Geometry Theorems, Formulas, and Concepts
Classified in Mathematics
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Polygon Interior Angles Theorem
The sum of the measures of interior angles of an n-gon is (n-2)x180
Interior angles of a quadrilateral:
The sum of the measures of the interior angles of a quadrilateral is 360o
Polygon Exterior Angles Theorem
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.
Angles have to measure up to 360. 360/n (n=#of sides)
Theorems
If a quadrilateral is a parallelogram, then its opposite sides are congruent
If a quadrilateral is a parallelogram, then its opposite angles are congruent
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary
If a quadrilateral is a parallelogram, then its diagonal bisects each other
If both pairs of opposite sides of a quadrilateral are congruent, if both pairs of opposite angles of a quadrilateral are congruent, if one pair of opposite sides of a quadrilateral are congruent and parallel, and If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
Rhombus
A quadrilateral is a rhombus only if it has four congruent sides.
A parallelogram is a rhombus if and only if its diagonals are perpendicular, if each diagonal bisects a pair of opposite angles.
Rectangle
A quadrilateral is a rectangle if and only if it has four right angles.
A parallelogram is a rectangle If and only if its diagonals are congruent.
Square
A quadrilateral is a square if and only it it is a rhombus and a rectangle.
Pythagorean Theorem
In a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
45-45-90 Theorem
The hypotenuse is √2 times as long as each leg
30-60-90 Theorem
In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.
Perimeters of similar polygons
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.
AA Similarity postulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
SSS Similarity Theorem
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180.
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the nonadjacent interior angles.
Formulas
Distance Formula
d=√(x2−x1)2+(y2−y1)2
Slope Formula
m=(x1−x2 ) / (y1−y2)
Pythagorean Theorem
c=a2+b2 (hypotenuse)2=(leg2)+(leg2)
Area triangle
1/2xbasexheight
Hypotenuse 45-45-90
= legx√2
Hypotenuse 30-60-90
= 2xshorter leg
Longer leg
= shorter leg x √3
tan= opposite/adjacent
cos=adjacent/hypotenuse
sin= opposite/hypotenuse Inverse tan= tan-1 Inverse sine = sin-1Inverse Cosine= cos-1`
