1.2.1: Resources and Key Concepts

Resources

Videos

• Triangles, their types, and the Pythagorean Theorem
  • Right Triangle Trigonometry - Triangles
• Special Triangles
  • Right Triangle Trigonometry - Special Triangles in Trigonometry

Key Concepts

Definitions

• Triangle: A three-sided polygon formed by three line segments bounding a portion of a plane. The line segments are its sides, the points where sides meet are its vertices, and the interior angles are its angles.
• Equilateral Triangle: A triangle where all three sides have the same length.
• Isosceles Triangle: A triangle where at least two sides have equal length. The angle between the two equal sides is the vertex angle, and the other two angles are the base angles.
• Scalene Triangle: A triangle where no sides have equal length.
• Obtuse Triangle: A triangle that contains one obtuse angle (an angle greater than 90°).
• Right Triangle: A triangle that contains one right angle (an angle of exactly 90°).
• Acute Triangle: A triangle in which all three angles are acute (less than 90°).
• 30°-60°-90° Triangle: A right triangle whose acute angles measure 30° and 60°.
• Isosceles Right Triangle: An isosceles triangle where the vertex angle is 90°.
• 45°-45°-90° Triangle: A common name for an isosceles right triangle, since its two other angles must be 45°.

Theorems

• Triangle Sum: The sum of the interior angles in a triangle is 180°.
• Triangle Inequality: In any triangle with side lengths \(p\), \(q\), and \(r\), the sum of the lengths of any two sides is greater than the length of the third side (\(p+q > r\)).
• Angles of an Equilateral Triangle: All the angles of an equilateral triangle are equal.
• Angles of an Isosceles Triangle: The base angles of an isosceles triangle are equal.
• Angles of an Obtuse Triangle: In an obtuse triangle, one angle must be greater than 90°, and the remaining two angles must each be less than 90°.
• Pythagorean Theorem: In a right triangle, if \(c\) is the length of the hypotenuse and the lengths of the two legs are \(a\) and \(b\), then \(a^2 + b^2 = c^2\).
• Converse of the Pythagorean Theorem: If the sides of a triangle satisfy the relationship \(a^2 + b^2 = c^2\), then the triangle must be a right triangle with hypotenuse \(c\).
• Side Relationships for a 30°-60°-90° Triangle: In any 30°-60°-90° triangle, the hypotenuse is twice the length of the shortest side (the side opposite the 30° angle), and the side opposite the 60° angle is \(\sqrt{3}\) times the length of the shortest side.
• Side Relationship for a 45°-45°-90° Triangle: In any 45°-45°-90° triangle, the hypotenuse is \(\sqrt{2}\) times the length of a leg.

Common Mistakes

• Misapplying the Pythagorean Theorem: The Pythagorean Theorem is only valid for right triangles. It cannot be used to find side lengths in acute or obtuse triangles.
• Confusing the Triangle Inequality with the Pythagorean Theorem: The Triangle Inequality theorem determines if a triangle can be formed from three side lengths. The Converse of the Pythagorean Theorem determines if a triangle is a right triangle. A triangle can exist without being a right triangle.
• Radical Notation Ambiguity: When a term includes a variable and a radical, it is best to write the radical last to avoid confusion. For example, write \(a\sqrt{3}\) instead of \(\sqrt{3}a\), as the latter might be misinterpreted as \(\sqrt{3a}\).
• Confusing Isosceles Triangle Angles: The vertex angle of an isosceles triangle is the angle between the two equal sides. The base angles are opposite the equal sides. The base angles are equal to each other, but not necessarily equal to the vertex angle.