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Triangle Theorems |
Right Triangle Theorems |
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is greater than the length
of the third side. |
If the alttitude is drawn to the hypotenuse of a right triangle,
then the two triangles formed are similar to the original triangle and to each
other. |
One side of a triangle is longer than a second side if and only
if
the angle opposite the first side is larger then the angle opposite the
second side. |
If the altitude is drawn to the hypotenuse of a right triangle,
then the length of the altitiude is the geometric mean of the lengths of the
segments of the hypothenuse. |
Exterior Angle of a Triangle Theorem
The measure of an exterior angle of a triangle
is equal to the sum of the meaures of the two interior angles not
adjacent
to it. |
If the altitude is drawn to the hypotenuse of a right triangle,
then the length of each leg is the geometric mean of the lengths of the hypotenuse
and the segment of the hypotenuse adjacent to that leg. |
SSS Theorem
If three sides of one triangle are congruent to the corresponding
sides of another triangle, then the triangles are congruent.
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In a 45-45-90 triangle,
the hyptenuse is square root of 2 times the length of each leg. |
SAS Theorem
If two sides and their included angle are congruent to the corresponding
sides of another triangle, then the triangles are congruent. |
In a 30-60-90 triangle
the length of the shorter leg and the length of the longer leg is square root
of times the length of the shorter leg. |
AAS Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding
parts of another trianlge
then the triangles are congruent. |
The Pythagorean Theorem
In a right triangle the square of the length of the hypotenuse
is equal to the sum of the squares of the lengths of the legs. |
HL Theorem
If the hypotenuse and a leg of one right triangle are congruent to the
corresponding parts of another right triangle
then the triangles are congruent. |
Converse of the Pythagorean Theorem
If a, b, and c are the lengthsof the sides of a triangle and a + b = c,
then the trianlge is a right triangle. |
If a point is on the perpendicular bisector of a segment
then the point is equaldistant from the endpoints of the segments. |
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The Isosceles Triangle Theorem
If two sides of a triangle are congruent
then the angles opposite the sides are congruent. |
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The Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent,
then the sides opposite the angles are congruent. |
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