Final answer:
The relationships between the angles in a triangle are defined by the angle sum property and trigonometric functions. In a right triangle, the roles of sine, cosine, and tangent come into play, relating the sides of the triangle to its angles. Furthermore, the Pythagorean theorem frames the relationship of the lengths of the sides in a right-angled triangle which can be interpreted in terms of angles using trigonometric identities.
Step-by-step explanation:
The relationship between the angles in a triangle is elucidated by several rules, such as angle sum property of a triangle and trigonometric relationships. The angle sum property states that the sum of all angles in a triangle is 180 degrees. On the other hand, when it comes to a right-angled triangle, we can resort to trigonometry to identify relationships between the angles.
In a right triangle, if we denote one of the acute angles as θ, the other angles can be defined in terms of this θ. The sides opposite and adjacent to θ will relate to each other through the tangent of θ, and their relationship to the hypotenuse (the side opposite the right angle) will be governed by the sine and cosine of θ. To elaborate, sin θ = (opposite side)/hypotenuse, cos θ = (adjacent side)/hypotenuse, and tan θ = (opposite side)/(adjacent side).
Moreover, Pythagorean theorem, given as a² + b² = c², relates the sides of a right-angled triangle, where a and b are the lengths of the legs, and c is the length of the hypotenuse. This relationship can also be spun into angle relationships using trigonometric identities.
Learn more about Angles in a Triangle