Hypotenuse-Leg Theorem

Learning Objectives Identify the hypotenuse and legs of a right triangle from a diagram or given information. State the three necessary conditions for applying the Hypotenuse-Leg (HL) Theorem. Differentiate between the HL Theorem and other triangle congruence postulates like SSS, SAS, ASA, and AAS. Construct a formal two-column proof to show two right triangles are congruent using the HL Theorem. Solve problems by applying the HL Theorem in various geometric figures, including those on a coordinate plane. Recognize and avoid common errors, such as misapplying the Side-Side-Angle (SSA) condition to non-right triangles. Ever wondered how a carpenter ensures a roof truss is perfectly stable and symmetrical? They rely on the power of rigid triangles! 📐 This tutorial focuses on...

TermDefinitionExample Right TriangleA triangle that has one angle measuring exactly 90 degrees.A triangle with angles 30°, 60°, and 90° is a right triangle. HypotenuseThe longest side of a right triangle. It is always the side directly opposite the 90-degree angle.In a right triangle with sides 3 cm, 4 cm, and 5 cm, the 5 cm side is the hypotenuse. Legs (of a Right Triangle)The two sides of a right triangle that form the right angle.In a right triangle with sides 3 cm, 4 cm, and 5 cm, the 3 cm and 4 cm sides are the legs. Congruent TrianglesTriangles that have the exact same size and shape. All corresponding sides and corresponding angles are equal in measure.If ΔABC has sides 5, 12, 13 and ΔXYZ has sides 5, 12, 13, and their corresponding angles are equal, then ΔABC ≅ ΔXYZ. Corresponding...

Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a corresponding leg of another right triangle, then the two triangles are congruent. This is a special theorem that ONLY works for right triangles. It is the one exception to the Side-Side-Angle (SSA) case, which is not a valid congruence theorem for other triangles. To use HL, you must establish three facts: 1. There are two right triangles. 2. The hypotenuses are congruent. 3. One pair of corresponding legs are congruent. Pythagorean Theorem In a right triangle with legs of lengths 'a' and 'b' and a hypotenuse of length 'c', the following relationship holds: a^2 + b^2 = c^2. This theorem is fundamental to all right trian...