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.