Triangles and Polygons | Theorems

Triangles and Polygons | Theorems

Theorems and Corollaries

LINES AND ANGLES

  1. If a straight line meets another straight line, the sum of the two adjacent angles is two right angles.
    Corollary: If any number of straight lines are drawn from a given point, the sum of the consecutive angles so formed is four right angles.
    Corollary: If a straight line meets another straight line, the bisectors of the two adjacent angles are at right angles to one another.
  2. If the sum of two adjacent angles is two right angles, their noncoincident arms are in the same straight line.
  3. If two straight lines intersect, the vertically opposite angles are equal.
  4. If a straight line cuts two other straight lines so as to make the alternate angles equal, the two straight lines are parallel.
  5. If a straight line cuts two other straight lines so as to make:
    1. two corresponding angles equal; or
    2. the interior angles, on the same side of the line, supplementary, the two straight lines are parallel.
  6. If a straight line intersects two parallel straight lines, it makes: (i) alternate angles equal; (ii) corresponding angles equal; (iii) two interior angles on the same side of the line supplementary.
    Corollary: Two angles whose respective arms are either parallel or perpendicular to one another are either equal or supplementary.
  7. Straight lines which are parallel to the same straight line are parallel to one another.

TRIANGLES AND THEIR CONGRUENCE

  1. If one side of a triangle is produced,
    (i) the exterior angle is equal to the sum of the interior non-adjacent angles;
    (ii) the sum of the three angles of a triangle is two right angles.
    Corollary: If two angles of one triangle are respectively equal to two angles of another triangle, the third angles are equal and the triangles are equiangular.
    Corollary: If one side of a triangle is produced, the exterior angle is greater than either of the interior non-adjacent angles.
    Corollary: The sum of any two angles of a triangle is less than two right angles.
  2. If all the sides of a polygon of n sides are produced in order, the sum of the exterior angles is four right angles.
    Corollary: The sum of all the interior angles of a polygon of n sides is (2n – 4) right angles.
  3. Two triangles are congruent if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other.
  4. Two triangles are congruent if two angles and a side of one triangle are respectively equal to two angles and the corresponding side of the other.
  5. If two sides of a triangle are equal, the angles opposite to these sides are equal.
    Corollary: The bisector of the vertex angle of an isosceJes triangle,
    (i) bisects the base;
    (ii) is perpendicular to the base.
    Corollary: An equilaterial triangle is also equiangular.
  6. If two angles of a triangle are equal, the sides which subtend these angles are equal.
    Corollary: An equiangular triangle is also equilateral.
  7. Two triangles are congruent if the three sides of one triangle are respectively equal to the three sides of the other.
  8. Two right-angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of the other.

INEQUALITIES

  1. If two sides of a triangle are unequal, the greater side has the greater angle opposite to it.
  2. If two angles of a triangle are unequal, the greater angle has the greater side opposite to it.
  3. Any two sides of a triangle are together greater than the third.
  4. If two triangles have two sides of the one respectively equal to two sides of the other and the included angles unequal, then the third side of that with the greater angle is greater than the third side of the other.
  5. If two triangles have two sides of the one respectively equal to two sides of the other, and the third sides unequal, then the angle contained by the sides of that with the greater base is greater than the corresponding angle of the other.
  6. Of all straight lines that can be drawn to a given straight line from a given external point,
    (i) the perpendicular is least;
    (ii) straight lines which make equal angles with the perpendicular are equal ;
    (iii) one making a greater angle with the perpendicular is greater than one making a lesser angle.
    Corollary: TWO and only two straight lines can be drawn to a given straight line from a given external point, which are equal to one another.

QUADRILATERALS AND OVER FOUR-SIDED POLYGONS

  1. The opposite sides and angles of a parallelogram are equal, each diagonal bisects the parallelogram, and the diagonals bisect one another.
    Corollary: The distance between a pair of parallel straight lines is everywhere the same.
    Corollary: The diagonals of a rhombus bisect each other at right angles.
    Corollary:. A square is equilateral.
  2. A quadrilateral is a parallelogram if
    (i) one pair of opposite sides are equal and parallel;
    (ii) both pairs of opposite sides are equal or parallel;
    (iii) both pairs of opposite angles are equal;
    (iv) the diagonals bisect one another.
  3. Two parallelograms are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other.
    Corollary: TWO rectangles having equal bases and equal altitudes are congruent.
  4. If three or more parallel straight lines intercept equal segments on one transversal, they intercept equal segments on every transversal.
    Corollary: A line parallel to a base of a trapezoid and bisecting a leg bisects the other leg also.
  5. If a line is drawn from the mid-point of one side of a triangle parallel to the second side , it bisects the third side. This line is called a mid-line of a triangle.
    Corollary: Conversely, a mid-line of a triangle is parallel to the third side and is equal to half its magnitude.
    Corollary: In any triangle, a mid-line between two sides and the median to the third side bisect each other.
  6. In a right triangle, the median from the right vertex to the hypotenuse is equal to half the hypotenuse.
  7. If one angle of a right triangle is 30°, the side opposite this angle is equal to half the hypotenuse.
    Corollary: Conversely, if one side of a right triangle is half the hypotenuse , the angle opposite to it is 30°.
  8. The median of a trapezoid is parallel to the parallel bases and is equal to half their sum.
    Corollary: The line joining the mid-points of the diagonals of a trapezoid is parallel to the parallel bases and is equal to half their difference.
  9. In an isosceles trapezoid, the base angles and the diagonals are equal to one another.

INTRODUCTION TO CONCURRENCY

  1. The perpendicular bisectors of the sides of a triangle are concurrent in a point equidistant from the vertices of the triangle which is the center of the circumscribed circle and called the circumcenter of the triangle.
  2. The bisectors of the angles of a triangle are concurrent in a point equidistant from the sides of the triangle which is the center of the inscribed circle and called the incenter of the triangle.
    Corollary: The bisector of any interior angle and the external bisectors of the other,two exterior angles are concurrent in a point outside the triangle which is equidistant from the sides ( or produced ) of the triangle and called an excenter of the triangle.
    Corollary: There are four points equidistant from the three sides of a triangle: one inside the triangle, which is the incenter, and three outside it, which are the excenters.
  3. The altitudes of a triangle are concurrent in a point called the orthocenter of the triangle.
  4. The medians of a triangle are concurrent in a point 2/3 the distance from each vertex to the mid-point of the opposite side. This point is called the centroid of the triangle.