Triangles and the Pythagorean Theorem

Study Notes

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

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.
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 ).

Description

This quiz covers important concepts related to triangles, including the Pythagorean Theorem, properties of triangles, and triangle congruence criteria. Test your knowledge on how to apply these principles and theorems in problem-solving.