Download GPS Geometry Definitions (Part 1) Conjecture – a conclusion made

GPS Geometry Definitions (Part 1)
Conjecture – a conclusion made using inductive reasoning.
Inductive Reasoning – the process by which a conclusion is made based on a pattern or
sequence.
Counterexample – an example that shows a statement is false.
Conditional Statements – statements where a conclusion can be made provided a
condition is met. “IF, THEN”
Hypothesis – the “if” portion of a conditional statement. The condition that must be
satisfied in order to determine a particular conclusion.
Conclusion – the “then” portion of a conditional statement. The conclusion that can be
made when the hypothesis is satisfied.
Negation – “NOT”
Converse – the statement made when the hypothesis becomes the conclusion and the
conclusion becomes the hypothesis. (Flip)
Inverse – the statement made when the hypothesis is negated and the conclusion is
negated. (Not – Not)
Contrapositive – the statement made when the hypothesis becomes the conclusion and is
negated, and the conclusion becomes the hypothesis and is negated. (Flip – Not – Not)
Equivalent Statements – Statements that have the same meaning.
Perpendicular Lines – Lines that meet to form one right angle.
Biconditional Statements – A conditional statement and it’s converse combined. If and
only if.
Deductive Reasoning – The logical process of making a conclusion based on a given set
of rules and the satisfaction of the hypothesis.
Law of Detachment – a rule of deductive reasoning that states a rule and confirms that the
hypothesis is true. The conclusion that is made through deductive reasoning uses the
Law of Detachment. ( If p then q; p; then q)
Law of Syllogism – a rule of deductive reasoning in which there are 2 conditional
statements and the hypothesis of the first statement allows a conclusion to be made which
will then satisfy the hypothesis of the second statement thus resulting in another
conclusion. (If p, then q; If q, then r; If p then r)
Proof – a logical argument that shows a statement is true.
Theorem – a rule in geometry that can be proven.
Adjacent angles – two angles that share a vertex and a side, but do not share any interior
points.
Linear Pair – two adjacent angles whose non-shared sides are opposite rays.
The distance from a point to a line is the length of the perpendicular segment from the
point to the line.
Transversal – a line that intersects two or more lines.
Hypotenuse – the longest side of a right triangle.
Leg of a right triangle – the sides that are not the hypotenuse.
Midsegment of a triangle – the segment that joins the midpoints of 2 sides of a triangle.
Perpendicular Bisector – a segment that bisects a side of a triangle and is perpendicular to
that side.
Equidistant – the same distance.
Concurrent – lines or segments that meet in a single point.
Point of concurrency – the point where lines meet (are concurrent)
Circumcenter – the point where the perpendicular bisectors meet in a triangle.
Angle bisector – a line, segment or ray that cuts an angle into two congruent parts.
Incenter – the point where the angle bisectors meet in a triangle.
Median – the segment that joins the midpoint of a side to the vertex opposite that side in a
triangle.
Centroid – the point where the medians meet in a triangle.
Altitude – the segment that connects a vertex of a triangle to the opposite side and is
perpendicular to the opposite side.
Orthocenter – the point where the altitudes of a triangle meet.
Indirect Proof – a proof made by contradiction.
Diagonal of a polygon – the segment that joins opposite vertices of a polygon.
Interior angles of a polygon – the angles formed by sides of the polygon.
Exterior angles of a polygon – an angle that forms a linear pair with an interior angle of a
polygon.
Quadrilateral – a four sided closed polygon.
Parallellogram – a quadrilateral with both pairs of opposite sides that are parallel.
Rhombus – a quadrilateral with 4 congruent sides.
Rectangle – a quadrilateral with 4 right angles .
Square – a quadrilateral with 4 right angles and 4 congruent sides.
Trapezoid – a quadrilateral with exactly 2 sides that are parallel.
Base Angles of a trapezoid – the angles formed by one of the two parallel sides and the
non-parallel sides.
Bases of a trapezoid – the parallel sides of the trapezoid.
Legs of a trapezoid – the non-parallel sides of the trapexoid.
Isosceles Trapezoid – a trapezoid with congruent legs.
Midsegment of a trapezoid – the segment joining the midpoints of the non-parallel sides
of a trapezoid.
Kite – a quadrilateral that has two pairs of congruent sides, but in which opposite sides
are not congruent.
Postulates and Theorems
Segment Addition Postulate – If B is between A and C, then AB + BC = AC.
Angle Addition Postulate – If AB is in the interior of  ACD, then
m  ABC + m  BCD = m  ACD.
Reflexive Property of Equality – a = a
Reflexive Property of Congruence – AB  AB
 ABC   ABC
Symmetric Property of Equality – If a = b, then b = a
Symmetric Property of Congruence – If AB  CD , then CD  AB .
If  ABC   DEF, then  DEF   ABC
Transitive Property of Equality – If a = b and b = c, then a = c.
Transitive Property of Congruence – If AB  CD and CD  EF , then AB  EF .
If  ABC   DEF and  DEF   XYZ, then  ABC   XYZ
Substitution Property of Equality – If a = b and a + c = d, then b + c = d.
Addition Property of Equality – If a = b, then a + c = b + c
Subtraction Property of Equality – If a = b, then a – c = b – c
Multiplication Property of Equality – If a = b, then ac = bc.
Division Property of Equality – If a = b, then
a b

c c
All right angles are congruent.
If 2 angles are supplementary to the same angle (or to congruent angles), then they are
congruent to each other.
If 2 angles are complementary to the same angle (or to congruent angles), then they are
congruent to each other.
Linear Pair Postulate – If 2 angles form a linear pair, then they are supplementary.
Vertical Angles are congruent.
If 2 lines intersect to form a linear pair of congruent angles, then the lines are
perpendicular.
If 2 lines are perpendicular, then they intersect to form 4 right angles.
If 2 sides of 2 adjacent acute angles are perpendicular, then the angles are
complementary.
If a transversal is perpendicular to one of 2 parallel lines, then it is perpendicular to the
other.
In a plane, if 2 lines are perpendicular to the same line, then they are parallel to each
other.
SSS Congruence – If 3 sides of one triangle are congruent to 3 sides of another triangle,
then the triangles are congruent.
SAS Congruence – If 2 sides and the included angle of one triangle are congruent to 2
sides and the included angle of another triangle, then the triangles are congruent.
HL Congruence – If the hypotenuse and leg of one right triangle are congruent to the
hypotenuse and leg of another right triangle, then the triangles are congruent.
ASA Congruence – If two angles and their included side of one triangle are congruent to
two angles and their included side of another triangle are congruent, then the triangles are
congruent.
AAS Congruence – If two angles and a non-included side of one triangle are congruent to
two angles and a non-included side of another triangle, then the triangles are congruent.
Midsegment Theorem – The segment connecting the midpoints of two sides of a triangle
is parallel to the third side and is half as long as that side.
In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant
from the endpoints of the segment.
In a plane, if a point is equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of the segment.
The perpendicular bisectors of a triangle intersect at a point that is equidistant from the
vertices of the triangle.
If a point is on the bisector of an angle, then it is equidistant from the 2 sides of the angle.
If a point is in the interior of an angle and is equidistant from the sides of the angle, then
it lies on the bisector of the angle.
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of
the triangle.
The medians of a triangle intersect at a point that is two thirds of the distance from each
vertex to the midpoint of the opposite side.
The lines containing the altitudes of a triangle are concurrent.
If one side of a triangle is longer than another side, then the angle opposite the longer side
is larger than the angle opposite the shorter side.
If one angle of a triangle is larger than another angle, then the side opposite the longer
angle is longer than the side opposite the smaller angle.
The sum of the lengths of any two sides of a triangle is greater than the length of the third
side.
The measure of an exterior angle of a triangle is greater than the measure of either of the
non-adjacent interior angles.
Hinge Theorem – If 2 sides of one triangle are congruent to 2 sides of another triangle,
and the included angle of the first is larger than the included angle of the second, then the
third side of the first is longer than the third side of the second.
Hinge Converse Theorem – If 2 sides of one triangle are congruent to 2 sides of a second
triangle and the third side of the first is greater than the third side of the second, then the
included angle of the first is greater than the included angle of the second.
The sum of the measures of the interior angles of a convex n-gon is (n-2)180.
The sum of the measures of the interior angles of a quadrilateral is 360 degrees.
If a quadrilateral is a parallelogram, then its’ opposite sides are congruent.
If a quadrilateral is a parallelogram, then its’ opposite angles are congruent.
If a quadrilateral is a parallelogram, then its’ consecutive angles are supplementary.
If a quadrilateral is a parallelogram, then its’ diagonals bisect each other.
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the
quadrilateral is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
A quadrilateral is a rhombus if and only if it has 4 congruent sides.
A quadrilateral is a rectangle if and only if it has 4 right angles.
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
A parallelogram is a rhombus if and only if its’ diagonals are perpendicular.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite
angles.
A parallelogram is a rectangle if and only if it’s diagonals are congruent.
If a trapezoid is isosceles, then each pair of base angles is congruent.
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
A trapezoid is isosceles if and only if its’ diagonals are congruent.
Midsegment Theorem for Trapezoids – the midsegment of a trapezoid is parallel to each
base and its length is one half the sum of the lengths of the bases.
If a quadrilateral is a kite, then its diagonals are perpendicular.
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.