Theorem Thursday 9 Flashcards
Isosceles trapezoid base angles converse
If a trapezoid has a pair of congruent base angles then it is an isosceles trapezoid
Isosceles trapezoid diagonals theorem
A trapezoid is isosceles if and only if it’s diagonals are congruent
Trapezoid mid segment theorem
The mid segment of a trapezoid is parallel to each base and the length is one half the sum of the lengths of the bases
Kite diagonals theorem
If a quadrilateral is a kite then it’s diagonals are perpendicular
Kite opposite angles theorem
If a quadrilateral is a kite then exactly one pair of opposite angles are congruent
Isosceles trapezoid base angles theorem
If a trapezoid is isosceles then each pair of base angles is congruent
Angle angle AA similarity theorem
If two angles of one triangle are congruent to two angles of another triangle the. The two triangles are similar
Side side side SSS similarity theorem
If the corresponding side lengths of two triangles are proportional then the triangles are similar
Side angle Side SAS similarity theorem
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of he sides including these angles are proportional then the triangles are similar
Areas of similar polygons
If two polygons are similar then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths
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
triangle proportionality theorem
if a line parallel to one side of a triangle intersects the other two sides then it divides the two sides proportionally
converse of the triangle proportionality theorem
if a line divides two sides of a triangle proportionally then it is parallel to the third side
three parallel lines theorem
if three parallel lines intersect two transversals then they divide the transversals proportionally
triangle angle bisector
if a ray bisects an angle of a triangle then it divides the opposite sides into segments whose lengths are proportional to the lengths of the other two sides
Right triangle similarity theorem
If the altitude is drawn to the hypotenuse of a right triangle then the two triangles formed are similar to the original triangle and to each other
Geometric mean (altitude) theorem
the altitude is the geometric mean of the lengths of the two segments of the hypotenuse
Geometric mean (leg) theorem
the length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg
Pythagorean theorem
in a right triangle a squared plus b squared equals c squared
converse of the Pythagorean theorem
if c squared equals a squared plus b squared then the triangle is right
Pythagorean inequalities theorem
for any triangle ABC where c is the length of the longest side, the following statements are true:
if c squared is less than a squared plus b squared then the triangle is acute
if c squared is greater then a squared plus b squared then the triangle is obtuse
