geometry

Proving Parallelograms

Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. One pair of opposite sides is both parallel and congruent. Both pairs of opposite angles are congruent. The diagonals bisect eachother.

Theorem 3.4: AIC → P

Given two coplanar lines and a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel.

Postulate (The Line Postulate

Through any two points there is exactly one line.

Postulate (The Protractor Postulate)

To every angle there corresponds a real number x between 0 and 180, not including 0, but including 180; i.e., 0<x<180.

(The Vertical Angle Theorem)

Vertical angles are congruent.

Definition: A square is

a parallelogram with four right angles and all four sides congruent.

A diagonal of a polygon is

a segment that joins nonconsecutive vertices of the polygon.

Definition: A midline of a triangle is

a segment whose endpoints are the midpoints of two sides of the triangle.

Reflexive Property Equality

a=a (Reflexive)

Theorem 3.13: If three parallel lines cut off congruent segments on one transversal, then they cut off

congruent segments on every transversal.

Theorem 2.10: Halves of congruent angles are

congruent.

Theorem 2.11: Halves of congruent segments are

congruent.

the sum of the exterior angles

is 360

Theorem 3.2: If two distinct coplanar lines m and n are parallel to a third line ℓ, then lines m and n are

parallel to each other.

Theorem 4.16: The median of a trapezoid is

parallel to the bases of the trapezoid and the length of the median is equal to half the sum of the lengths of the bases.

Definition: A rhombus is a

parallelogram with all four sides congruent.

there is a ratio of proportionality or a scale factor between

similar polygons

Definition: The median of a trapezoid is

the segment joining the midpoints of the legs of the trapezoid.

Theorem 3.10: The sum of the measures of the interior angles of a triangle is

180

(The Right Angle Theorem)

All right angles are congruent.

Theorem 3.14: The sum of the measures of the interior angles of a convex polygon with n sides is

(n−2)⋅180

Definition: Transversal

A line that intersects two different lines in two distinct points is called a transversal.

Indirect Proof

Assume opposite of the Prove Solve for contradictions

Characteristics of Parallelograms

Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. The diagonals bisect each other. Consecutive angles are supplementary.

Postulate (The Angle Addition Postulate) 2

Postulate: If C does not lie on line AOB, then m∠AOC+m∠COB=180.

Theorem 2.16: The Angle Bisector Characterization Theorem (ABC)

The bisector of an angle is the set of all points in the interior of the angle that are equidistant from the sides of the angle.

Theorem 4.13: The Midline Theorem

The segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length of the third side.

Postulate (The Ruler Postulate):

The set of real numbers and the points on a line can be put into a one-to-one correspondence. A scale can be obtained by assigning 0 to one point and 1 to another point.

Theorem 3.20: The Triangle Inequality

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Theorem 4.3: The diagonals of a parallelogram

bisect each other.

A trapezoid is an isosceles trapezoid if its

legs are congruent.

Reflexive Property of Congruence

line AB≅line AB (Reflexive)

The measure of angle ABC is denoted

m∠ABC. Note we are distinguishing between the angle as a set of points and its measure. We will write m∠1=90, not ∠1=90.

Vertex angle

the last angle

Ratio

the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.

For any two lines in space, one of the following conditions must exist:

1) the lines must intersect in a single point and therefore must be coplanar; 2) the lines do not intersect and are not coplanar, in which case they are called skew lines; 3) the lines do not intersect but are coplanar, in which case they are parallel lines.

Theorem 4.9: If ℓ and m are two parallel lines and points A and B are on ℓ, then the distance from A to line m is equal to the distance from

B to line m.

examples of how to incorporate auxiliary lines into proofs

Connecting two points: Drawing a median: Drawing an angle bisector: Extending a line segment:

the midpoint of the hypotenuse

For any right triangle ABC, the midpoint of the hypotenuse BC is equidistant from the 3 vertices A, B, C. (Comment: This is one direction of the Carpenter Locus Theorem. The other direction says that if BC is a segment with midpoint O and if A is a point with OA = OB = OC, then angle BAC is a right angle.)

Postulate: The Parallel Postulate

Given a point not on a line, there is exactly one line through the given point and parallel to the given line.

Theorem 3.5: CAC → P

Given two coplanar lines and a transversal, if a pair of corresponding angles are congruent, then the lines are parallel.

Postulate: The Side-Angle-Side Postulate (SAS)

Given △ABC and △DEF, if AB⎯⎯⎯⎯⎯⎯⎯≅DE⎯⎯⎯⎯⎯⎯⎯⎯, AC⎯⎯⎯⎯⎯⎯⎯⎯≅DF⎯⎯⎯⎯⎯⎯⎯⎯, and ∠BAC≅∠EDF, then △ABC≅△DEF.

Postulate: The Side-Side-Side Postulate (SSS)

Given △ABC and △DEF, if line AB ≅ line DE, line BC ≅ line EF, and line AC≅ line DF, then △ABC≅△DEF.

Postulate (The Segment Addition Postulate)

If B is between A and C, then AB+BC=AC.

Postulate (The Angle Addition Postulate)

If C lies in the interior of ∠AOB, then m∠AOC+m∠COB=m∠AOB

(The Midpoint Theorem)

If M is the midpoint of line AB, then 2AM=AB and AM=1/2AB.

Corollary:

If a line goes through the midpoint of one side of a triangle and is parallel to a second side, then the line goes through the midpoint of the third side of the triangle.

Theorem 5.2: The Side-Splitter Theorem

If a line is parallel to a side of a triangle and intersects the other two sides in two points, then it divides those two sides proportionally.

Corollary:

If a line through the midpoint of one side of a triangle is parallel to a second side of the triangle, then it passes through the midpoint of the third side.

Theorem 5.4: The Triangle Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the adjacent two sides of the triangle.

Transitive Property of Equality

If a=b and b=c, then a=c (Transitive)

Addition Property Equality

If a=b and c=d, then a+c=b+d (Add. Prop.)

Mulitplication Property Equality

If a=b and c=d, then ac=bd (Mult. Prop.)

Substitution Property of Equality

If a=b, then either may be substituted for the other. (Substitution)

Theorem 5.5: SAS Similarity Theorem (OPTIONAL)

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are proportional, then the triangles are similar.

Transitive Property of Congruence

If line AB≅ line CD, and line CD≅ line CD≅ line EF, then line AB≅ line EF (Transitive)

(The Angle Bisector Theorem)

If ray BD is the bisector of ∠ABC, then 2m∠ABD=m∠ABC and m∠ABD=1/2m∠ABC.

Theorem 1.8: If the exterior sides of two adjacent, acute angles are perpendicular, then the adjacent angles are......

If the exterior sides of two adjacent, acute angles are perpendicular, then the adjacent angles are complementary. Another version of this theorem is "If A is in the interior of right angle ∠XYZ, then ∠XYAand ∠ZYA are complementary."

Theorem 2.15: The Hypotenuse-Leg Theorem (HL)

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.

geometric mean

If the means in a proportion are equal 3/x=x/12 — then we call x the geometric mean or the mean proportional of 3 and 12.

Theorem 5.6: SSS Similarity Theorem (OPTIONAL)

If three sides of one triangle are proportional to the three corresponding sides of a second triangle, then the two triangles are similar.

Postulate: The Angle-Side-Angle Postulate (ASA)

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

(The Complement Theorem)

If two angles are complements of congruent angles or of the same angle, then the two angles are congruent CCAC

(The Supplement Theorem)

If two angles are supplements of congruent angles or of the same angle, then the two angles are congruent. SCAC

The Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

AA Postulate:

If two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar.

Theorem 3.12: The No-Choice Theorem

If two angles of one triangle are congruent, respectively, to two angles of a second triangle, then the third angles are congruent. If two angles of one triangle are congruent, respectively, to two angles of a second triangle, then the third angles are congruent.

Theorem 1.6: If two lines are perpendicular, then the adjacent angles....

If two lines are perpendicular, then the adjacent angles they form are congruent. This is more frequently expressed as "Adjacent angles formed by perpendicular lines are congruent."

Theorem 1.7: If two lines form congruent adjacent angle, then the lines .....

If two lines form congruent adjacent angles, then the lines are perpendicular. Note that this is the converse of Theorem 1.6.

Theorem 3.8: P → AIC

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Theorem 3.7: P → CAC

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

The Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Definition: perpendicular bisector

In a plane, the perpendicular bisector of a segment is the line that is perpendicular to the segment at its midpoint.

Theorem 2.12: The Perpendicular Bisector Characterization Theorem (PBC)

In a plane, the perpendicular bisector of a segment is the set of all points that are equidistant from the endpoints of the segment.

Postulate: The Perpendicular Postulate

In a plane, through a point on a line, there is one and only one line perpendicular to the given line.

base angles

The congruent angles of an isosceles triangle

Theorem 3.11: The Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.

Theorem 3.1: Exterior Angle Inequality

The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

If a:b=c:d, what are the extremes what are the means

This is derived from the colon notation, where there are numbers written on the far outside, and numbers written in between. If a:b=c:d, then we call a and d the extremes of the proportion, while b and c are the means.

Definition: Parallel

Two lines are parallel if and only if they are coplanar and do not intersect.

Definitions: An angle

a plane figure formed by two different rays that have the same endpoint. The rays are called the sides of the angle and the common endpoint is called the vertex.

Definition An isosceles triangle

a triangle that has two of its sides congruent. The congruent sides are called the legs of the triangle.

Distributive Property Equality

a(b+c)=ab+ac (Distrib. Prop.)

Coplanar points

are points that all lie on some plane

Collinear points

are points that lie on a line.

Definition: Congruent segments

are segments that have the same length.

Adjacent angles

are two angles in a plane that have a common vertex and a common side but no common interior points.

Definition: Supplementary angles

are two angles whose measures sum to 180.

Definition: Complementary angles

are two angles whose measures sum to 90.

Definitions: Vertical angles

are two angles whose sides form pairs of opposite rays.

Definition: Perpendicular line

are two lines that form right angles.

Theorem 2.7: The medians drawn to the legs of an isosceles triangle are

congruent.

Theorem 2.8: The angle bisectors of the base angles of an isosceles triangle are

congruent.

Theorem 2.9: The altitudes drawn to the legs of an isosceles triangle are

congruent.

Theorem 4.15: The base angles of an isosceles trapezoid are

congruent.

Theorem 4.1: Opposite sides of a parallelogram are

congruent.

Theorem 4.2: Opposite angles of a parallelogram are

congruent.

A proportion is an

equality between two ratios

Corollary 2.3: Every equilateral triangle is

equiangular.

Corollary 2.4: Every equiangular triangle is

equilateral

If all three sides are congruent, then the triangle is an

equilateral triangle.

Postulate: The Angle-Angle-Side Postulate (AAS)

f two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

Definition: Two lines are perpendicular...

if and only if they form right angles.

Definition: A bisector of a segment

is a line, ray, or segment that intersects the segment at its midpoint.

Definition: A parallelogram

is a quadrilateral in which both pairs of opposite sides are parallel.

Definition: The bisector of an angle

is a ray or segment in the interior of the angle that divides the angle into two congruent adjacent angles.

An altitude

is a segment drawn from a vertex of the triangle perpendicular to the line that contains the opposite side; the endpoints of an altitude are one of the vertices of the triangle and a point on the opposite side (extended, if necessary).

A median of a triangle

is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

Definition: The midpoint of a segment

is the point that divides the segment into two congruent parts.

If at least two of the sides of the triangle are congruent, then the triangle is an

isosceles triangle

Definition: A rectangle is a

parallelogram with four right angles.

Theorem 2.14: In a plane, if each of two points is equidistant from the endpoints of a segment, then the line they lie on is the

perpendicular bisector of the segment.

Theorem 2.5: The bisector of the vertex angle of an isosceles triangle is

perpendicular to the base.

Theorem 2.13: If a point is not on a line, then there is a line through the point which is

perpendicular to the given line.

Definition: A trapezoid is a

quadrilateral with exactly one pair of opposite sides parallel.

Theorem 5.1: If a line is parallel to one side of a triangle and intersects the other two sides in two points, then the line cuts off a triangle that is

similar to the given triangle.

Theorem 3.9: If two parallel lines are cut by a transversal, then same-side interior angles are

supplementary.

Theorem 4.4: Consecutive angles of a parallelogram are

supplementary.

Theorem 2.6: The median drawn from the vertex angle of an isosceles triangle is

the angle bisector of the vertex angle.

Theorem 4.10: If a quadrilateral is a rectangle, then

the diagonals are congruent.

Theorem 4.11: If a quadrilateral is a rhombus, then

the diagonals are perpendicular to each other.

Theorem 4.12: If a quadrilateral is a rhombus, then

the diagonals bisect the angles of the rhombus and the diagonals are perpendicular

Theorem 4.14: If M is the midpoint of hypotenuse AB of right triangle ABC,

then MA=MB=MC.

Theorem 3.18: If two sides of a triangle are not congruent,

then the angles opposite those sides are not congruent, and the larger angle is opposite the longer side.

Theorem 3.6: Given two coplanar lines and a transversal, if a pair of same-side interior angles are supplementary,

then the lines are parallel.

Theorem 3.3: In a plane, if two lines m and n are perpendicular to a third line ℓ,

then the lines m and n are parallel to each other.

Theorem 4.5: If both pairs of opposite sides of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

Theorem 3.19: If two angles of a triangle are not congruent,

then the sides opposite those angles are not congruent, and the longer side is opposite the larger angle.

Theorem 5.3: If three parallel lines intersect two transversals,

then those transversals are cut proportionally.