8th Grade Geometry Final Review Sheet
(by Vasu Jayanthi)
Properties:
•	Addition Property of Equality: If a=b and c=d, then a+c=b+d
•	Subtraction Property of Equality: If a=b and c=d, then a-c=b-d
•	Multiplication Property of Equality: If a=b, then ac=bc
•	Division Property of Equality: If a=b and c is not equal to 0, then a/c=b/c
•	Transitive Property of Equality: If a=b and b=c, then a=c
•	Reflexive Property of Equality: a=a
•	Symmetric Property of Equality: If a=b, then b=a
•	Substitution Property of Equality: If a=b, then either a or b may be substituted for the other in any equation or inequality.
•	Distributive Property: a(b+c)=ab+ac
•	Associative Property of Addition: a+(b+c)=(a+b)+c
•	Associative Property of Multiplication: a(bc)=(ab)c
•	Commutative Property of Addition: a+b=b+a
•	Commutative Property of Multiplication: ab=ba
The transitive, reflexive, and symmetric properties carry over to congruence and similarity.
•	Cross Product Property of Proportions: If a/b=c/d, then ad=bc.
•	Means-Extremes Property of Proportions: If a/b=c/d, then a/c=b/d and d/b=c/a.
•	Reciprocal Property of Proportions: If a/b=c/d, then b/a=d/c
•	If a/b=c/d, then (a+b)/b=(c+d)/d
•	If a/b=c/d=e/f=…, then (a+c+e+…)/(b+d+f+…)=a/b
Postulates:
•	Ruler Postulate: 
•	The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1.
•	Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.
•	Segment Addition Postulate: If B is between A and C, then AB+BC=AC
•	Protractor Postulate: On line AB in a given plane, choose any point O between A and B. Consider ray OA and ray OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with real numbers from 0 to 180 in such a way that>
•	Ray OA is paired with 0, and ray OB with 180.
•	If ray OP is paired with x, and ray OQ with y, then the measure of angle POQ= the absolute value of x-y.
•	Angle Addition Postulate: If point B lies in the interior of angle AOC, then the measure of angle AOB + the measure of angle BOC= the measure of angle AOC. If angle AOC is a straight angle and B is any point not on line AC, then the measure of angle AOB + the measure of angle BOC= 180.
•	A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.
•	Between any two points there exists exactly one line.
•	Through any three points, there is at least one plane, and through any three noncollinear points there is exactly one plane.
•	If two points are in a plane, then the line that contains those points is in that plane.
•	If two planes intersect, then their intersection is a line.
•	Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
•	Corresponding Angles Converse: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
•	Parallel Postulate: Through a point outside a line, there is exactly one line parallel to the given line.
•	Perpendicular Postulate: Through a point outside a line, there is exactly one point perpendicular to the given line. 
•	SSS Congruence Postulate: If three sides on one triangle are congruent to three sides on another triangle, then the triangles are congruent.
•	SAS Congruence Postulate: If two sides and the included angles of one triangle are congruent to two sides and the included angles of another triangle, then the triangles are congruent.
•	ASA Congruence Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
•	AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
•	Area Congruence Postulate: If two figures are congruent, then they have the same area.
•	Area Addition Postulate: The area of a region is the sum of the areas of the non-overlapping parts.

Theorems:
•	If two lines intersect, then they intersect in exactly one point.
•	Through a line and a point not in the line there is exactly one plane.
•	If two lines intersect, then exactly one plane contains the lines.
•	Midpoint Theorem: If M is the midpoint of segment AB, then AM=MB=1/2 AB
•	Angle Bisector Theorem 1: If ray BX is the bisector of angle ABC, then the measure of angle ABX=the measure of angle XBC= ½ the measure of angle ABC.
•	Vertical Angles Theorem: Vertical angles are congruent.
•	Two lines are perpendicular if and only if they form congruent adjacent angles.
•	If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. 
•	Congruent Supplements Theorem: If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.
•	Congruent Supplements Converse: If two angles are congruent, then they are supplementary to the same or congruent angles. 
•	Congruent Complements Theorem: If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. 
•	Congruent Complements Converse: If two angles are congruent, then they are complementary to the same or congruent angles. 
•	If two parallel planes are cut by a third plane, then the lines of intersection are parallel.
•	Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
•	Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
•	Same-side Interior Angles Theorem: If two parallel lines are cut by a transversal, then same-side interior angles are congruent. 
•	Same-side Exterior Angles Theorem: If two parallel lines are cut by a transversal, then same-side Exterior angles are congruent. 
•	Alternate Interior Angles Converse: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
•	Alternate Exterior Angles Converse: If two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel.
•	Same-side Interior Angles Converse: If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.
•	Same-side Exterior Angles Converse: If two lines are cut by a transversal and same-side exterior angles are supplementary, then the lines are parallel.
•	In a plane, two lines perpendicular to the same line are parallel.
•	Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it is also perpendicular to the other line
•	Two lines parallel to a third line are parallel to each other. 
•	Triangle Sum Theorem: The sum of the measures of the angles of a triangle is 180.
•	Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
•	Each angle of an equiangular triangle has measure 60.
•	An equilateral triangle is equiangular.
•	An equiangular triangle is equilateral.
•	An equilateral triangle has three 60° angles.
•	In a triangle there can be at most one right angle or obtuse angle.
•	The acute angles of a right triangle are complementary. 
•	Exterior Angle of a Triangle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. 
•	The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.
•	Base Angles Theorem: Two sides of a triangle are congruent is and only if the angles opposite those sides are congruent. OR Base angles of an isosceles triangle are congruent, and if two angles of a triangle are congruent, then the sides opposite those angles are congruent. 
•	The bisector of the vertex angles of an isosceles triangle is perpendicular to the base at its midpoint. 
•	AAS Congruence Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
•	HL Congruence Theorem: If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
•	Perpendicular Bisector Theorem: A point lies on the perpendicular bisector of a segment if and only if the point is equidistant from the endpoints of the segment.
•	Angle Bisector Theorem 2: A point lies on the bisector of an angle if and only if the point is equidistant from the sides of the angle.
•	Opposite sides of a parallelogram are congruent.
•	Opposite angles of a parallelogram are congruent.
•	Diagonals of a parallelogram bisect each other.
•	If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
•	If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.
•	If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
•	If two lines are parallel, then all points on one line are equidistant from the other line.
•	If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. 
•	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.
•	Triangle Midsegment Theorem: The segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.
•	The diagonals of a rectangle are congruent.
•	The diagonals of a rhombus are perpendicular.
•	Each diagonal of a rhombus bisects two angles of the rhombus.
•	The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
•	If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
•	If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
•	Base angles of an isosceles trapezoid are congruent.
•	Trapezoid Midsegment Theorem: The midsegment (or median) or a trapezoid is parallel to the bases and has a length equal to the average of the base lengths.
•	Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
•	If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.\
•	If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.
•	The perpendicular segment from a point to a line is the shortest segment from the point to the line.
•	The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
•	The Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
•	 SAS Similarity Theorem: If one 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.
•	SSS Similarity Theorem: If the sides of two triangles 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.
•	Proportional Transversal Theorem: If three (or more) parallel lines intersect two (or more) 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 proportional to the other two sides. 
•	If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
•	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. 
•	When the altitude is draw 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. 
•	Pythagorean Theorem: In a right triangle, the square of the hypotenuse length is equal to the sum of the squares of the lengths of the legs.
•	Pythagorean Converse: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the two shorter sides, then the triangle is a right triangle. If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the two shorter sides, then the triangle is an obtuse triangle. If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the two shorter sides, then the triangle is an acute triangle.
•	In a 45-45-90 right triangle, the hypotenuse is square root of 2 times as long as a leg.
•	In a 30-60-90 right triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is square root of 3 times as long as the shorter leg.
Formulas:
•	The sum of the measures of the angles of a convex polygon with n sides is (n-2)180.
•	Tangent= Opposite/adjacent
•	Sine= Opposite/hypotenuse
•	Cosine= Adjacent/hypotenuse
•	Area of a square= the square of a side length
•	Area of a rectangle= bh or lw
•	Area of a triangle= ½ bh
•	Area of a parallelogram= bh
•	Area of a rhombus= ½ d1d2
•	Area of a trapezoid= ½ (b1+b2)h
•	Area of a regular polygon= ½ Pa
•	
•	Circumference= 
•	Length of an arc= 
•	Area of a sector= 
•	Lateral Area of a right prism= Ph
•	Volume of a right prism= Bh
•	Surface Area of a right prism= 2B+Ph
•	Lateral Area of a pyramid= ½ Pl
•	Volume of a Pyramid= 1/3 Bh
•	Surface Area of a Pyramid= B+LA
•	Lateral Area of a Cylinder= 2
•	Volume of a Cylinder= 
•	Surface area of a cylinder= 2
•	Lateral Area of a Cone= 
•	Volume of a cone= 1/3 
•	Surface Area of a cone= 
•	Surface Area of a sphere= 4
•	Volume of a sphere=