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Transcript
Theorem and postulate list for Scholarship Geometry Chapter 1 Through any two points there is exactly one line. Through any three noncollinear points there is exactly one plane containing them If two points lie in a plane, then the line containing those points lies in the plane. If two lines intersect, then they intersect in exactly one point. If two planes intersect, then they intersect in exactly one line. Segment Addition Postulate Angle Addition Postulate Midpoint Formula: The midpoint M of AB with endpoints A(x1, y1) and B(x2, y2) x x2 y1 y2 is: M( 1 , ) 2 2 Distance Formula: d (x2 x1 )2 (y2 y1 )2 Chapter 2 Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property All right angles are congruent. The sum of the measures of the angles of a linear pair is 180. Two angles that form a linear pair are supplementary. Vertical angles are congruent. If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. Chapter 3 If two parallel lines are cut by a transversal, then corresponding angles are congruent. If two parallel lines are cut by a transversal then alternate interior angles are congruent. If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. If two parallel lines are cut by a transversal, then same side interior angles are supplementary. If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. If two lines are cut by a transversal so that same side interior angles are supplementary, then the lines are parallel Chapter 4 The sum of the measures of the angles of a triangle is 180. The measure of each angle of an equiangular triangle is 60. The acute angles of a right triangle are complementary. The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. SAS (Side-Angle-Side) ASA (Angle-Side-Angle) AAS (Angle-Side-Side) SSS (Side-Side-Side) HL (Hypotenuse-Leg) CPCTC If two sides of a triangle are congruent, then the angles opposite those sides are congruent. If three sides of a triangle are congruent, then the three angles are also congruent. If a triangle is equilateral, then it is equiangular. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If three angles of a triangle are congruent, then the three sides are also congruent. If a triangle is equiangular, then it is equilateral. Chapter 5 If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it is on a perpendicular bisector of the segment. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. 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. The circumcenter of a triangle (where perpendicular bisectors of the sides meet) is equidistant from the vertices of the triangle. The incenter of a triangle (where angle bisectors meet) is equidistant from the sides of the triangle. The centroid of a triangle is located 2/3 the distance from the vertex to the midpoint of the opposite side. The midsegment of a triangle is parallel to the third side of the triangle and its length is half the length of the third side. If two sides of a triangle are not congruent, then the largest angle is opposite the longest side. If two angles of a triangle are not congruent, then the longest side is opposite the largest angle. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. THE Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangles is equal to the square of the length of the hypotenuse. Better remembered as a2 + b2 = c2. Converse of the Pythagorean Theorem and its Inequalities If a ≤ b ≤ c are the lengths of sides of a triangle and: 1) c2 = a2 + b2, then the triangle is a right triangle; 2) c2 < a2 + b2, then the triangle is an acute triangle; 3) c2 > a2 + b2, then the triangle is an obtuse triangle. In a 45-45-90 triangle, both legs are congruent, and hypotenuse is a leg times 2. In a 30-60-90 triangle, the hypotenuse is two times the shorter leg, and the longer leg is the shorter leg times 3 . Chapter 7 AA Similarity Postulate SSS Similarity Postulate SAS Similarity Postulate If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. The bisector of an angle of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. Chapter 6 The sum of the interior angle measures of a convex polygon with n sides is: S = (n – 2) 180. Each angle of a regular polygon of n sides is: a = (n – 2)180 n The sum of the measures of the exterior angles of a polygon, one angle at each vertex, is 360 The measure of each exterior angle of a regular polygon of n sides is 360 . n If a quadrilateral is a parallelogram, then opposite sides are congruent. If a quadrilateral is a parallelogram, then opposite angles are congruent. If a quadrilateral is a parallelogram, then consecutive angles are supplementary. If a quadrilateral is a parallelogram, then its diagonals bisect each other. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If an angle of a quadrilateral is supplementary to both of its consecutive angles, 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 opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. If a parallelogram is a rectangle, then its diagonals are congruent. If a parallelogram is a rhombus, then its diagonals are perpendicular. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If one pair of consecutive sides of a parallelogram is congruent, then the parallelogram is a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. If a quadrilateral is an isosceles trapezoid, then its base angles are congruent. A trapezoid is isosceles if and only if its diagonals are congruent. If a quadrilateral is a kite, then one pair of opposite angles are congruent and one diagonal bisects a pair of opposite angles. If a quadrilateral is a kite, then its diagonals are perpendicular and one diagonal bisects the other (but not both). Chapter 9 ~ Prect = 2l + 2w ~ Arect = lw (or bh) ~ Psq = 4s ~ Asq = s2 ~ Apar = bh ~ Atri = ½bh ~ Atrap = ½h(b1 + b2) ~ Arhom = ½d1d2 ~ Akite = ½d1d2 ~ C = 2πr (or C = dπ) ~ Acir = πr2 ~ Areg polygon = ½ap ~ Lateral Area of a prism LAprism = Ph P = perimeter of the base and h = height of the prism ~ Surface Area of a prism SAprism = LA + 2B LA = lateral area and B = area of the base ~ Volume of a rectangular prism Vrect prism = lwh l = length of base, w = width of base, h = height of prism ~ Volume of any prism Vprism = Bh B = area of base, h = height of prism Chapter 11 If a line is tangent to a circle, then it is perpendicular to eh radius drawn to the point of tangency. If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. If two segments are tangent to a circle from the same exterior point, then the segments are congruent. The measure of an inscribed angle is half the measure of its intercepted arc. If inscribed angles of a circle intercept the same arc, then the angles are congruent. If a tangent and a chord (or secant) intersect at a point on a circle, then the measure of the angle formed is half the measure of the intercepted arc. If two chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. If a tangent and a secant, two tangents or two secants intersect at a point in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs. ~ Arc length = m 2 r 360 1 m = arc measure (degrees) r = radius ~ Area of a sector = m 360 r 2 1 m = arc measure (degrees) r = radius