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Math
Geometry
Geometry S1- Postulates, Theorems, & Corollaries
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reasons used in proofs: given, definitions, postulates, properties, theorems, corollaries
Terms in this set (92)
segment addition postulate
if B is between A and C, then AB+BC=AC
angle addition postulate
if point B lies in ∠AOC, m∠AOB + m∠BOC = m∠AOC
postulate 5
a line contains at least two points; a plane contains at least three points not all on one line; space contains at least four points not all in one plane
Postulate 1-1
through any two points there is exactly one line
postulate 7
through any three points there is at least one plane; through any three non-collinear points there is exactly one plane
postulate 8
if two points are in a plane, then the line that contains the points is in that plane
postulate 9
if two planes intersect, then their intersection is a line
Postulate 1-2
if two lines intersect, then they intersect in exactly one point
Postulate 1-4
through any thhree noncollinear points, rhere is exactly one plane.
theorem 3
if two lines intersect, then exactly one plane contains the lines
angle bisector theorem
if ray BX is the angle bisector of ∠ABC, then m∠ABX=½m∠ABC and m∠XBC=½m∠ABC
theorem 4
vertical angles are congruent
theorem 5
if two lines are perpendicular, they form congruent adjacent angles
theorem 6
if two lines form congruent adjacent angles, they are perpendicular
theorem 7
if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary
theorem 8
if two angles are supplements of congruent angles, then they are congruent
theorem 9
if two angles are complements of congruent angles, then they are congruent
theorem 10
if two parallel planes are cut by a third plane, then the lines of intersection are parallel
postulate 10
if two parallel lines are cut by a transversal, then corresponding angles are congruent
theorem 11
if two parallel lines are cut by a transversal, then alternate interior angles are congruent
theorem 12
if two parallel lines are cut by a transversal, then same side interior angles are supplementary
theorem 13
if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also
theorem 14
if alternating interior angles are congruent, then the lines are parallel
theorem 15
if same side interior angles are supplementary, then the lines are parallel
theorem 16
in a plane two lines perpendicular to the same line are parallel
theorem 17
two lines parallel to a third line are parallel to each other
theorem 18
through a point outside a line, there is exactly one line parallel to the given line
theorem 19
through a point outside a line, there is exactly one line perpendicular to the given line
theorem 20
the sum of the measures of the angles of a triangle is 180°
corollary 1
if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent
corollary 2
each angle of an equiangular triangle has measure 60
corollary 3
in a triangle, there can be at most one right angle or obtuse angle
corollary 4
the acute angles of a right triangle are complementary
theorem 21
the measure of an exterior angle equals the sum of the measures of the two remote interior angles
side-side-side
if given that three sides of one triangle were congruent to three sides of a second triangle, then the triangles are congruent
side-angle-side
if the two sides and the included angle are congruent, then the triangles are congruent
angle-side-angle
if the two angles and the included side are congruent, then the triangles are congruent
CPCTC
corresponding parts of congruent triangles are congruent
definition
a line is perpendicular to a plane iff they intersect and the line is perpendicular to all lines in the plane which pass through the point of intersection
isosceles triangle theorem (PIC)
if two sides of a triangle are congruent, then the angles opposite of those sides are congruent
angle-angle-side
if the two angles and a non-included side are congruent, then the triangles are congruent
hypotenuse leg
if the hypotenuse and a leg of one right triangle are congruent, then the triangles are congruent
theorem 22
if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segments
theorem 23
if a point is equidistant from the endpoints of segment, then the point lies on the perpendicular bisector of the segment
theorem 24
if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
theorem 25
if a point is equidistant from the sides of an angle, then the point lies on the bisector of an angle
theorem 26
opposite sides of a parallelogram are congruent
theorem 27
opposite angles of a parallelogram are congruent
theorem 28
diagonals of a parallelogram bisect each other
theorem 29 (PIC)
if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
theorem 30 (PIC)
if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram
theorem 31 (PIC)
if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
theorem 32 (PIC)
if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
theorem 33
if two lines are parallel, then all points on one line are equidistant from the other line
theorem 34
if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
theorem 35
a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side
theorem 36
the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side
theorem 37
the diagonals of a rectangle are congruent
theorem 38
the diagonals of a rhombus are perpendicular
theorem 39
each diagonal of a rhombus bisects two angles of the rhombus
theorem 40
if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle
theorem 41
if two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
theorem 42
the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
theorem 43
the median of a trapezoid:
- parallel to the bases
- has length equal to the average of the bases
theorem 44
the base angles of an isosceles trapezoid are congruent
theorem 45
the diagonals of an isosceles trapezoid are congruent
theorem 46
the bisectors of the angles of a triangle intersect at a point that is equidistant from the three sides of a triangle (point of intersection is called incenter)
theorem 47
the perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the three vertices of the triangle (point is called circumcenter)
theorem 48
the lines that contain the altitudes of a triangle intersect at a point (point is called orthocenter)
theorem 49
the medians of a triangle intersect at a point that is two thirds the distance from each vertex to the midpoint of the opposite side (point of intersection is called centroid)
exterior angle inequality theorem
the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle
triangle inequality theorem
the sum of the lengths of any two sides of a triangle is greater than the length of the third side
theorem 50
if one side of a triangle is greater than a second side, then the angle opposite the first side is greater than the angle opposite the second side
theorem 51
if one angle of a triangle is greater than a second angle, then the side opposite the first angle is greater than the side opposite the second angle
SAS inequality
if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle
SSS inequality
if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second triangle
AA similarity postulate
if two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar
SSS similarity postulate
if the sides of two triangles are in proportion, then the triangles are similar
SAS similarity postulate
if an angle of one triangle is congruent to an angle of another triangle, and the sides including those angles are in proportion, then the triangles are similar
triangle proportionality theorem
if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally
corollary to triangle proportionality theorem
if three lines intersect two transversals, then they divide the transversals proportionally
triangle angle bisector theorem
if a ray bisects an angle of a triangle, then it divides the opposite side into segments which are proportional to the other sides
theorem 52
if the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and each other
corollary 1
when the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse
corollary 2
when the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg
the pythagorean theorem
in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs a²+b²=c²
converse of the pythagorean theorem
if the square of one side of a triangle is equal to the sum of squares of the other two sides, then the triangle is a right triangle
theorem 53
if a²+b²=c², then m∠c=90° and ∆ABC is right
theorem 54
if a²+b²=c², then m∠c<90° and ∆ABC is acute
theorem 55
if a²+b²=c², then m∠c>90° and ∆ABC is obtuse
45-45-90 theorem
in a 45-45-90 triangle, the hypotenuse is √2 times as long as a leg
30-60-90 theorem
in a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is √3 times as long as the shorter leg
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