Equations, Theorems and Postulates from Geometry
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| 810823635 | Area of a Square | s² | |
| 810823636 | Perimeter of a Square | 4s | |
| 810823637 | Perimeter of a Rectangle | 2b+2h | |
| 810823638 | Area of a Rectangle | bh | |
| 810823639 | Area of a Parallelogram | bh | |
| 810823640 | Area of a Triangle | ½bh | |
| 810823641 | Area of a Trapezoid | ½h(b₁+b₂) | |
| 810823642 | Area of a Regular Polygon | ½ap | |
| 810823643 | Area of a Rhombus | ½d₁d₂ | |
| 810823644 | Triangle Angle Sum | m∠A+m∠B+m∠C=180 | |
| 810823645 | Pythagorean Theorem | a²+b²=c² | |
| 810823646 | 45°-45°-90° Triangle Side Ratio | 1:1:1√2 | |
| 810823647 | 30°-60°-90° Triangle Side Ratio | 1:√3:2 | |
| 810823648 | Area of a Circle | πr² | |
| 810823649 | Circumference of a Circle | 2πr | |
| 810823650 | Length of an Arc | measure of sector÷360 × 2πr | |
| 810823651 | Area of a Sector of a Circle | measure of sector ÷360 × πr² | |
| 810823652 | Equation of a Circle | (x-h)²+(y-k)²=r² | |
| 810823653 | Distance Formula | √(x₂-x₁)²+(y₂-y₁)² | |
| 810823654 | Midpoint Formula | (x₁+x₂)÷2 , (y₁+y₂)÷2 | |
| 810823655 | Slope Formula | (y₂-y₁)/ (x₂-x₁) | |
| 810823656 | Slop Intercept Form | y=mx+b | |
| 810823657 | Lateral Area of a Right Prism | ph | |
| 810823658 | Surface Area of a Right Prism | L.A. +2B | |
| 810823659 | Volume of a Right Prism | Bh | |
| 810823660 | Lateral Area of a Right Cylinder | 2πrh | |
| 810823661 | Surface Area of a Right Cylinder | L.A.+ 2B | |
| 810823662 | Volume of a Right Cylinder | πr²h | |
| 810823663 | Lateral Area of a Right Pyramid | ½pl | |
| 810823664 | Surface Area of a Right Pyramid | L.A. + B | |
| 810823665 | Lateral Area of a Right Cone | πrl | |
| 810823666 | Surface Area of a Right Cone | L.A. + B | |
| 810823667 | Volume of a Right Cone | (1÷3)Bh | |
| 810823668 | Surface Area of a Sphere | 4πr² | |
| 810823669 | Volume of a Sphere | (4/3)πr³ | |
| 810823670 | Postulate 1-1 | Through any two points there is exactly one line | |
| 810823671 | Postulate 1-2 | If two lines intersect then they intersect in exactly one point | |
| 810823672 | Postulate 1-3 | if two planes intersect then they intersect in exactly one line | |
| 810823673 | Postulate 1-4 | Through any three noncollinear points there is exactly one plane | |
| 810823674 | Segment Addition Postulate | If three points, A, B, and C, are collinear and B is between A and C, then AB+BC=AC | |
| 810899891 | Angle Addition Postulate | if point B lies in the interior of ∠AOC then m∠AOB+m∠BOC=m∠AOC | |
| 810899892 | Postulate 1-9 | If two figures are congruent, then their areas are equal. | |
| 810899893 | Postulate 1-10 | The area of a region is the sum of the areas of its non overlapping parts | |
| 810899894 | Law of Detatchment | If a conditional is true and its Hypothesis is true, then its conclusion is true. | |
| 810899895 | Law of Syllogism | If p→q and q→r are true statements then p→r is a true statement. | |
| 810899896 | Vertical Angles Theorem | Vertical angles are congruent | |
| 810899897 | Congruent Supplements Theorem | If two angles are supplements of the same angle, or congruent angles, the the two angles are congruent. | |
| 810899898 | Congruent Complements Theorem | If two angles are complements of the same angles. or congruent angles, then the two angles are congruent. | |
| 810899899 | Theorem 2-5 | If two angles are congruent and supplementary, then each is a right angle. | |
| 810899900 | Corresponding Angles Postulate | If a transversal intersects two parallel lines, then corresponding angles are congruent. | |
| 810899901 | Alternate Interior Angles Theorem | If a transversal intersects two parallel lines then the alternate interior angles are congruent. | |
| 811271565 | Converse of the Corresponding Angles Postulate | If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. | |
| 811271566 | Converse of the Alternate Interior Angles Theorem | If two lines and a transversal form congruent alternate interior angles, then the two lines are parallel. | |
| 811271567 | Converse of the Same Side Interior Angles Theorem | If two lines and a transversal form same side interior angles that are supplementary, then the two lines are parallel. | |
| 811271568 | Theorem 3-5 | If two lines are parallel to the same line, then the two lines are parallel. | |
| 811271569 | Theorem 3-6 | In a plane if two lines are perpendicular to the same line, then they are parallel to each other. | |
| 811271570 | Triangle Exterior Angle Theorem | The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. | |
| 811271571 | Corollary to the Triangle Exterior Angle Theorem | The measure of an exterior angle of a triangle is greater than either of its remote interior angles. | |
| 811271572 | Parallel Postulate | Through a point not on a line, there is one and only one line parallel to a given line. | |
| 811271573 | Spherical Geometry Parallel Postulate | Through a point not on a line, there is no line parallel to the given line. | |
| 811271574 | Polygon Angle Sum Theorem | The sum of the angles of a n-gon is : 180(n-2) | |
| 811271575 | Polygon Exterior Angle Sum Theorem | The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. | |
| 811271576 | Theorem 4-1 | If the two angles of one triangle are congruent to two angles of another triangle then the third angles are congruent. | |
| 811271577 | Side-Side-Side Postulate | If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent | |
| 811271578 | Side-Angle-Side Postulate | If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle then the two triangles are congruent. | |
| 811271579 | Angle-Side-Angle Postulate | If the two angles and the included side of one triangle are congruent to two angles and the included side of of another triangle, then the two triangles are congruent. | |
| 811271580 | Angle-Angle-Side 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. | |
| 811271581 | Isosceles Triangle Theorem | If two sides of a triangle are congruent then the angles opposite those sides are congruent. | |
| 811271582 | Corollary to the Isosceles Triangle Theorem | If a triangle is equilateral then the triangle is equiangular. | |
| 811271583 | Converse of the Isosceles Triangle Theorem | If two angles of a triangle are congruent, then the sides opposite the angles are congruent. | |
| 811271584 | Corollary to the Converse of the Isosceles Triangle Theorem | If a triangle is equiangular then the triangle is equilateral. | |
| 811284734 | Theorem 4-5 | The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. | |
| 811284735 | Hypotenuse-Leg Theorem | If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. | |
| 811284736 | Triangle Midsegment Theorem | If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half its length. | |
| 811284737 | Perpendicular Bisector Theorem | If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. | |
| 811284738 | Converse of the Perpendicular Bisector Theorem | If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. | |
| 811284739 | Angle Bisector Theorem | If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. | |
| 811284740 | Converse of the angle Bisector Theorem | If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. | |
| 811284741 | Theorem 5-6 | The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the verticies. | |
| 811284742 | Theorem 5-7 | The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. | |
| 811284743 | Theorem 5-8 | The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. | |
| 811284744 | Theorem 5-9 | The lines that contain the altitudes of a triangle are concurrent. | |
| 811284745 | Theorem 5-10 | If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. | |
| 811284746 | Theorem 5-11 | If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. | |
| 811284747 | Triangle Inequality Theorem | The sum of the length of any two sides of a triangle is greater than the length of the third side. | |
| 811308259 | Theorem 6-1 | Opposite sides of a parallelogram are congruent. | |
| 811308260 | Theorem 6-2 | Opposite angles of parallelograms are congruent. | |
| 811308261 | Theorem 6-3 | Diagonals of a parallelogram bisect each other. | |
| 811308262 | Theorem 6-4 | If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. | |
| 811308263 | Theorem 6-5 | If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. | |
| 811308264 | Theorem 6-7 | If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. | |
| 811308265 | Theorem 6-8 | If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. | |
| 811308266 | Theorem 6-9 | Each diagonal of a rhombus bisects two angles of the rhombus. | |
| 811308267 | Theorem 6-10 | The diagonals of a rhombus are perpendicular. | |
| 811308268 | Theorem 6-11 | The diagonals of a rectangle are congruent. | |
| 811308269 | Theorem 6-12 | If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. | |
| 811308270 | Theorem 6-13 | If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. | |
| 811308271 | Theorem 6-14 | If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. | |
| 811308272 | Theorem 6-15 | The base angles of an isosceles trapezoid are congruent. | |
| 811308273 | Theorem 6-16 | The diagonals of an isosceles trapezoid are congruent. | |
| 811340435 | Theorem 6-17 | The diagonals of a kite are perpendicular. | |
| 811340436 | Theorem 6-18 | 1) The midsegment of a trapezoid is parallel to its bases. 2) The length of a midsegment of a trapezoid is half the sum of the bases. | |
| 811340437 | Arc Addition Postulate | The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. | |
| 811340438 | Angle-Angle Similarity Postulate | If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. | |
| 811340439 | Side-Angle-Side Similarity Theorem | If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are congruent. | |
| 811340440 | Side-Side-Side Similarity | If the corresponding sides of two triangles are proportional, then the triangles are similar. | |
| 811340441 | Theorem 8-3 | The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. | |
| 811340442 | Corollaries to Theorem 8-3 | 1)The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. 2)The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse. | |
| 811364404 | Side-Splitter Theorem | If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. | |
| 811364405 | Corollary to the Side-Splitter Theorem | If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. | |
| 811364406 | Converse of the Side-Splitter Theorem | If a line divides two sides of a triangle proportionally, then it is parallel to the third side. | |
| 811364407 | Triangle-Angle-Bisector Theorem | If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other sides of the triangle. | |
| 811364408 | Perimeters and Areas of Similar Figures | If the similarity ratio of two similar figures is a÷b, then (1) the ratio of their perimeters is a÷b and (2) the ratio of their areas is a²÷b². | |
| 811364409 | Area of a Triangle given SAS | The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. Area of ABC = ½bc(sin A) | |
| 811364410 | Cavalieri's Principle | If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume. | |
| 811364411 | Areas and Volumes of Similar Solids | If the similarity ratio of two similar solids is a : b, then (1) the ratio of their corresponding areas is a² : b², and (2) the ratio of their volumes is a³ : b³. | |
| 811364412 | Theorem 11-1 | If a line is tangent to a circle, then then the line is perpendicular to the radius drawn to the point of tangency. | |
| 811364413 | Theorem 11-2 | If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. | |
| 811364414 | Theorem 11-3 | The two segments tangent to a circle from a point outside the circle are congruent. | |
| 811364415 | Theorem 11-4 | Within a circle or in congruent circles (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles. | |
| 811364416 | Theorem 11-5 | Within a circle or in congruent circles (1) Chords equidistant from the center are congruent. (2) Congruent chords are equidistant from the center. | |
| 811364417 | Theorem 11-6 | In a circle, a diameter that is perpendicular to a cord bisects the cord and its arcs. | |
| 811364418 | Theorem 11-7 | In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord. | |
| 811529481 | Theorem 11-8 | In a circle, the perpendicular bisector of a chord contains the center of the circle. | |
| 811529482 | Inscribed Angle Theorem | The measure of an inscribed angle is half the measure of its intercepted arc. | |
| 811529483 | Corollaries to the Inscribed Angle Theorem | 1)Two inscribed angles that intercept the same arc are congruent. 2)An angle inscribed in a semicircle is a right angle. 3)The opposite angles of a quadrilateral inscribed in a circle are supplementary. | |
| 811529484 | Theorem 11-10 | The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. | |
| 811529485 | Theorem 11-11 | The measure of an angle formed by two lines that (1) intersect inside a circle is half the sum of the measures of the intercepted arcs. m∠1 = ½(x + y) (2) intersect outside a circle is half the difference of the measures of the intercepted arcs. m∠1 = ½(x − y) | |
| 811540834 | Theorem 11-12 | For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle. | |
| 811540835 | Theorem 12-1 | A translation or rotation is a composition of two reflections. | |
| 811540836 | Theorem 12-2 | A composition of reflections in two parallel lines is a rotation. | |
| 811540837 | Theorem 12-3 | A composition of reflections in two intersecting lines is a rotation. | |
| 811540838 | Fundamental Theorem of Isometries | In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections. | |
| 811540839 | Isometry Classification Theorem | There are only four isometries. They are rotation, reflection, translation, and glide reflection. | |
| 811540840 | Theorem 12-6 | Every Triangle Tessellates. | |
| 811540841 | Theorem 12-7 | Every quadrilateral tessellates |
