Triangles with Non Congruent Sides

We have shown that in an isosceles triangle, the angles opposite the congruent sides are congruent. This was the Isosceles Triangle Theorem which we proved two weeks ago.

In this post, we are gong to learn a slightly related theorem: a theorem that states that if in a triangle, two sides are not congruent, then the angles opposite these sides are not congruent and the angle opposite to the larger side is the larger angle.

 triangle

In the triangle above, $latex \overline{AB}$ is greater than $latex \overline{BC}$. We will show that they are not equal and $latex \angle C > \angle B$. Continue reading

Proof that Equilateral Triangles are Equiangular

In the previous post, we have proved the Isosceles Triangle Theorem. The theorem states that the angles opposite to the two congruent sides of an isosceles triangle are congruent.  In this post, we use the said theorem to prove that equilateral triangles are equiangular.

 equilateral triangle

Theorem

Equilateral triangles are equilangular.

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The Proof of the Isosceles Triangle Theorem

An isosceles triangle is a triangle whose two sides are congruent. In the figure below, $latex ABC$ is an isosceles triangle and $latex \overline{AB} \cong \overline{BC}$.

The Isosceles Triangle Theorem states that the angles opposite to the two congruent sides of an isosceles triangle are congruent. In the figure above, the theorem states that since $latex AB \cong BC$, $latex \angle A \cong \angle C$.

The proof to this theorem uses the SSS triangle congruence. The SSS Triangle Congruence Theorem states that if the three corresponding sides of two triangles are congruent, then the two triangles are congruent. Continue reading