How do angles in a quad add up to 540. when split into triangles
To understand why the sum of angles in a quadrilateral (quad) is 540 degrees, we need to consider the concept of triangles.
A quadrilateral can be split into two triangles by drawing a diagonal from one vertex to the opposite vertex. Let's call the quadrilateral ABCD, with the diagonal connecting vertices A and C.
When we split the quad into two triangles (ABC and CDA), let's examine the individual angles in each triangle.
Triangle ABC:
- Angle A: One angle of the quad.
- Angle B: One angle of the quad.
- Angle C: The diagonal of the quad splits this angle into two smaller angles, let's call them C₁ and C₂.
Triangle CDA:
- Angle C: One angle of the quad.
- Angle D: One angle of the quad.
- Angle A: The diagonal of the quad splits this angle into two smaller angles, let's call them A₁ and A₂.
So, combining the angles from both triangles, we have:
Angle A + Angle B + Angle C + Angle D + C₁ + C₂ + A₁ + A₂ = 540 degrees.
In each triangle, the sum of the angles is always 180 degrees (as it is true for all triangles). Therefore, C₁ + C₂ = 180 degrees, and A₁ + A₂ = 180 degrees.
Substituting these values into the equation:
Angle A + Angle B + Angle C + Angle D + 180 + 180 = 540 degrees.
Simplifying the equation:
Angle A + Angle B + Angle C + Angle D = 540 - 360
Angle A + Angle B + Angle C + Angle D = 180 degrees.
Therefore, the sum of angles in a quadrilateral is always 180 degrees.