47 terms

theorems and postulates chapters 5-7

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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
perpendicular bisector theorem
if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
angle bisector theorem
if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
converse of the angle bisector theorem
if a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.
theorem 5-6
the perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices
theorem 5-7
the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides
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
theorem 5-9
the lines that contain the altitudes of a triangle are concurrent
corollary to the triangle exterior angle theorem
the measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles
theorem 5-10
if the two sides of a triangle are not congruent then the larger angle lies opposite the longer side.
theorem 5-11
if two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
triangle inequality theorem
the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
theorem 6-1
opposite sides of a parallelogram are congruent
theorem 6-2
opposite angles of a parallelogram are congruent
theorem 6-3
the diagonals of a parallelogram bisect each other
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
theorem 6-5
if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
theorem 6-6
if one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram
theorem 6-7
if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
theorem 6-8
if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
theorem 6-9
each diagonal of a rhombus bisects two angles of the rhombus
theorem 6-10
the diagonals of a rhombus are perpendicular.
theorem 6-11
the diagonals of a rectangle are congruent
theorem 6-12
if one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus
theorem 6-13
if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
theorem 6-14
if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle
theorem 6-15
the base angles of an isosceles trapezoid are congruent
theorem 6-16
the diagonals of an isosceles trapezoid are congruent
theorem 6-17
the diagonals of a kite are perpendicular
trapezoid midsegment theorem
1) the midsegment of a trapezoid is parallel to the bases.
20 the length of the midsegment of a trapezoid is half the sum of the lengths of the bases
area of a rectangle
the area of a rectangle is the product of its base and height. a=bh
area of parallelogram
the area of a parallelogram is the product of a base and the corresponding height. a=bh
area of a triangle
the area of a triangle is half the product of base and the corresponding height a=1/2bh
Pythagorean theorem
in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a(squared) +b(squared)=c(squared)
theorem 7-6
if the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuse.
theorem 7-6
if the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuse. is c2>a2+b2 the triangle is obtuse
theorem 7-7
if c2<a2+b2 then it is acute
45 45 90 triangle theorem
in a 45 45 90 triangle both legs are congruent and the length of the hypotenuse is square root of 2 times the length of a leg
hypotenuse=square root of 2 times leg
30 60 90 triangle theorem
in a 30 60 90 triangle, the length of the hypotenuse is twice the length of the shorter leg. the length of the longer leg is square root of 3 times the length of the shorter leg.
hypotenuse=2 times shorter leg
longer leg= square root of three times shorter leg
area of trapezoid
a=1/2h(b1+b2)
area of a rhombus or kite
a=1/2d(1)d(2)
area of a regular polygon
a=1/2ap
arc addition postulate
the measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs.
circumference of a circle
c=pie times diameter or C=pie times radius squared
arc length
the length of an arc of a circle is the product of the ratio measure of the arc divided by 360 and the circumference of the circle
length of arc ab=measure of arc ab over 360 times 2pieR
arc of a circle
a=pie R squared
area of a sector of a circle
the area of a sector of a s=circle is the product of the ratio measure of the arc over 360 and the circle.