Notes on Congruent Triangles and Related Concepts

Types of Triangles

  • By Length of Sides:

    • Equilateral Triangle:

    • All sides are equal.

    • Isosceles Triangle:

    • Two sides are equal.

    • Scalene Triangle:

    • No sides are equal.

  • By Angles:

    • Acute Angled Triangle:

    • All angles are less than 90°.

    • Obtuse Angled Triangle:

    • One angle is greater than 90°.

    • Right Angled Triangle:

    • One angle is exactly 90°.

Properties of Triangles

  • The sum of the interior angles of a triangle is always 180°.

    • m∠A+m∠B+m∠C=180°m\angle A + m\angle B + m\angle C = 180°

  • Exterior Angle of a Triangle: This is equal to the sum of the two opposite interior angles.

    • m∠Aexterior=m∠B+m∠Cm\angle A_{exterior} = m\angle B + m\angle C

  • Exterior Angle Theorem:

    • The exterior angle is equal to the sum of the two non-adjacent interior angles.

    • The sum of the measures of the three exterior angles of any triangle equals 360°.

Pythagorean Theorem

  • In a right-angled triangle, the relation between the sides is defined by:

    • Formula:
      c2=a2+b2c^2 = a^2 + b^2
      Where:

    • c² = length of the hypotenuse (side opposite the right angle)

    • a², b² = lengths of the other two sides.

Secondary Parts of a Triangle

  • Median:

    • A segment joining the midpoint of a triangle's base to the opposite vertex.

  • Altitude:

    • Perpendicular segment from a vertex to the line containing the opposite side.

  • Angle Bisector:

    • A segment that bisects (divides into two equal parts) an angle of the triangle.

Congruent Triangles

  • Definition: Two triangles are congruent if they have the same shape and size. All corresponding sides and angles must be equal.

Congruence Postulates

  • SSS Congruence Postulate:

    • If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

  • SAS Congruence Postulate:

    • If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

  • ASA Congruence Postulate:

    • If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

  • AAS Congruence Theorem:

    • If two angles and a non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent.

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

  • This theorem states that once two triangles are proven to be congruent, all corresponding sides and angles are also congruent.

  • This is typically used at the end of geometric proofs to finalize congruence of specific sides or angles.