Synthetic Geometry Flashcards
a statement accepted without proof, as a basis for argument
axiom
a statement deduced from the axioms by logical argument
theorem
a statement that follows directly from a theorem
corollary
a theorem or corollary in reverse
a converse of a theorem or corollary
a series of statements each following logically from the preceding one, starting at an axiom or previously proven theorem and ending with the statement to be proven
proof
=>
implies
<=>
is equivalent to
form of proof that establishes the truth of a statement by proving that the statement being false leads to a contradiction. since a statement must be either true or false, showing its falsity to be impossible proves that the statement must be true
proof by contradiction
two points axiom
there is exactly one line through any two given points
number of degrees in an angle is always between..
0º and 360º
ordinary angle
less than 180º
straight angle
180º
how many rays from point A at dº
2; clockwise and anticlockwise
if an angle is cut and two, the sum of the two is
the same as the original angle
3 combinations of congruent triangles
SAS
ASA
SSS
explain SAS
2 sides and angles in between
explain ASA
2 angles and any side
AAA
~ similar triangle
theorem; vertically opposite angles
are equal
theorem; isosceles triangles (2)
- the angles opposite the equal sides are equal
2. conversely, if 2 angles are equal then the triangle is isosceles
theorem; alternate angles
2 lines are parallel if and only if, alternate angles are equal
theorem; sum of angles in triangle
180º
theorem; corresponding angles
2 lines are parallel if and only if, for any transversal , corresponding angles are equal
theorem; exterior angle
each exterior angle of a triangle is equal to the sum of the interior opposite angles
theorem; greater side
the angle opposite the greater of two sides is greater than the angle opposite the lesser
greater angle theorem
the side opposite the greater of two angles is greater than the side opposite the lesser angle
theorem; triangle inequality
the sum of two sides of a triangle is greater than the third
theorem; equals in a parallelogram
in a parallelogram, opposite sides are equal and opposite angles are equal, converses also equal
corollary; divides diagonal
a diagonal divides a parallelogram in 2 congruent triangles
theorem; bisect parallelogram
the diagonals of a parallelogram bisect each other
theorem; transversal cutting parallel lines
if 3 parallel lines cut off equal segments on some transversal line they will cut off equal segments on any other transversal
theorem; lines parallel to the base of the triangle
let ABC be a triangle, if a line l is parallel to BC and cuts AB in the ratio m:n, then it also cuts AC in the same ratio
theorem; similar triangles have proportional sides
if 2 triangles ABC and A’B’C’ are similar, then their sides are proportional in order:
lABl lBCl lCAl
—— = —— = ——-
lA’B’l lB’C’l lC’A’l
theorem; area of parallelogram
base by the height
theorem; same arc angles
angle at the centre of a circle standing on a given arc is twice the angle standing on the same arc at the edge of the circle
corollary; same arc same angles
all angles at points of the circle, standing on the same arc, are equal
corollary, semicircle angle
each angle in a semi circe is a right angle
theorem; cyclic quadrilateral
in a cyclic quadrilateral, opposite angles are equal
if two angles share a common tangent at one point
then the two centres of the circles are co-linear
the perpendicular bisector of a chord
passes through the midpoint
