Geometry

Postulates

1. a Every line contains at least two points
b Every plane contains at least three noncollinear points
c Space contains at least four noncoplanar points
2. Through any two distinct points there is exactly one line
3. Through any three noncollinear points there is exactly one plane.
4. If two points lie in a plane, then the line containing them also lies in the plane.
5. If two distinct panes intersect, then they intersect in a line.
6. Linear Pair Postulate: If two angles are a linear pair, then they are supplementary
7. Every segment has exactly one midpoint
8. Every angle has exactly one bisector
9. Angle Addition Postulate: If BX is between BA and BC, then ABX + XBC = ABC
10. Perpendicular Postulate
· Given a line and a point not on the line, there is one and only one line, through the point, perpendicular to given line.
· In a plane, given a line and point on the line, there is one and only one line perpendicular to the given line at the given point.
11. If two parallel lines are cut by a transversal, then the corresponding angles are congruent
12. If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. 13. Parallel Postulate: Through a point not on a given line, there is one and only one line parallel to the given line.

Similar and Congruent Triangles
14. SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. 15. SSS Postulate: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
16. ASA 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 two triangles are congruent.
17. Hypotenuse-Leg (HL) Postulate: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two right triangles are congruent.
18. AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
19. SSS Similarity Postulate: If all pairs of corresponding sides of two triangles are proportional, then the two triangles are similar. 20. SAS Similarity Postulate: If one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional, then the two triangles are similar.
21. Arc Addition Postulate: If X is a point on an arc with endpoints A and B, then AX + XB = AXB
22. Area Congruent Postulate: If two figures are congruent, then they have equal areas.
23. Area Addition Postulate: When any polygonal region is divided into nonoverlapping regions, its area is equal to the sum of the areas of these regions.

Perimeter/Area of Rectangles
Perimeter/Area of Squares
24. Rectangle Area Postulate: The area A of any rectangle is equal to the product of the length l and the width w. A=lw

Perimeter/Circumference/Area Circumference/Area of Circles
25. Circumference Postulate: The circumference C of any circle is equal to the product of π and twice the radius r. C=2πr. 26. Circle Area Postulate: The area A of any circle is equal to the product of π and the square of the radius r. A=πr2

Surface Area/Volume Rectangular Prism
27. Right Prism Volume Postulate: The volume V of any right prism is the product of B, the area of a base, and the height h of the prism. V=Bh.
28. Volume Addition Postulate: When any solid is divided into nonoverlapping solids, then its volume is equal to the sum of the volumes of these solids.

• Pythagorean Theorem
• Trigonometry
• Trigonometry Ratios & Scientific Calculator
• Ferris Wheel Tracing Sinusoidal Curve
• Rectangle Maximum Area/Constant Perimeter
• Rectangle Minimum Perimeter/Constant Area
• Draw Ellipse
• Transformations
• Protractor
• Cosine Law
• Sine Law

Formulas

• Simple and Compound Interest
• Sine Law
• Cosine Law
• Quadratic Formula

Theorems

1 Angles and Deductive Reasoning

• If two distinct lines intersect, then they intersect in exactly one point.
• There is exactly one plane containing a given line and a point not on the line.
• If two angles are supplementary to the samt angle, then they are congruent.
• If two angles are complementary to the same angle, then they are congruent.

2 Parallel Lines and Proof

• Any two right angles are congruent.
• Supplementary Angles If two angles are both congruent and supplementary, then they right angles.
• Supplements of congruent angles are congruent.
• Complements of congruent angles are congruent.
• Vertical angles are congruent.
• If one angle of a linear pair is right angle, then other anglis also a right angle.
• If two lines intersect to form one right angle, then they form four right angles.
• If two lines are perpendicular, then they form congruent adjacent angles.
• If two lines intersect to form congruent adjacent angles, then the lines are perpendicular.
• If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
• If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary.
• If two lines are cut by a transversal so that alternate interior angles are congruent, then the line are parallel.
• If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, then the lines are parallel.
• In a plane, if two lines are both perpendicular to a third line, then the two lines are parallel to each other.
• Triangle Sum Theorem: The sum of the measures of the angles of any triangle is 1800. Sum of 3 Angles in a Triangle
• If two angles of one triangles are congruent to two angles of another triangle, then the third angles are congruent.
• Exterior Angle Theorem: For any triangle, the measure of an exterior angle is equal to the sum of the measures of its remote interior angles.

3 Congruent Triangles

• AAS Theorem: If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent.
• Leg-Leg (LL) Theorem: If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent.
• Leg-Acute Angle (LA) Theorem: If a leg and an acute angle of one right triangle are congruent, then the two right triangles are congruent.
• Hypotense-Acute Angle (HA) Theorem: If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the right triangles are congruent.
• Isosceles triangle Theorem: The base angles of an isosceles triangle are congruent.
• Equilateral Triangle Theorem: If a triangle is equilateral, then it is equiangular.
• The measure of each angle of any equilateral triangle is 60o.
• If two angles of a triangle are congruent, then they are the base angles of an isosceles triangle.
• Every equiangular triangle is equilateral.

4 Quadilaterlas

• Inscribed Angles
  Polygon Angles
• In any quadrilateral, the sum of the measures of the angles is 360o.
• A diagonal of amy parallelogram forms two congruent triangles.
• Both pairs of opposite sides of a parallelogram are congruent.
• Both pairs of opposote angles of a parallelogram are congruent.
• The diagonals of any parallelogram bisect each other.
• If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
• If one pair of oppsite sides of a quadrilateral are both parallel and quadrilateral is parallelogram.
• Every rectangle is a parallelogram.
• The diagonals of any rectangle are congruent.
• Any rhombus with one right angle is a rectangle.
• Every rhombus is a parallelogram.
• The diagonals of any rhombus form four congruent triangles.
• The diagonals of any rhombus bisect the angles of the rhombus.
• Every square is a rectangle.
• The median of a trapezoid is parallel to the bases, and its measure equals one-half the sum of the measures of the the bases.
• Each pair of base angles of an isosceles trapezoid are congruent.
• Let n be the number of sides of any polygon. Then the sum S of the measures of its angles is giveng by the formula S=(n-2)180o.
• Let n be the number od sides of any equiangular polygon. Then the measure x of each angle is given by the formula x=(n-2) 180o/n.
• For any polygon, the sum of the measures of the exterior angles, one at each vertex, is 360o.
• For any equiangular polygon with n sides, the measures of any exterior angles is 360o/n.

5 Similar Polygons

• Similar and Congruent Triangles
• A segment joining the midpoints of two sides of a triangle is parallel to the thrid side, and the measure is one-half the measure of the third side.
• In similar triangles, corresponding altitudes are proportional to corresponding sides.
• If a segment is parallel to one side of a triangle and intersects the other sides in two points, then the triangle formed is similar to the original triangle.
• If segment is parallel to one side of a triangle and intersects the other sides in two points, then the segment divides those two sides proportionally.
• If three parallel lines intersect two transversals, then they divide the transversal proportionally.
• In similar triangles, corresponding medians are proportional to corresponding sides.
• The perimeters of similar polygons are proportional to corresponding sides.

6 Right Triangles

• Similar and Congruent Triangles
• The altitude to the hypotenuse of a right triangle forms two right thright triangles that are similar to each other and to the original triangle.
• The measure of an altitude to the hypotenuse of a right thriangle is the geometric mean between the measures of the two segments of the hypotenuse.
• Pythagorean Theorem: In a right triangle, the square of the measure of thre hypotenuse equals the sum of the squares of the measures of the two legs.
• Coverse of the Pythagorean Theorem: In a triangle, if the sum of the squares of the measures of two sides is equal to the square of the measure of the third side, then the triangle is a right triangle.
• 45-45-90 Triangle Theorem: In an isosceles right triangle, the measure of the hypotenuse is √2 times the measure of a leg.
• 30-60-90 Triangle Theorme:In a 30-60-90 triangle, the measure of the hypotenuse is 2 times that of the leg opposite the 30o angle. The measure of the other leg is √3 times that of the leg opposite the 30o angle.

7 Circles

• All radii of a circle are congruent.
• If a radius of a circle is perpendicular to a chod, then the radius bisects the chord.
• In a circle or congruent circles, if two chorcs are congruent, then they are the same distance from the center.
• In a circle or in congruent circles, if two chords are the same distance from the center, then they are congruent.
• If a line is tangent to a circle, then the line is perpendicular to the radius containing the point of tangency.
• If a line in the plane of a circle is perpencular to a radius at its endpoint on the circle, then the line is tangent to the circle at that point.
• Two tangent segments to a circle from the same exterior ponit are congruent.
• For a circle or for congruent circles, if two minor arcs are congruent, then thier central angles are congruent.
• For a circle or for congruent circles, if two central angles are congruent, then their arcs are congruent.
• For a circle or for congruent circle, if two minor arcs are congruent, then their chords are congruent.
• For a circle or for congruent circle, if two chords are congruent, then thier minor arcs are congruent.
• The measure of an inscribed angle is equal to one-half the degree measure of its intercepted arc.
• any angle inscribed in a segmicircle is a right angle.
• If a quadrilateral is inscribed in circle, then both pairs of opposite angles are supplementary.
• When a tangent and a secant intersect on a circle, the measure of the angle formed is equal to one-half the degree measure of the intercepted arc.
• When two secants intersect inside a circle, the measure of each angle formed is equal to one-half the sum of the degree measure of the intercepted arcs.
• When two secants, two tangents, or a secant and a tangent intersect outside a circle, the measure of the angle formed is equal to one-half the differnce of the degree measure of the intercepted arcs.
• When two chords intersect inside a circle, the product of the measures of the two segments of onr chord is equal to the product of the measure of the two segments of the other chord.
• When two secant segments have a common endpoint outside a circle, the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.
• When a tangent segment and a secant segment have a common endpoint outside a circle, the square of the measure of the tangent segment is equal to the product of the seasures of the secant segment and its exetrnal part.

8 Area

• Perimeter/Area of Quadrilaterals
  Perimeter/Area of Rhombuses
  Perimeter/Area of Trapezoids
  Perimeter/Area of Regular Polygons
• Square Area Theorem: Perimeter/Area of Rectangles and Squares The area A of any square is equal to the square of the length s of a side. A=s2.
• Right Triangle Area Theorem: The area A of any right triangle is equal to one-half the product of the lengths s1 and s2 of the legs. A=1/2 s1s2.
• Triangle Area Theorem: Perimeter/Area of Triangles The area A of any triangle is equal to one-hale the product of any base b and corresponding height h. A=1/2 bh.
• Parallelogram Area Theorem: Perimeter/Area of Parallelograms The area A of any parallelogram is equal to the product of any base b and the correesponding height h. A=bh.
• Rhombus Area Theorem: The area A of any rhombus is equal to one-half the product of the lengths d1 and d2 of its diagonals. A= 1/2 d1d2
• Trapezoid Area Theorem: The area A of any trapezoid is equal to one-half the product of the height h and the sum of the bases, b1 and b2. A = 1/2h(bi+b2).
• Regular Poygon Centeral Angle Theorem: The Measure of each centeral angle regular polygon with n sides is 306/n.
• Regular Polygon Area Thereom: The area A of any regular polygon with perimeter P and apothem of measure a is given by the formula A =1/2aP.
• Arc Length Theorem: For a circle with radius r, the length L od any arc with degree measure n0 is given by the formula L=n/360(2πr).
• Sector Area Theorem: The area A of any sector with an arc that has degree measure n and with radius r is given by the formula A=nπr2/360.
• The ratio of the circumferences of two circles is equal to the ratio of their radii.
• The ratio of the the areas of two circles is equal to the square of the the ratio of their radii.

9 Coordinate Geometry

• Midpoint Formula: The midpoint of the segment joining points with coordinates(x1,y1) and (x2,y2) is the point with coordinates [(x1+x2), (y1+y2)].
• Any two lines that have equal slopes are parallel.
• Any two nonvertical parallel lines have equal slopes.
• Any two lines whose slopes are negative reciprocals of each other are perpendicular.
• Any two perpendicular lines, neither of which is vertical, have slopes that are negative reciprocals of each other.
• Distance Formula: The distance d between any two points with coordinates(x1,y1) and (x2,y2) is given by the formula d=[(x2-x1)2+(y2-y1)2]1/2.
• The equation of a circle with radius r and center at the origin is x2+y2=r2.
• The equation of a circle with radius and center with coordinates(h,k) is (x-h)2+(y-k)2=r2.

10 Triangle Inequalities and Space Geometry

• A triangle cannot have some more than one right angle.
• 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 larger angle is longer than the side oppsite the smaller angle.
• The hypotenuse is the lognest side of any right triangle.
• The perpendicular segment from a point to a line is the shortest segment from the point to the line.
• Triangle Inequality Theorem: The sum of the measures of any two sides of any triangle is greater than the measure of the third side.
• Suppose that two sides of one triangle are congruent to two sides of another triangle, and the included angle of one triangle is larger than the included angle of the other. Then the third side of the triangle with the larger included angle is longer than the third side of the other triangle.
• Suppose that two sides of one triangle are congruent to two sides of another triangle, and the third side of one triangle is longer than the third side of the other triangle. Then the included angle of the triangle with the longer thrid side is larger than the included angle of the other triangle.
• If any plane intersects any two parallel planes, then it intersects them in two parallel lines.
• If a line intersects a plane that does not contain it, then the intersection is exactly one point.
• If one line is perpendicular to each of two intersecting lines at their ponits of intersection, then it is perpendicular to the plane containing them.
• Two lines perpendicular to the same plane are parallel.
• Two planes perpendicular to the same line are parallel.
• Any two plane angles of a dihedral angle are congruent.

11 Areas and Volumes of Solids: Surface Area/Volume; Perimeter and Area

• Right Prism Lateral Area Theorem: The lateral area L of any right prism is equal to the product of the perimeter P of a base and the height h of the prism. L=Ph
• Pyramid Volume Theorem: The volum V of any pyramid with height h and a base with area B is given by the formula V= 1/3 Bh
• Regular Pyramid Lateral Area Theorem: The lateral area L of any regular pyramid with a base that has perimeter P and with slant height l is given by the formula L=1/2Pl.
• Cyclinder Volume Theorem: The volume V of any cyclinder with radius r and height h is equal to the product of the area of a base and the height. V=πr2 h.
• Cyclinder Area Theorem: For any right circular cylinder with radius r and height h, the lateral area L and the total area T are given by the formulas L= 2 πrh and T = L + 2πr2.
• Cone Volume Theorem: The volume V of any cone with radius r and height h is given by the formula V =1/3 πr2h.
• Cone Area Theorem: For any right circlar cone with radius r and slant height l, the lateral area L and the total area T are given by the formulas L=πrl and T = L + πr2.
• Sphere Volume and area Theorem: For any sphere with radius r, the volume V and the area A are given by the formulas V =4/3πr3 and A = 4πr2.
• Similar Solids Area and Volume Theorem: Suppose two similar solids ahve corresponding segments that have lenghts in the ratio R. Then the ratio of the total areas of the solids, taken in the samt order, is R2 and the ratio of the volumes, taken in the same order, is R3.

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