The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles.
Video: External Angle Theorem
In several high school treatments of geometry, the term "exterior angle theorom" has been applied to a different result, namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel postulate will be referred to as the "High school exterior angle theorem" (HSEAT) to distinguish it from Euclid's exterior angle theorem.
Exterior angles
Video: Exterior Angles of a Polygon
A triangle has three corners, called vertices. The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments). Only one of these angles contains the third side of the triangle in its interior, and this angle is called an interior angle of the triangle. In the picture below, the angles ∠ABC, ∠BCA and ∠CAB are the three interior angles of the triangle. An exterior angle is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle ∠ACD is an exterior angle.
Euclid's exterior angle theorem
The proof of Proposition 1.16 given by Euclid is often cited as one place where Euclid gives a flawed proof.
Euclid proves the exterior angle theorem by:
construct the midpoint E of segment AC,
draw the ray BE,
construct the point F on ray BE so that E is (also) the midpoint of B and F,
draw the segment FC.
By congruent triangles we can conclude that ∠ BAC = ∠ ECF and ∠ ECF is smaller than ∠ ECD, ∠ ECD = ∠ ACD therefore ∠ BAC is smaller than ∠ ACD and the same can be done for the angle ∠ CBA by bisecting BC.
The flaw lies in the assumption that a point (F, above) lies "inside" angle (∠ ACD). No reason is given for this assertion, but the accompanying diagram makes it look like a true statement. When a complete set of axioms for Euclidean geometry is used (see Foundations of geometry) this assertion of Euclid can be proved.
High school exterior angle theorem
Video: Exterior Angle Theorem (High School EAT) | Two Opposite Interior Angles Add to the Exterior Angle
The high school exterior angle theorem (HSEAT) says that .....
An angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interiorangle (or internal angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.
If every internal angle of a simple polygon is less than 180°, the polygon is called convex.
In contrast, an exteriorangle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.
Video: What is the Relation of an Exterior Angle of a Triangle with its Interior Angles?
Properties
The sum of the internal angle and the external angle on the same vertex is 180°.
The sum of all the internal angles of a simple polygon is 180(n–2)° where ......
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom.
This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology.
A solid bounded by a cylindrical surface and two parallel planes is called a (solid) cylinder. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a base of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder, otherwise it is called an oblique cylinder. If the bases are disks (regions whose boundary is a circle) the cylinder is called a circular cylinder. In some elementary treatments, a cylinder always means a circular cylinder.
The height (or altitude) of a cylinder is the perpendicular distance between its bases.
Right circular cylinders
Video: What is a Right Circular Cone?
The bare term cylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder.
A right circular cylinder can also be thought of as the solid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.
Cylindric sections
Cylindric section
A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram. Such a cylindric section of a right cylinder is a rectangle.
Video: Cylindrical sections
A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section. If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic or hyperbolic respectively.
Volume
Video: Volume Of A Cylinder
If the base of a circular cylinder has a radius r and the cylinder has height h, then its volume is given by
V = πr2h.
This formula holds whether or not the cylinder is a right cylinder.
Surface area
Video: Surface Area of a Cylinder
Having radius r and altitude (height) h, the surface area of a ......
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.
The term oblong is occasionally used to refer to a non-square rectangle.A rectangle with vertices ABCD would be denoted as ABCD.
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
Rectangle
Rectangle
Video: Area of a Rectangle
Type
quadrilateral, parallelogram, orthotope
Edges
and
vertices
4
Schläfli symbol
{ } × { }
Coxeter diagram
Symmetry group
Dihedral (D2), [2], (*22), order 4
Dual polygon
rhombus
Properties
convex, isogonal, cyclic Opposite angles and sides are congruent
Characterizations
Video: Understanding Quadrilaterals - Properties of Rectangles
A convex quadrilateral is a rectangle if and only if it is any one of the following:
a parallelogram with at least one right angle
a parallelogram with diagonals of equal length
a parallelogram ABCD where triangles ABD and DCA are congruent
an equiangular quadrilateral
a quadrilateral with four right angles
a convex quadrilateral with successive sides a, b, c, d whose area is .:fn.1
a convex quadrilateral with successive sides a, b, c, d whose area is
Classification
A rectangle is a special case of both parallelogram and trapezoid. A square is a special case of a rectangle.
Traditional hierarchy
A rectangle is a special case of a parallelogram in which each pair .....
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.
Formulas
Height
For any isosceles triangle, the following six line segments coincide:
the altitude, a line segment from the apex perpendicular to the base,
the angle bisector from the apex to the base,
the median from the apex to the midpoint of the base,
the perpendicular bisector of the base within the triangle,
the segment within the triangle of the unique axis of symmetry of the triangle, and
the segment within the triangle of the Euler line of the triangle.
Video: Height of Isosceles Triangle
Their common length is the height of the triangle. If the triangle has equal sides of length and base of length , the general triangle formulas for the lengths of these segments all simplify to
This formula can also be derived from the Pythagorean theorem using the fact that .....
The rhombus has a square as a special case, and is a special case of a kite and parallelogram.
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.
Characterizations
A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:
a parallelogram in which a diagonal bisects an interior angle
a parallelogram in which at least two consecutive sides are equal in length
a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
a quadrilateral with four sides of equal length (by definition)
a quadrilateral in which the diagonals are perpendicular and bisect each other
a quadrilateral in which each diagonal bisects two opposite interior angles
a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent
a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point
Basic properties
Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:
Opposite angles of a rhombus have equal measure.
The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
Its diagonals bisect opposite angles.
The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus
Not every parallelogram is a rhombus, though any ......
In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners.
Video: What is a Quadrilateral? – Geometric Shapes
Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.
The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side".
Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is
This is a special case of the n-gon interior angle sum formula (n − 2) × 180°
Simple quadrilaterals
Any quadrilateral that is not self-intersecting is a simple quadrilateral.
Video: Quadrilaterals - 04 Simple and Complex Quadrilateral
Convex quadrilaterals
Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes American English.
Video: Convex and Concave Quadrilateral
In a convex quadrilateral, all interior angles are less than 180° and .....
In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional.
Video: Similar Triangles
It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of which is necessary and sufficient for two triangles to be similar:
The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:
If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar.
All the corresponding sides have lengths in the same ratio:
AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.
ABCD.
.:fn.1
of the triangle. If the triangle has equal sides of length 
and base of length 
, the general triangle formulas for the lengths of these segments all simplify to
(half the product of the diagonals)
