Understanding the Triangle ABC
In triangle ABC, we are given two equal altitudes, RE and CF. Altitudes are the perpendicular segments from a vertex to the line containing the opposite side. The congruence of these altitudes can help us demonstrate that triangle ABC is isosceles.
Using RHS Congruence Rule
The RHS (Right angle-Hypotenuse-Side) congruence rule states that if the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, then the two triangles are congruent.
Steps to Prove Isosceles Triangle
- Identify Triangles: Consider triangles AER and ACF, where RE and CF are the altitudes from points E and F respectively.
- Right Angles: Both triangles AER and ACF include a right angle (formed by the altitudes), making them right triangles.
- Equal Altitudes: Since RE = CF (given), this serves as one of the conditions for the RHS rule.
- Common Hypotenuse: The segment AC is common to both triangles AER and ACF, serving as the hypotenuse.
Applying RHS Congruence
With one right angle, the equal altitudes, and the common hypotenuse, triangles AER and ACF can be proven congruent by the RHS rule.
Conclusion: Triangle ABC is Isosceles
Since triangles AER and ACF are congruent, it follows that AE = AF. This implies that sides AB and AC are equal. Therefore, triangle ABC is an isosceles triangle with AB = AC.
This proof effectively demonstrates that equal altitudes from the same vertex lead to an isosceles triangle using the RHS congruence rule.