Triangles
There are three types of triangles based on how many sides or angles that are equal. Those are, the Equilateral Triangle with all three sides equal, and all angles equal (always \(60°\)), the Isosceles Triangle with all two sides equal and two angles equal, and lastly, the Scalene Triangle with none of the sides equal and none of the angles equal.
Equilateral Triangle
Isosceles Triangle
Scalene Triangle
Once we classify the triangle, we can use this information to solve for missing sides/angles.
Sum of Interior Angles
In a triangle, the three interior angles always add to \(180°\):
\(\angle \text{A} + \angle \text{B} + \angle \text{C} = 180°\)
Refer to the diagram below. A triangle with angles \(\angle \text{A},\ \angle \text{B}\) and \(\angle \text{C}\) is extended with a parallel line \(\vec{\text{DE}}\), such that \(\vec{\text{AC}}\) is parallel to \(\vec{\text{DE}}\). From this we can use a "Z" pattern rule to see that the three angles make a straight line around point \(\text{B}\). We know that a straight line is \(180°\) so the sum of the interior angles of the triangle must also be \(180°\)!
In an equilateral triangle, all three sides are the same. Therefore, we can determine that the missing length, \(\boldsymbol{c = 3}\).
Since the sum of interior angles in any triangle is \(180^{\circ}\), we can substitute the known angles and isolate for \(\angle \text{C}\) to determine its value:
\(\angle \text{A} + \angle \text{B} + \angle \text{C} = 180°\)
\(116 + 35 + \angle \text{C} = 180\)
\(\angle \text{C} = 180 - (116 + 35)\)
\(\angle \text{C} = 180 - 151\)
\(\angle \text{C} = 29^{\circ}\)
Therefore, we can determine that the value of the third angle, \(\boldsymbol{\angle {\textbf{C}} = 29^{\circ}}\).
To solve this problem we first define the side length of the equilateral triangle as \(x\). Each side is the same and we know the height. We can focus on half the equilateral triangle with base, \(x/2\), height, \( h = 2\sqrt{3}\) and hypoteneuse, \(x\).
Since half of an Equilateral Triangle represents a Right Triangle, we can use the Pythagorean Theorem to determine the side length!
\(c^2 = a^2 + b^2\)
\(x^2= \left(\cfrac{x}{2}\right)^2+ (2√3)^2\)
\(x^2=\cfrac{x^2}{4}+ 12\)
\(\cfrac{3x^2}{4} = 12\)
\(x^2 = 16\)
\(\sqrt{x^2} = \sqrt{16}\)
\(x = 4\)
Therefore, we can determine that the side length of the triangle is \(\boldsymbol{x = 4}\).