Right triangles are the cornerstone of geometry, offering unique properties that make solving problems a breeze. From the Hypotenuse-Leg theorem to the Pythagorean theorem, these tools help us prove congruence and find missing side lengths.

Special right triangles, like 45-45-90 and 30-60-90, have fixed ratios between their sides. This makes them super useful for quick calculations and proofs. Knowing these relationships can save you time and effort in geometric problem-solving.

Congruence in Right Triangles

Hypotenuse-Leg theorem for congruence

• States if hypotenuse and one leg of right triangle congruent to hypotenuse and corresponding leg of another right triangle, then two triangles are congruent
  • Example: If $\overline{AC} \cong \overline{DF}$ and $\overline{AB} \cong \overline{DE}$, then $\triangle ABC \cong \triangle DEF$
• To apply theorem:
  • Identify hypotenuse and one leg in each right triangle
  • Check if corresponding parts are congruent
  • If conditions met, conclude triangles are congruent

Pythagorean theorem in triangle problems

• States in right triangle, square of length of hypotenuse equal to sum of squares of lengths of other two sides
  • Algebraically, $a^2 + b^2 = c^2$, where $c$ is length of hypotenuse and $a$ and $b$ are lengths of other two sides
• To use theorem with triangle congruence:
  • Identify right triangles in problem
  • Use theorem to find missing side lengths
  • Apply appropriate congruence postulate or theorem to prove triangles congruent (HL, LL, ASA, SAS, SSS)

Leg-Leg theorem for right triangles

• States if legs of one right triangle congruent to corresponding legs of another right triangle, then two triangles are congruent
  • Example: If $\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$, then $\triangle ABC \cong \triangle DEF$
• To apply theorem:
  • Identify legs in each right triangle
  • Check if corresponding legs are congruent
  • If conditions met, conclude triangles are congruent

Properties of special right triangles

45-45-90 triangles

• Have two 45° angles and one 90° angle
• Legs are congruent, length of hypotenuse is $\sqrt{2}$ times length of leg
  • If leg length is $x$, then hypotenuse length is $x\sqrt{2}$

30-60-90 triangles

• Have one 30° angle, one 60° angle, and one 90° angle
• Side opposite 30° angle is half length of hypotenuse, side opposite 60° angle is $\sqrt{3}$ times length of shorter leg
  • If shorter leg length is $x$, then hypotenuse length is $2x$, and longer leg length is $x\sqrt{3}$
• To apply properties of special right triangles:
  1. Identify type of special right triangle based on given angles or side lengths
  2. Use relationships between sides to find missing lengths
  3. Apply appropriate congruence postulate or theorem to prove triangles congruent, if necessary