A1.4 Triangles in VS .NET

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A1.4 Triangles
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A triangle is a plane closed figure formed by three line segments that intersect each other at their endpoints. Triangles that have no two sides with the same length are called scalene triangles, those with at least two sides having the same length are called isosceles triangles, and those with all three sides having the same length are called equilateral triangles. If a triangle contains a right angle, it is a right triangle. A triangle that does not contain a right angle is called an oblique triangle. Two triangles are said to be congruent when they have the same size and the same shape. When two triangles are congruent, the pairs of corresponding sides have the same length and the pairs of corresponding angles are equal. Triangles that have the same shape are said to be similar. Similar triangles have corresponding sides that are proportional in length and corresponding angles that are equal. A median of a triangle is a line segment from a vertex to the midpoint of the opposite side. An altitude of a triangle is a line segment from one vertex perpendicular to the opposite side.
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Properties and Theorems
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If three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent. If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent. If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, the triangles are congruent. If two angles and a nonincluded side of one triangle are equal to the corresponding two angles and nonincluded side of another triangle, the triangles are congruent. If the hypotenuse and leg of one right triangle are equal to the corresponding sides of another right triangle, then the two right triangles are congruent. If two angles of one triangle are equal to the corresponding two angles of another triangle, the triangles are similar. The sum of the measures of the angles of a triangle is 180 . An exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. The acute angles of a right triangle are complementary. The measure of each angle of an equiangular triangle is 60 . If two sides of a triangle are equal, then the angles opposite these sides are equal. If a triangle is equilateral, then it is also equiangular. If two angles of a triangle are equal, then the sides opposite these angles are equal. If a triangle is equiangular, then it is also equilateral. The altitude to the base of an isosceles triangle bisects the base and the vertex angle. The median to the base of an isosceles triangle bisects the vertex angle and is perpendicular to the base.
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APPENDIX 1 Geometry
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The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. In a right triangle, the square of the length of the hypotenuse c is equal to the sum of the squares of the lengths of the two legs a and b; i.e., c2 a2 b2 (Pythagorean theorem). In a 45 45 right triangle, the length of the hypotenuse c is equal to 22 times the length of a leg 22a. a; i.e., c In a 30 60 right triangle, the length of the hypotenuse c is equal to 2 times the length of the leg a opposite the 30 angle; i.e., c 2a. Also, the length of the leg b opposite the 60 angle is equal to 23 times the length of the leg a opposite the 30 angle; i.e., b 23a. The midpoint of the hypotenuse of a right triangle is equidistant from all three vertices of the triangle. If the square of the length of one side c of a triangle is equal to the sum of the squares of the lengths of the other two sides a and b of the triangle, i.e., c2 a2 b2, then the triangle is a right triangle. The area K of a triangle is one-half the product of its altitude a and base b; i.e., K
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