Corollaries

Content

1-1 (LL)

Two right triangles are equivalent if the legs of one are equal to the legs of the other

2-1 (LA)

Two right triangles are equivalent if a leg and the adjacent acute angle of one are equal to a leg and the adjacent acute angle of the other

3-1

An equilateral triangle is also equiangular

5-1

Only one perpendicular can be drawn to a line from a point outside the line

6-1

Equilateral is also equiangular ? [polygons?]

7-1

When two straight lines are cut by a transversal, if two corresponding angles are equal, then the two straight lines are parallel

7-2

Two lines perpendicular to the same line are parallel

8-1

If two parallel lines are cut by a transversal, then the corresponding angles are equal

8-2

If two parallel lines are cut by a transversal, then the sum of the two interior angles on the same side of the transversal is equal to a straight angle

8-3

When two straight lines are cut by a transversal, if the sum of the interior angles on the same side of the transversal doesn't equal a straight angle, then the two lines aren't parallel

8-4

A straight line perpendicular to one of two parallel lines is perpendicular to the other also

8-5

Lines perpendicular to non parallel lines aren't parallel

9-1

If two angles have sides parallel right to left and left to right, the angles are supplementary

10-1

If two angles have sides perpendicular right to left and left to right, the angles are supplementary

11-1

An external angle of a triangle is equal to the sum of the two opposite interior angles

11-2

In any triangle, there can be but one right angle or one obtuse angle

11-3

In any right triangle, the two acute angles are complimentary

11-4

If an acute angle of one right triangle and an acute angle of another right triangle [are equal], then the remaining acute angles are equal

11-5 (HA)

Two right triangles are equivalent if the hypotenuse and acute angle of one triangle are equal respectively to the hypotenuse and acute angle of the other

11-6 (LA)

Two right triangles are equivalent if a leg and either acute ange of one triangle are equal respectively to a leg and corresponding acute angle of the other triangle

11-7

If two angle of one triangle are equal respectively to two angles of another triangle, then the third angles are equal

11-8 (SAA)

Two triangles are equivalent if a side and any two angles are equal respectively

11-9

Each angle of an equilateral triangle equals 60 degrees

12-1

The perpendicular from a vertex to the base of an isosceles triangle bisects the base and the vertex angle

14-1

The perpendicular from a point to a line is the shortest line that can be drawn from the point to the line

14-2

If a line is the shortest line that can be drawn from a given point to a given line, then it is perpendicular

15-1

If 2 straight lines drawn from a point in a perpendicular to a line cut of from unequal segments from the foot of the perpendicular, the segment that cuts off the greater segment is greater

16-1

If 2 straight lines from a point in a perpendicular to a line are unequal, the smaller segment cuts of the smaller segment from the foot of the perpendicular

19-1

All sides of a rhombus are equal , all sides of a square are equal

19-2

Diagonals of a parallelogram divides it into 2 congruent triangles

19-3

Parallel lines included between parallel lines are equal

19-4

2 parallel lines everywhere are the same distance apart

24-1

A line parallel to one side of a triangle and bisecting the second side bisects the third side

26-1

In an equiangular polygon of n sides, each angle = [(n-2)180(degrees) / n] straight angles

26-2

The sum of external angles of a polygon , made by extending each of its sides in succession, is equal to 2 straight angles

27-1

2 points each equidistant from the ends of a line determine the perpendicular bisector of that line

29-1

The 3 altitudes of a triangle meet in a point

32-1a

If in the same circle or congruent circles 2 central angles are unequal, then the greater angle intercepts the greater minor arc

32-1b

If in the same circle or in congruent circles 2 arcs are unequal, then the greater minor arc subtends the greater central angle

35-1

A diameter that bisects a chord (not a diameter) is perpendicular to the chord

35-2

A perpendicular bisector of a chord passes through the center of a circle

35-3

A perpendicular drawn to any chord from the center bisects the chord and the arcs subtended by it

39-1

A straight perpendicular line to a radius or diameter at its endpoint on a circle is tangent to the circle (?)

39-2

A perpendicular (?) to tangent at the point of tangency passes through the center of the circle

39-3

A perpendicular drawn from the center of the circle to a tangent passes through the point of tangency

40-1

Tangents drawn to a circle from an external point makes equal angles with a line joining that if (?) with center of the circle (?)

42-1

All angles inscribed in the same segment or in congruent segments are equal (?)

42-2

An angle inscribed in a segment whose arc is equal to a semicircle is a right angle (?)

42-3

An angle inscribed in a segment whose arc is smaller than a semicircle is an obtuse angle . An angle inscribed in a segment whose arc is greater than a semicircle is an acute angle

48-1

The locus of points equidistant from 2 given intersecting lines is a pair of lines bisecting angles formed by the given lines

53-1

If a line drawn through 2 sides of a triangle parallel to the third, then either side is to 1 of its segments as the other side is to its corresponding segment (?)

53-2

Corresponding segments cut off on 2 transversal by series of parallel lines are proportional

54-1

If AD:DC = BC:EC or AC:AD=BC:BE, then DE is parallel to AB

57-1

If 2 angles of one triangle equal respectively to 2 angles of another, then the triangles are similar

57-2

2 right triangles are similar if an acute angle of one are equal to the acute angle of the other triangle

58-1 (Leg, Leg ?)

If the legs of 1 right triangle are proportional to the legs of another, the triangles are similar

61-1

If a perpendicular line dropped from any point on a circle upon a diameter , then the perpendicular is a mean proportional between the segments of the diameter

62-1

c^2 - a^2 = b^2

63-1

(x) of segments of any chord through a fixed point is constant (?)

64-1

(x) of any secant from a fixed point outside a circle and its external segment is constant (?)

64-2

2 or more secants are drawn to a circle from a fixed point outside the circle, then the product of 1 secant and its external segment is equal to (x) of any other secant and its external segment (?)

68-1

2 rectangles are to each other as (x) of their bases and their altitudes (?)

68-2

2 rectangles having equal bases are to each other as their altitudes

68-3

2 rectangles having equal bases and equal altitudes are equal

68-4

The area of a square is equal to the square of any of its sides: Square = s^2

69-1

any 2 parallelograms are to each other as the (x)'s of their bases and altitudes (?)

69-2a

2 parallelograms having equal bases are to each other as their altitudes

69-2b

Altitudes to bases (?)

69-3

2 parallelograms having equal bases and altitudes are equal

70-1

Any 2 triangles are to each other as the (x) of their bases and their altitudes (?)

70-2

(?)

70-3

2 triangles having equal bases and altitudes are equal

70-4

A triangle equals 1/2 of a parallelogram having the same base and altitude (?)

71-1

The area of a trapezoid = (x) of its altitude and median (?)

74-1

Areas of 2 similar polygons are to each other as squares of any 2 corresponding lines

75-1

Square of 1 leg equals square of the hypotenuse minus the square of the other leg of a right triangle

75-2

If similar polygons are constructed on 3 sides of a right triangle , the sum of polygons on the legs is equal to the polygon on the hypotenuse (?)

77-1

If the midpoints of arcs subtended by the sides of a regular inscribed polygon are joined top adjacent vertex (?) then a regular inscribed polygon of double the number of sides is formed

77-2

If tangents are drawn at the midpoints of arcs of adjacent points of tangency of sides of regular circumscribed polygon , then regular circumscribed polygon of double number of sides is formed (?)

78-1

In any regular polygon of n sides an angle at center = 360 degrees divided by n, any angle of the polygon = (n - 2) * 180 degrees / n

80-1

Perimeters of 2 regular polygons of same number of sides have same lat (?) as any 2 corresponding sides

80-2

Areas of any 2 regular polygons of same number of sides are to each other as squares of their radii or as squares of apothems

82-1

c / c' = d / d'

82-2

Ratio of any circumference to its diameter is constant

82-3

c = pi * d; c = 2 * pi * r

82-4

s / c = theta (?) / 360

83-1

area of a circle (k) = pi * r^2; k = 1/4 * pi * d^2

83-2

k / k' = r^2 / r'^2 = d^2 / d'^2 = c^2 / c'^2

83-3

Area of the sector of a circle is to the area of the circle as the angle of the sector is 360 degrees (?)

83-4

Area of the sector of a circle = 1/2 (r * length of arc)

83-5

Area of the segment of a circle whose arc is a minor arc is equal to the area of the corresponding sector minus the area of a triangle formed by the 2 radii and chord of the segment

83-6

Similar sectors are to each other as squares of their radii