Prove that the sum of the angles of Quadrilateral is 360°.
Once you have proven that the sum of the angles in a triangle is 180° .....
in your quad, draw in any diagonal to create two triangles.
So ......
To prove that the sum of the angles of a quadrilateral is 360°, we can use the fact that the sum of the angles in a triangle is 180°.
1. Start with any quadrilateral. Label its four angles as A, B, C, and D.
2. Create a diagonal by connecting two non-adjacent vertices of the quadrilateral. This will divide the quadrilateral into two triangles.
3. Label the newly formed angles of the triangles as α, β, γ, and δ.
4. Using the fact that the sum of the angles in a triangle is 180°, we can write the following equations for each of the triangles:
Triangle 1: α + β + A = 180° (Equation 1)
Triangle 2: γ + δ + D = 180° (Equation 2)
5. Sum up the equations from step 4 to eliminate the variables α, β, γ, and δ:
(α + β + A) + (γ + δ + D) = 180° + 180°
α + β + γ + δ + A + D = 360° (Equation 3)
6. Now, notice that the sum of the angles A, B, C, and D of the quadrilateral is also equal to the sum of the angles α, β, γ, and δ:
A + B + C + D = α + β + γ + δ (Equation 4)
7. Substitute Equation 4 into Equation 3:
A + B + C + D = 360°
8. Therefore, we have shown that the sum of the angles of a quadrilateral is indeed 360°.
By following these steps, we have proven that the sum of the angles in any quadrilateral is 360°.