Another Proof of the Hypotenuse Leg Theorem

Given two triangles, if their hypotenuse are congruent, and one pair of their legs are congruent then the two triangles are congruent. In this post, we are going to prove this theorem.

In the figure below, ABC and DEF are right triangles with right angles at C and F, respectively.

hypotenuse leg 1

It is given that $latex \overline{AB} \cong \overline{DE}$ and $latex \overline{AC} \cong \overline{DF}$. We are going to prove that $latex \triangle ABC \cong \triangle DEF$.  Continue reading

Sample Proof on Triangle Congruence Part 3

Recall that for real numbers a, b, and c are real numbers, if $latex a = b$ and $latex b = c$, then $latex a = c$. This is called Transitive Property of Equality. This is also the same with congruence. If $latex A$, $latex B$, and $latex C$ are polygons, and if $latex A$ is congruent to $latex B$, and $latex B$ is congruent $latex C$, then $latex A$ is congruent to $latex C$. This is called the Transitive Property of Congruence. We will use this to prove the following problem.

Given: $latex \overline{BE} \cong \overline{DC}$ and $latex \overline{BD} \cong \overline{CA}$.

Prove: $latex \triangle DBE \cong \triangle CAB$.

Proof

It is given that $latex \overline{BE} \cong \overline{DC}$.

triangle congruence

Now, by reflexive property, that is a segment is congruent to itself, $latex \overline{BD} \cong \overline{BD}$.  Continue reading

The Proof of the Hypotenuse Leg Theorem

The Hypotenuse Leg Theorem states that if the hypotenuse and one leg of a triangle are congruent to the hypotenuse and leg of another triangle, then the two triangles are congruent.

hypotenuse leg theorem

In the figure above, $latex ABC$ and $latex DEF$ are right triangles with right angles at $latex C$ and $latex F$ and with $latex AB \cong DE$ and $latex BE \cong EF$. We are going to show that the two triangles are congruent.  Continue reading