Theorem 4 7 If Two Angles example essay topic

i got this from a geometry book Theorem 1-1 Vertical Angles Theorem Vertical angles are congruent. Theorem 1-2 Congruent Supplements Theorem If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. Theorem 1-3 Congruent Complements Theorem If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. Theorem 2-1 Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180.

Theorem 2-2 Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. Theorem 2-3 Polygon Interior Angle-Sum Theorem The sum of the measures of the interior angles of an n-go is (n-2) 180. Theorem 2-4 Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. Theorem 2-5 Two lines parallel to a third are parallel to each other. Theorem 2-6 In a plane, two lines perpendicular to a third line are parallel to each other. Theorem 3-1 A composition of reflections in two parallel lines is a translation.

Theorem 3-2 A composition of reflections in two intersecting lines is a rotation. Theorem 3-4 Isometry Classification Theorem There are only four isometries. They are reflection, translation, rotation, and glide reflection. Theorem 4-1 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. Theorem 4-2 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Theorem 4-3 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

Theorem 4-4 If a triangle is a right triangle, then the acute angles are complementary. Theorem 4-5 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Theorem 4-6 All right angles are congruent. Theorem 4-7 If two angles are congruent and supplementary, then each is a right angle.

Theorem 4-8 Triangle Midsegment Theorem If a segment joins the midpoint of two sides of a triangle, then the segment is parallel to the third side and half its length. Theorem 4-9 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Theorem 4-10 If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side. Theorem 4-11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. Theorem 4-12 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 4-13 Converse Of Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Theorem 4-14 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Theorem 4-15 Converse of Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector. Theorem 4-16 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem 4-17 The bisector of the angles of a triangle are concurrent at a point equidistant from the sides. Theorem 4-18 The lines that contain the altitude of a triangle are concurrent. Theorem 4-19 The medium of a triangle are concurrent.

Theorem 5-1 Area Of A Rectangle The area of a rectangle is the product of its base and height. A = bh Theorem 5-2 Area of a parallelogram The area of a parallelogram is the product of any base and its corresponding height. A = bh Theorem 5-3 Area of a Triangle The area of a triangle is half the product of any base and the corresponding height. A = 1/2 bh Theorem 5-4 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. A 2+b 2+c 2 Theorem 5-5 Converse Of The Pythagorean Theorem If the square of the length of one 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. Theorem 5-645-45-90 Triangle Theorem In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is radical 2 times the length of a leg.

Hypotenuse = radical 2 times leg Theorem 5-730-60-90 Triangle Theorem In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is radical 3 times the length of the shorter leg. Hypotenuse = 2 times shorter leg Longer leg = radical 3 times shorter leg Theorem 5-8 Area Of A Trapezoid The area of a trapezoid is half the product of the height and the sum of the lengths of the bases. A = 1/2 h (b 1+b 2) Theorem 5-9 Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter. A = 1/2 ap Theorem 5-10 Circumference of a Circle The circumference of a circle is pie (3.14) times the diameter. C = pie (3.14) d or C = 2 pie r Theorem 5-11 Arc Length The length of an arc of a circle is the product of the ratio (measure of the arc over 360) and the circumference of the circle.

Length of arc AB = measure of arc AB over 360 times 2 pie r Theorem 5-12 Area of a Circle The area of a circle is the product of pie and the square of the radius. A = pie r 2 Theorem 5-13 Area of a Sector of a Circle The area of a sector of a circle is the product of the ratio (measure of the arc over 360) and the area of the circle. Area of sector AOB = measure of arc AB over 360 times pie r 2 Theorem 6-1 Lateral and Surface Areas of a Right Prism The lateral area of a right prism is the product of the perimeter of the base and the height. L.A. = ph The surface area of a right prism is the sum of the lateral area and the area of the bases. S.A. = L.A. +2 BTheorem 6-2 Lateral and Surface Areas of a Right Cylinder The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. L.A. = 2 pie rh or L.A. = pie dh The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. S.A. = L.A. + 2 BTheorem 6-3 Lateral and Surface Areas of a Regular Pyramid The lateral area of a regular pyramid is half the product of the perimeter of the base and the slant height. L.A. = 1/2 pl The surface area of a regular pyramid is the sum of the lateral area and the area of the base. S.A. = L.A. + B Theorem 6-4 Lateral and Surface Area of a Right Cone The lateral area of a right cone is half the product of the circumference of the base and the slant height. L.A. = 1/2 times 2 pie l or L.A. = pie r l The surface area of a right cone is the sum of the lateral area and the area of the base. S.A. = L.A. + BTheorem 6-5 Cavalier's Principle If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume. Theorem 6-6 Volume of a Prism The volume of a prism is the product of the area of a base and the height of the prism.

V = Bh Theorem 6-7 Volume of a Cylinder The volume of a cylinder is the product of the area of a base and the height of the cylinder. V = Bh or V = pie r 2 h Theorem 6-8 Volume of a Pyramid The volume of a pyramid is one third the product of the area of the base and the height of the pyramid. V = 1/3 Bh Theorem 6-9 Volume of a Cone The volume of a cone is one third the product of the area of the base and the height of the cone. V = 1/3 Bh or V = 1/3 pie r 2 h Theorem 6-10 Surface Area of a Sphere The surface area of a sphere is four times the product of pie and the square of the radius of the sphere. S.A. = 4 pie r 2 Theorem 6-11 Volume of a Sphere The volume of a sphere is 4/3 the product of pie and the cube of the radius of the sphere. V = 4 pie r 3 Theorem 7-1 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Theorem 7-2 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.

Theorem 7-3 Converse of Alternate Interior Angles Theorem If two lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the lines are parallel. Theorem 7-4 Converse of Same-Side Interior Angles Theorem If two lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the lines are parallel. Theorem 8-1 Angle-Angle Side Theorem (AAS Theorem) If two angles and a non included side of one triangle are congruent to two angles and the corresponding non included side of another triangle, then the triangle are congruent. Theorem 8-2 Hypotenuse-Leg Theorem (HL Theorem) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Theorem 9-1 Opposite sides of a parallelogram are congruent.

Theorem 9-2 Opposite angles of a parallelogram are congruent. Theorem 9-3 The diagonals of a parallelogram bisect each other. Theorem 9-4 If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Theorem 9-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 9-6 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 9-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 9-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 9-9 Each diagonal of a rhombus bisects two angles of the rhombus. Theorem 9-10 The diagonals of a rhombus are perpendicular. Theorem 9-11 The area of a rhombus is equal to half the product of the lengths of its diagonals.

Theorem 9-12 The diagonals of a rectangle are congruent. Theorem 9-13 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 9-14 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 9-15 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Theorem 9-16 Base angles of an isosceles trapezoid are congruent.

Theorem 9-17 The diagonals of an isosceles trapezoid are congruent. Theorem 9-18 The diagonals of a kite are perpendicular. Theorem 9-19 Trapezoid Midsegment Theorem The mid segment of a trapezoid is (1) parallel to the bases and (2) half as long as the sum of the lengths of the bases. Theorem 10-1 Side-Angle-Side Similarity Theorem (SAS ~ Theorem) If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. Theorem 10-2 Side-Side-Side Similarity Theorem ( ~ Theorem) If the corresponding sides of two triangles are proportional, then the triangles are similar. Theorem 10-3 The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

Theorem 10-4 Side-Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Theorem 10-5 Triangle-Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. Theorem 10-6 Perimeters and Areas of Similar Figures If the similarity ratio of two similar figures is a: b, then (1) the ratio of their perimeters is a: b, and (2) the ratio of their areas is a 2: b 2. Theorem 10-7 Areas and Volumes of Similar Solids If the similarity ratio of two similar solids is a: b, then (1) the ratio of their areas is a 2: b 2, and (2) the ratio of their volumes is a 3: b 3. Theorem 11-1 Area of Triangle Given SAS The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. Area of triangle = 1/2 times side length times side length times sine of included angle Theorem 12-1 Equations of a Circle The standard form of an equation of a circle with center (h, k) and radius r is (x + h) 2 + (y - k) 2 Theorem 12-2 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Theorem 12-3 Converse of Theorem 12-2 If a line in the same plane as a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. Theorem 12-4 Two segments tangent to a circle from a point outside the circle are congruent. Theorem 12-5 In the same circle or in congruent circles, (1) congruent central angles have congruent arcs, and (2) congruent arcs have congruent central angles. Theorem 12-6 In the same circle or in congruent circles, (1) congruent chords have congruent arcs, and (2) congruent arcs have congruent chords. Theorem 12-7 A diameter that is perpendicular to a chord bisects the chord and its arc.

Theorem 12-8 The perpendicular bisector of a chord contains the center of the circle. Theorem 12-9 In the same circle or in congruent circles, (1) chords equidistant from the center are congruent, and (2) congruent chords are equidistant from the center. Theorem 12-10 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. Theorem 12-11 The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.

Theorem 12-12 The measure of an angle formed by two chords that intersects inside a circle is half the sum of the measures of the intercepted arcs. Theorem 12-13 The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle is half the difference of the measures of the intercepted arcs. Theorem 12-14 If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. Theorem 12-15 If two secant segments are drawn from a point outside a circle, the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. Theorem 12-16 If a tangent and a secant are drawn from a point outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.