Geometry: Properties of Triangles

Angles

• A triangle has three angles, which always add up to 180°.
• Angles can be classified as:
  • Acute (less than 90°)
  • Right (exactly 90°)
  • Obtuse (greater than 90°)
  • Straight (exactly 180°)
• The sum of the interior angles of a triangle is always 180°.
• Exterior angles of a triangle are equal to the sum of the two non-adjacent interior angles.

Sides

• A triangle has three sides, which can be classified as:
  • Equal (in an equilateral triangle)
  • Unequal (in a scalene triangle)
  • Two equal sides (in an isosceles triangle)
• The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
• The length of the third side is always less than the sum of the lengths of the other two sides.

Congruence

• Two triangles are congruent if they have the same shape and size.
• Congruence can be determined by:
  • SSS (side-side-side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
  • SAS (side-angle-side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
  • ASA (angle-side-angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
  • AAS (angle-angle-side): If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, the triangles are congruent.

Similarities

• Two triangles are similar if they have the same shape but not necessarily the same size.
• Similarity can be determined by:
  • AA (angle-angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
  • SSS (side-side-side): If the ratios of corresponding sides of two triangles are equal, the triangles are similar.
  • SAS (side-angle-side): If the ratios of two sides and the included angle of one triangle are equal to the ratios of two sides and the included angle of another triangle, the triangles are similar.

Trigonometry

• Trigonometry is the study of the relationships between the sides and angles of triangles.
• Important trigonometric concepts include:
  • Sine (sin): Opposite side / Hypotenuse
  • Cosine (cos): Adjacent side / Hypotenuse
  • Tangent (tan): Opposite side / Adjacent side
  • Pythagorean identity: sin^2(A) + cos^2(A) = 1
  • Trigonometric identities: sin(A+B) = sin(A)cos(B) + cos(A)sin(B), etc.

Angles

• A triangle has three angles, which always add up to 180°.
• Angles can be classified as acute (less than 90°), right (exactly 90°), obtuse (greater than 90°), or straight (exactly 180°).
• The sum of the interior angles of a triangle is always 180°.
• Exterior angles of a triangle are equal to the sum of the two non-adjacent interior angles.

Sides

• A triangle has three sides, which can be classified as equal (in an equilateral triangle), unequal (in a scalene triangle), or two equal sides (in an isosceles triangle).
• The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
• The length of the third side is always less than the sum of the lengths of the other two sides.

Congruence

• Two triangles are congruent if they have the same shape and size.
• Congruence can be determined by:
  • SSS (side-side-side): If three sides of one triangle are equal to three sides of another triangle.
  • SAS (side-angle-side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.
  • ASA (angle-side-angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
  • AAS (angle-angle-side): If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle.

Similarities

• Two triangles are similar if they have the same shape but not necessarily the same size.
• Similarity can be determined by:
  • AA (angle-angle): If two angles of one triangle are equal to two angles of another triangle.
  • SSS (side-side-side): If the ratios of corresponding sides of two triangles are equal.
  • SAS (side-angle-side): If the ratios of two sides and the included angle of one triangle are equal to the ratios of two sides and the included angle of another triangle.

Trigonometry

• Trigonometry is the study of the relationships between the sides and angles of triangles.
• Important trigonometric concepts include:
  • Sine (sin): Opposite side / Hypotenuse.
  • Cosine (cos): Adjacent side / Hypotenuse.
  • Tangent (tan): Opposite side / Adjacent side.
  • Pythagorean identity: sin^2(A) + cos^2(A) = 1.
  • Trigonometric identities: sin(A+B) = sin(A)cos(B) + cos(A)sin(B), etc.