 Q03-02

Answer: Put the sides in order from smallest to largest: 7, 9, 16.

Add the smaller two sides: 7 + 9 = 16.

Ask yourself, “Do these two sides add up to a number LARGER than the largest side?”  If yes, it’s a triangle.  If no, it’s not a triangle.

7 + 9 > 16 is false.  Because 16 is not greater than 16.  So, this is NOT a triangle.

  • Ex 6. A triangle has two sides of lengths 7 and 9. What is the range of the length of the third side?
    • Answer: Remember the rule: Large – Small < x < Large + Small
      • 9 – 7 < x < 9 + 7
      • 2 < x < 16
      • The length of the 3rd side, x, could be 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, or 15 (or any number between 2 and 16).

 

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