Triangles and the Pythagorean Theorem

Triangles

Pythagorean Theorem

• Formula: ( a^2 + b^2 = c^2 )
  • Where ( c ) is the hypotenuse of a right triangle, and ( a ) and ( b ) are the lengths of the other two sides.
• Applicable only to right triangles.
• Used to determine the length of one side when the other two sides are known.

Properties of Triangles

• The sum of the interior angles is always ( 180^\circ ).
• The exterior angle is equal to the sum of the two opposite interior angles.
• The shortest side is opposite the smallest angle; the longest side is opposite the largest angle.
• Triangle Inequality Theorem: The sum of the lengths of any two sides must be greater than the length of the third side.

Triangle Congruence

• Two triangles are congruent if they have:
  • SSS: Three sides equal.
  • SAS: Two sides and the included angle equal.
  • ASA: Two angles and the included side equal.
  • AAS: Two angles and a non-included side equal.
  • HL: The hypotenuse and one leg equal (only for right triangles).
• Congruent triangles have the same shape and size.

Types of Triangles

1. Based on Sides:

   • Equilateral: All sides are equal; all angles are ( 60^\circ ).
   • Isosceles: Two sides are equal; two angles are equal.
   • Scalene: All sides and angles are different.

2. Based on Angles:

   • Acute: All angles are less than ( 90^\circ ).
   • Right: One angle is exactly ( 90^\circ ).
   • Obtuse: One angle is greater than ( 90^\circ ).

Pythagorean Theorem

• The Pythagorean Theorem is expressed by the formula ( a^2 + b^2 = c^2 ).
• Hypotenuse ( c ) is the longest side of a right triangle, while ( a ) and ( b ) represent the other two sides.
• This theorem is applicable solely to right triangles, allowing the calculation of an unknown side length when the other two sides are known.

Properties of Triangles

• The interior angles of any triangle always total ( 180^\circ ).
• An exterior angle of a triangle is equivalent to the sum of the two non-adjacent interior angles.
• The shortest side of a triangle is always opposite the smallest angle, and the longest side is opposite the largest angle.
• According to the Triangle Inequality Theorem, the combined lengths of any two sides must exceed the length of the remaining side.

Triangle Congruence

• Two triangles are congruent if they meet specific criteria:
  • SSS (Side-Side-Side): All three sides are equal in length.
  • SAS (Side-Angle-Side): Two sides and the included angle are equal.
  • ASA (Angle-Side-Angle): Two angles and the included side are equal.
  • AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
  • HL (Hypotenuse-Leg): Hypotenuse and one leg are equal (specific to right triangles).
• Congruent triangles share identical shape and size characteristics.

Types of Triangles

• By Sides:

  • Equilateral: All sides and angles are equal; each angle measures ( 60^\circ ).
  • Isosceles: Two sides are equal in length, resulting in two equal angles.
  • Scalene: All sides and angles are distinct and differ from one another.

• By Angles:

  • Acute: All angles measure less than ( 90^\circ ).
  • Right: Contains one angle that is precisely ( 90^\circ ).
  • Obtuse: Has one angle that exceeds ( 90^\circ ).