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# Triangles

### Theorems and definitions

Similar Triangles 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 △CBD ~ ￼ABC △￼ACD~￼ABC △￼CBD~￼ACD
Corollarry 1 when the altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric mean between the segments of the hypotenuse
Corollary 2 when the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg
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 C²=a²+b²
Pythagorean Triple Is a set of three positive integers a, b, and c that satisfy the equation C²=a²+b²
Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a RIGHT triangle if C^2=a^2+b^2 then (triangle)ABC is a right triangle
Acute Triangles Theorem If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. C^2<a^2+b^2
Obtuse Triangle Theorem 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, then the triangle is obtuse C²>a²+b²
45⁰- 45⁰ - 90⁰ In a 45 45 90 triangle, the hypotenuse is root 2 times as long as each leg
60⁰-90⁰ the hypotenuse is 2 times as long as the shorter leg and the longer leg is root 3 times as long as shorter leg
Trigonometric Ratios sin= side opposite <A / hypotenuse = a/c cos= side adjacent <A / hypotenuse = b/c tan= side opposite <A / side adjacent <A = a/b
Angle Elevation is the angle that your line of sight makes with a horizontal line when you stand and look up at a point in the distance
Triangle Sum Theorem the sum of the measure of the interior angles of a triangle is 180⁰
Exterior Angles Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of two non-adjacent interior angels
Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent
Side Side Side Congruence Postulate (SSS) If three sides of one triangle are congruent to three sides of second triangle, then the two triangles are congruent
Side Angle Side (SAS) Congruence Postulate If two dies and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent
Angle Side Angle (ASA) Congruence Postulate If two angles and he included side of one triangle is congruent to two angles and the included side of a second triangle, then the two triangles are congruent
Angle Angle Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent
Corresponding Parts of Congruent Triangles are Congruent (CPCTC) If two triangles are congruent, then all corresponding parts f those triangles are also congruent
Base Angle two angles that are adjacent to the base of the triangle
Base Angles Theorem If two sides of a triangles are congruent, then the angles opposite them are congruent
Corollary to the Base Angles Theorem If a triangle is equilateral, then it is equiangular
Corollary to Theorem If a triangle is equiangular, then it is equilateral
Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right (RT) triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent
Created by: McChicklet