Question

Given \(\triangle A B C\) and \(\triangle D E F\), name the corresponding parts needed to prove that \(\triangle A B C \cong \triangle D E F\) by each theorem. HA

Short answer

Right angles \(\angle C\) and \(\angle F\), hypotenuses \(\overline{AB}\) and \(\overline{DE}\), plus angles \(\angle A\) and \(\angle D\) ensure congruence by HA.

Step-by-step solution

Identify the Parts for Hypotenuse-Angle Theorem

The Hypotenuse-Angle (HA) Theorem states that if one angle and the hypotenuse of two right triangles are congruent, then the triangles are congruent. Here, identify the right angles in both triangles, say \( \angle C \) and \( \angle F \), and recognize that these are congruent since both are right angles.

Confirm the Hypotenuses Are Congruent

The hypotenuse is the side opposite the right angle in a right triangle. For \( \triangle ABC \), the hypotenuse is \( \overline{AB} \), and for \( \triangle DEF \), it is \( \overline{DE} \). To use the HA theorem, these hypotenuses must be congruent: \( \overline{AB} \cong \overline{DE} \).

Ensure a Pair of Angles is Congruent

In addition to the right angles being congruent, at least one other pair of angles must be congruent. Let's choose \( \angle A \) and \( \angle D \). Confirm that these angles are congruent: \( \angle A \cong \angle D \).

State the Congruence of the Triangles

With the right angles (\( \angle C \cong \angle F \)), the congruent hypotenuses (\( \overline{AB} \cong \overline{DE} \)), and an additional pair of congruent angles (\( \angle A \cong \angle D \)), you can now apply the Hypotenuse-Angle Theorem to conclude that \( \triangle ABC \cong \triangle DEF \).

Key concepts

Hypotenuse-Angle Theorem

The Hypotenuse-Angle (HA) Theorem is a useful tool in geometry for proving the congruence of two right triangles. Right triangles are special because they have specific properties that other triangles don't. The HA Theorem states that if the hypotenuse and one corresponding angle (apart from the right angle) of two right triangles are congruent, then the triangles themselves are congruent.

This theorem relies on the fact that in a right triangle, the hypotenuse is the longest side and opposite the right angle. Identifying congruent angles and hypotenuses between two triangles is crucial. For example, if in two right triangles, one pair of angles and the hypotenuses are known to be identical, it can be concluded that the entire triangles are congruent. Therefore, finding the congruence of these elements simplifies the problem-solving process in geometry.

Right Triangle

A right triangle is defined as a triangle that includes a right angle, which measures exactly 90 degrees. It is distinct due to this angle, and it possesses other special properties useful in geometric theorems.

In such triangles, the side opposite the right angle is known as the hypotenuse, notably the longest side of the triangle. The other two sides are referred to as the legs. These features make the right triangle an important subject of study in geometry.

The structure of a right triangle allows certain theorems, like the Hypotenuse-Angle Theorem, to be applied effectively. These theorems help in proving congruence, which is where two triangles are identical in shape and size.

Triangle Congruence Proof

Proving two triangles are congruent means showing they have identical size and shape. There are several methods to achieve this, one of which is employing the Hypotenuse-Angle Theorem for right triangles.

To use this theorem in a proof, first identify the right angles in each triangle. Having both angles at 90 degrees instantly confirms they are congruent. Next, verify that the hypotenuses of the triangles are congruent; they must have equal length. Finally, find a pair of corresponding angles that matches. With these conditions in check, the triangles can be declared congruent based on the Hypotenuse-Angle Theorem.

This structured approach ensures clarity in proofs and helps establish a solid understanding of triangle congruence in geometry.