Math Guide

Geometric Theorems Triangle

Geometric theorems triangle shows a plane triangle or simply a triangle is a geometrical figure formed by three lines in a plane. Geometry is  the most important branches of Mathematics. Geometric theorems triangle gives the idea of Various geometrical shapes and figures in our daily life such as articles in the houses, wells, buildings, bridges. in triangle we have following types.

  • scalene triangle.
  • isosceles triangle.
  • equilateral triangle.
  • acute triangle.
  • obtuse triangle.

My forthcoming post is on Area of a Triangular Prism and Area of an Ellipse, will give you more understanding about Geometry.

Geometry theorems triangles-Theorem 1:


prove that  the exterior angle  formed is equal to the sum of  interior opposite angles  whose side of triangle is produced in Geometric theorems triangle .

Proof:

 

 InGeometric theorems triangle ABC is a given triangle. Produce BC and take a point X on the extension of BC as in figure. Now m∠ACX  is  an external angle and ∠A and ∠B are interior opposite angles. We have to show that m∠ACX = ∠A + ∠B. Now ∠ACX and ∠C are supplementary.

                              ∴ ∠ACX+ ∠C = 180°.

                          But ∠A + ∠B +∠C = 180°.

                    ∴∠ACX+ ∠C = ∠A + ∠B +∠C

Canceling ∠C on both sides, in  Geometric theorems triangle we get 

                              ∠ACX =∠A + ∠B




prove that Two triangles are congruent if any two of angles and the included side of one triangle are equal to the two angles and the included side of the other triangle in Geometric theorems triangle.


proof:

Draw two triangles ABC and DEF such that ∠A=∠D,∠B=∠E and BC = EF

Since ABC is a Geometric theorems triangle, ∠A+∠B+∠C=180° (1)

Since DEF is a triangle, ∠D+∠E+∠F=180°

But ∠D=∠A, ∠E=∠B ∴∠A+∠B+∠F=180° (2)

From (1) and (2), we get =∠A+∠B+∠C=∠A+∠B+∠F.

                            ∴ ∠C=∠F

 

Now,in Geometric theorems triangle ABC and DEF, we observe that the side BCand the angles ∠B and on it correspond to the side EF and the angles ∠Eand ∠F on it. Hence by ASA criterion, ΔABC ≡ ΔDEF. Thus, we have the following theorem.