Geometry 1/2

Midpoint Formula

Midpoint Formula

(x1 + x2)/2 + (y1 + y2)/2

Distance Formula

√(x2 - x1)2 + (y2 - y1)2

Deductive Reasoning

Logical approach.

Factual.

Going from general ideas to specific conclusions.

Following a general theory.

Ex.

Idea = All men are mortal.

Observation: Jason is a man.

Conclusion: Jason is mortal.

Inductive Reasoning

Using patterns.

Going from specifics to generalized statements.

Observations.

cReAtiVE

Ex.

Observations: I break out when I eat blueberries.

Analysis: This is a symptom of being allergic.

Theory: I'm allergic to blueberries.

Law of Detachment

If a conditional, and its hypothesis is true, then so is the conclusion.

If p → q is a true statement and p is true, then so is q.

Law of Syllogism

If two statements are true, a third can be derived from it.

If p → q and q → r are both true, then p → r

Conditional - OG statement

If p, then q. p → q

Converse

If q, then p. q → p

Inverse

~p → ~q (~ means negative statement)

Contrapositive

If not q, then not p. ~q → ~p

Biconditional

p, if and only if q. p ↔ q

Addition Property

If a = b, then a + c = b + c

Basically PEMDAS on both sides addition

Subtraction Property

If a = b, then a - c = b - c

Basically PEMDAS on both sides subtraction

Multiplication Property

If a = b, then ac = bc

Basically PEMDAS on both sides multiplication

Division Property

If a = b and c ≠ 0 , then a/c = b/c

Basically PEMDAS on both sides division

Reflexive Property

Any real number a, a = a

Symmetric Property

a = b, then b = a

Transitive Property

a = b an b = c, then a = c

Substitution Property

a = b, a can be used for b in any equation and vice-versa.

Ex.

a + c = 7

b + c = 7

They are both equal and therefore the same.

Segment Addition Postulate

If X is on segment AB, and A-X-B (X is between A and B), then AX + XB = AB

Postulate 1

Through any two points, there is exactly one line.

Postulate 4

If two lines intersect, then they intersect in exactly one point.

Postulate 5

Through any three noncollinear points, there is exactly one plane.

Postulate 6

If two planes intersect, then their intersection is a line. +

Supplementary Angles

A pair of angles whose sum is 180 degrees.

Complementary Angles

A pair of angles whose sum is 90 degrees.

Adjacent Angles

Angles that share a common side and vertex.

Vertical Angles

Angle opposite when two lines cross. These angles are congruent.

Linear Pair Angles

Two angles that are adjacent and supplementary.

Theorem 3.8

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

Theorem 3.9

If two are perpendicular, then they intersect to form four right angles.

Theorem 3.10

If two sides of adjacent angles are perpendicular, then the angles are complementary.

Theorem 3.11

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Lines Perpendicular to a Transversal Theorem

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

Equilateral Triangle - Side

Has three equal sides.

Isosceles Triangle - Side

Has two equal sides.

Scalene Triangle - Side

Has no equal sides.

Acute Triangle - Angle

Has three angles less than 90 degrees.

Right Triangle - Angle

Has one angle that is exactly 90 degrees.

Obtuse Triangle - Angle

Has one angle greater than 90 degrees.

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle then the third angles are also congruent.

Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite to them are also congruent.

Converse of Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite to them are also congruent.

Ex. If angles B and C are congruent, then segment AB is congruent to segment AC.

Corollary to the Base Angles Theorem

If a triangle is equilateral, then it is equiangular.

Corollary to the Converse of Base Angles Theorem

If a triangle is equiangular, then it is equilateral.

Side - Side - Side (SSS) Triangle Congruence Postulate

If all corresponding sides are congruent in both triangles, then those triangles are congruent.

Side - Angle - Side (SAS) Triangle Congruence Postulate

If two corresponding sides and one corresponding angle of each triangle are congruent to the other.

Angle - Side - Angle (ASA) Triangle Congruence Theorem

If two angles and its included side are congruent in both triangles, then both the triangles themselves are congruent.

Angle - Angle - Side (AAS) Triangle Congruence Theorem

If two angles and a non-included side are congruent in the two triangles, then both triangles themselves are congruent.

Hypotenuse Leg (HL) Theorem

Two right triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the other right triangle's hypotenuse and leg side.

Corresponding Parts of Congruent Triangles are Congruent(CPCTC)

Once two triangles are proven to be congruent, then every side and angle will also be congruent.

Triangle Midsegment Theorem

Each side of a triangle has a midpoint. When you connect this midpoint, another triangle can be formed. Each side of this new triangle will be parallel to a side in the overall triangle. This means, that whatever the length of the larger triangle side is, the midsegment counterpart will be half of that + vice-versa.

Perpendicular Bisector Theorem

In a plane, if a point is on the perpendicular bisector of a then it is equidistant from the endpoints of the segment.

Converse of the Perpendicular Bisector Theorem

In a plane, if a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector.

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Converse of the Angle Bisector Theorem

If a point is in the interior of an angle and it is equidistant from the two sides of the angle, then it lies on the bisector of the angle.

Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from all sides of the triangle.

Centroid Theorem

The centroid of a triangle is located 2/3rds of the distance from each vertex to the midpoint of the opposite side.

Concurrency of Medians of a 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.

Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent.

Theorem 5.10

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.

Theorem 5.11

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller 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.

Hinge Theorem

If two sides of one triangle are congruent to two 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.

Converse of the Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is larger than the third side of the second, then the included angle of the first is longer than the included angle of the second.