Prove the hypotenuse leg theorem.

Steps to Solve

1. Understand the Hypotenuse Leg Theorem

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

2. Label the Triangles

Let triangle $ABC$ be one right triangle where $\angle C$ is the right angle. Label the sides as follows: $AC$ is leg $a$, $BC$ is leg $b$, and $AB$ is the hypotenuse $c$. For the other triangle, let it be triangle $DEF$, where $\angle F$ is the right angle. Label the sides: $DE$ is leg $a'$, $EF$ is leg $b'$, and $DF$ is hypotenuse $c'$.

3. Set the Corresponding Sides as Congruent

According to the theorem, we need to establish that:

• $AC \cong DE$
• $AB \cong DF$

This means:

• $a = a'$
• $c = c'$

4. Use the Pythagorean Theorem

For both triangles, apply the Pythagorean theorem: For triangle $ABC$:

$$ c^2 = a^2 + b^2 $$

For triangle $DEF$:

$$ c'^2 = a'^2 + b'^2 $$

5. Substitute Congruent Sides into the Equation

Since we established that $a = a'$ and $c = c'$, replace $a'$ and $c'$ in the second triangle's equation:

$$ c^2 = a^2 + b'^2 $$

6. Equating the Two Equations

Now, we have two equations:

1. $c^2 = a^2 + b^2$ (for triangle $ABC$)

2. $c^2 = a^2 + b'^2$ (for triangle $DEF$)

Since both equal $c^2$, we can equate the remaining parts:

$$ a^2 + b^2 = a^2 + b'^2 $$

7. Simplifying the Equation

Subtract $a^2$ from both sides:

$$ b^2 = b'^2 $$

This implies that:

$$ b = b' $$

8. Conclusion of Congruency

Since we have shown that:

• $AC \cong DE$ (one leg is congruent)
• $AB \cong DF$ (the hypotenuse is congruent)
• $BC \cong EF$ (the other leg is congruent)

Therefore, by the Side-Side-Side (SSS) congruency criterion, triangle $ABC \cong DEF$.