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Math
Geometry
Postulates and Theorems
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Terms in this set (50)
Segment Congruence Postulate
If two segments have the same length as measured by a fair ruler, then the segments are congruent (≅). Also, if two segments are congruent, then they have the same length as measured by a fair ruler.
Angle Congruence Postulate
If two angles have the same measure as measured by a protractor, then the angles are congruent. Also, if two angles are congruent, then they have the same measure as measured by a protractor
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary angles
Vertical Angles Theorem
If two angles form a pair of vertical angles, then they are congruent
Corresponding Angles Postulate
If two parallel lines are intersected by a transveral, then corresponding angles are congruent
Alternate Interior Angles Theorem
If two parallel lines are interseceted by a transversal, then the alternate interior angles are congruent
Alternate Exterior Angles Theorem
If two parallel lines are intersected by a transversal, then the alternate ecterior angles are congruent
Same-Side interior Angles Theorem
If two parallel lines are intersected my a transversal, then the same-side interior angles are supplementary
Converse of the Corresponding Angles Postulate
If two coplanar lines are intersected nu a transversla and the corresponding angles are congruent, then the lines are parallel
Converse of the Alternate Interior Angles Theorem
If two coplaner lines are intersected by a transversal and the alternate interior angles are congruent, then the lines are parallel
Converse of the Alternate Exterior Angles Theorem
If two coplanar lines are intersected by a transversal and the alternate ecterior angles are congruent, then the lines are parallel
Converse of the Same-Side Interior Angles Theorem
If two coplanar lines are intersected by a transversal and the same-side interior angles are supplementary, then the lines are parallel
Parallel Postulate
Given a line and a point not on the line, there is one and onlu oneline that contains the givent point and is parallel to the given line
Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180⁰
Exterior Angle Theorem
The measure of an ecterior angle of a triangle is equal to the sim of the measures fo the remote interior angles
Parallel Lines Theorem
Two coplanar nonvertical lines are parallel if and onlu if they have the same slope. Any two vertical lines are parallel
Perpendicular Lines Theorem
Two coplanar nonvertical lines are perpendicular if and only if the product of their slopes equals -1. Any vertical line is perpendicular to any horizontal line
Polygon Congruence Postulate
Two polygons are congruent if and only if there is a correspondence between their sides and angles so that all pairs of correspondign angles are congruent and all pairs of corresponging sides are congruent
SSS Congruence Postulate
If the three sides of one triangle are congruent to the three sides of another triangle, the two triangles are congruent
SAS Congruence Postulate
If two sides and the included angle in one triangle are congruent to two sides and the included angle and the included angle in another triangle, then the two 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 in another triangle, then the two triangles are congruent
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent
Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent
AAS Congruence Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the two triangles are congruent
Hypotenuse-leg (HL) Congruence Theorem
If the hypotenuse and a leg of one right triangel are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congrunet
The quadrilateral is a parallelogram if
-Two pairs of opposite sides of a quadrilateral are congruent
or
-Two opposite sides of a quadrilateral are parallel and congruent
or
-The diagonals of a quadrilateral bisect each other
The parallelogram is a rectangle if
-One angle of a parallelogram is a right angle
or
-One of the diagonals of a parallelogram is congruent
The parallelogram is a rhombus if
-The diagonals of a parallelogram are perpendicular
or
-Two adjacent sides of a parallelogram are congruent
or
-The diagonals of a parallelogram bisect the angles of the parallelogram
Triangle Inequality Theorem
The sum of the lengths of any teo sides of a triangle is grater than the length of the third side
Area Sum Postulate
If a figure is made up of nonoverlapping regions, then the area of the figure is the sum of the areas of the regions
Polygon Similarity Postulate
Two polygons are similar if and only if there is a correspondence between their anlges and their sides so that all corresponging angles are congruent and all corresponding sides are proportianal
AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then theriangles oare similar
SSS Similarity Theorem
If the three sides of one trianlge are proplortional to the three side of another triangle, then the triangles are similar
SAS Similarity Theorem
If two sides of one triangle are proportional to two sides of another triangle and if their included angles are congruent, then the traiangles are similar
Side Splitting Theorem
A line parallel to one side of a triangle divides the other two sides proportionally
Chords and Arcs Theorem
In a circle or in congruent circles, the arcs of congruent chords are congruent
Converse of the Chords and Arcs Theorem
In a circle or in congruent circles, the chords of congruent arcs are congruent
The Tangent Theorem
A line that is tangent to a circle is perpendicular to a radius of the circle at the point of tangency
The Converse of the Tangent Theorem
A line that is perpendicular to a radius of a circle at its endpoints on the circle is tangent to the circle
The Radius and Chord Theorem
A radius that is perpendicular to a chord of a circle bisects the chord
Inscribed Angle Theorem
An angle inscribed in a circle had a measure that equals one-half the measure of its intercepted arc
Right Angle Corollary
An angle that is inscribed in a simicircle is a right anlge
Circle
Circumference
Area
Length of an arc with degree measure
C=πd=2πr
A=πr²
L=(m/360⁰)2πr
Cone
Volume
Surface Area
V=⅓πr²h
S=πrl+πr²
Cylinder
Volume
Surface Area
V=πr²h
S=2πrh+2πr²
Slope
m= rise = y₂-y₁
run x₂-x₁
Coordinates of midpoint
x₁+x₂ , y₁+y₂
2 2
Distance
d=√(x₂-x₁)²+(y₂-y₁)²
Triangle: General
Sum of interior angles
Area
Length of midsegment
Law of Sines
Law of Cosines
m∠A + m∠B + m∠C= 180°
A=½bh
length= ½ length of parallel side
sin A = sin B = sin C
a b c
a²=b²+c²-2bc cos A
Triangle: Right
Pythagorean Theorem
Tangent
Sine
Cosine
a²+b²=c²
tan A= opposite= a
adjacent b
sin A= opposite= a
hypotenuse c
cos A= adjacent= b
hypotenuse c
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