Synthetic Geometry Flashcards

1
Q

a statement accepted without proof, as a basis for argument

A

axiom

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2
Q

a statement deduced from the axioms by logical argument

A

theorem

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3
Q

a statement that follows directly from a theorem

A

corollary

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4
Q

a theorem or corollary in reverse

A

a converse of a theorem or corollary

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5
Q

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

A

proof

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6
Q

=>

A

implies

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7
Q

<=>

A

is equivalent to

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8
Q

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

A

proof by contradiction

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9
Q

two points axiom

A

there is exactly one line through any two given points

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10
Q

number of degrees in an angle is always between..

A

0º and 360º

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11
Q

ordinary angle

A

less than 180º

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12
Q

straight angle

A

180º

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13
Q

how many rays from point A at dº

A

2; clockwise and anticlockwise

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14
Q

if an angle is cut and two, the sum of the two is

A

the same as the original angle

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15
Q

3 combinations of congruent triangles

A

SAS
ASA
SSS

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16
Q

explain SAS

A

2 sides and angles in between

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17
Q

explain ASA

A

2 angles and any side

18
Q

AAA

A

~ similar triangle

19
Q

theorem; vertically opposite angles

20
Q

theorem; isosceles triangles (2)

A
  1. the angles opposite the equal sides are equal

2. conversely, if 2 angles are equal then the triangle is isosceles

21
Q

theorem; alternate angles

A

2 lines are parallel if and only if, alternate angles are equal

22
Q

theorem; sum of angles in triangle

23
Q

theorem; corresponding angles

A

2 lines are parallel if and only if, for any transversal , corresponding angles are equal

24
Q

theorem; exterior angle

A

each exterior angle of a triangle is equal to the sum of the interior opposite angles

25
theorem; greater side
the angle opposite the greater of two sides is greater than the angle opposite the lesser
26
greater angle theorem
the side opposite the greater of two angles is greater than the side opposite the lesser angle
27
theorem; triangle inequality
the sum of two sides of a triangle is greater than the third
28
theorem; equals in a parallelogram
in a parallelogram, opposite sides are equal and opposite angles are equal, converses also equal
29
corollary; divides diagonal
a diagonal divides a parallelogram in 2 congruent triangles
30
theorem; bisect parallelogram
the diagonals of a parallelogram bisect each other
31
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
32
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
33
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
34
theorem; area of parallelogram
base by the height
35
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
36
corollary; same arc same angles
all angles at points of the circle, standing on the same arc, are equal
37
corollary, semicircle angle
each angle in a semi circe is a right angle
38
theorem; cyclic quadrilateral
in a cyclic quadrilateral, opposite angles are equal
39
if two angles share a common tangent at one point
then the two centres of the circles are co-linear
40
the perpendicular bisector of a chord
passes through the midpoint