# theorems and postulates chapters 5-7

## 47 terms · geometry sukah

### 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

a=1/2h(b1+b2)

a=1/2d(1)d(2)

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

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.