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Example research essay topic: Opposite Sides One Side - 1,501 words

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... triangle, then the triangles are congruent. A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection. CPCTC - Corresponding parts of congruent triangles are congruent.

Recall that an isosceles triangle has two congruent sides. These congruent sides are called are called legs and the third side is called the base. The angles at the base are called base angles and the angle opposite the base is called the vertex angle of the isosceles triangle. Theorem 3 - 1 The Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those angles are congruent. An equilateral triangle is also equiangular. An equilateral triangle has three 60 angles.

The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. An equiangular triangle is also equilateral. If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. In a right triangle the side opposite the right angle is called the hypotenuse (hyp. ). The other two sides are called legs.

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. A median of a triangle is a segment from a vertex to the midpoint of the opposite side. An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. A perpendicular bisector of a segment is a line (or ray or segment) that is perpendicular to the segment at its midpoint. If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

The distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line (or plane). If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Opposite sides of a parallelogram are congruent. If two lines are parallel, then all points on one line are equidistant from the other line.

Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A line that contains the midpoint of one side of a triangle and is parallel to another side bisects the third side. A quadrilateral with four right angles is a rectangle. Since both pairs of opposite angles are congruent, every rectangle is a parallelogram. A quadrilateral with four congruent sides is a rhombus.

Since both pairs of opposite sides are congruent, every rhombus is a parallelogram. A quadrilateral with four right angles and four congruent sides is a square. The diagonals of a rectangle are congruent. The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects two angles of the rhombus. The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. A quadrilateral with exactly one pair of parallel sides is called a trapezoid. The parallel sides are called bases: the other sides are legs. A trapezoid with congruent legs is called isosceles. Base angles of an isosceles trapezoid are congruent.

The median of a trapezoid is the segment that joins the midpoints of the legs. (2) has a length equal to half the sum of the lengths of the bases. The segment that joins the midpoints of two sides of a triangle (1) is parallel to the third side; (2) has a length equal to half the length of the third side. In an indirect proof you begin by assuming temporarily that the conclusion is not true. Then you reason logically until you reach a contradiction of the hypothesis or another known fact. If a b and c d, then a + c b + d. If a = b + c, and c 0, then a b.

If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle. The perpendicular segment from a point to a line is the shortest segment from the point to the line. The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Theorem 4 - 20 The Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second. The ratio of one number to another is the quotient when the first number is divided by the second. A proportion is an equation stating that two ratios are equal. Two polygons are similar if their vertices can be paired so that: (1) Corresponding angles are congruent. (2) Corresponding sides are in proportion. (Their lengths have the same ratio. ) The ratio of the lengths of two corresponding sides is called the scale factor of the similarity. Postulate 15 AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. If the sides of two triangles are in proportion, then the two triangles are similar. Points L and M lie on -AB and -CD, respectively. If AL = CM, we say that -AB and -CD are divided proportionally. LB MD Theorem 5 - 3 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

If three parallel lines intersect two transversal's, then they divide the transversal's proportionally. Theorem 5 - 4 Triangle Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. Suppose r, s, and t are positive numbers with r = s. Then s is called the geometric mean between r and t. s t When you write radical expressions you should write them in simplest form. This means writing them so that 1.

No radicand has a factor, other than 1, that is a perfect square. 3. No fraction has a denominator that contains a radical. 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. When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

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. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Bibliography:


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Research essay sample on Opposite Sides One Side

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