Euclidean Geometry (Before Midterm)

• Euclid’s 5th Axiom

• Triangle Congruence: SAS, ASA, SSS, AAS

• Angles and Parallel Lines

  • vertical angels are congruent

  • corresponding angles are congruent

  • alternate interior angles are congruent

  • supplementary angles add to \(180^\circ\)

Quadrilaterals: 4 sided figure in the plane, where the edges are straight lines.

Parallelogram: Both pairs of opposite sides are parallel.

Trapezoid: At least 1 pair of opposite sides are parallel.

Rhombus: Parallelogram, all sides are same length.

Rectangle: Parallelogram whose internal angles are all right angles.

Square: Rectangle whose sides are all equal length.

Parallelogram Theorem: Let ABCD be a parallelogram. Then the following are equivalent:

1. ABCD is a parallelogram. (opposite sides are parallel)

2. \(\angle DAB \cong \angle BCD\) and \(\angle ABC\cong \angle CDA\) (angles that are across from each other are congruent)

3. \(AB=CD\) and \(BC=DA\) (opposite sides have equal measure)

4. \(\overline{AC}\) and \(\overline{BD}\) bisect each other. (diagnals bisect each other)

Axioms of Area:

1. To every polygonal region (space enclosed by straight lines in the plane) there corresponds a unique positive number called \(\underline{area}\).

2. If 2 tirangles are congruent their areas are equal.

3. If \(R=R_1\cup R_2\) and \(R_1\cap R_2\) is a finite number of segments or points, then the area of \(R\) is the sum of the areas of \(R_1+R_2\).

4. The area of a rectangle is its base times its height.

Similarity

Dilation: shrink or expand by a scaling factor, k, from a center point, P.

Similarity: 2 figures are similar if one can be superimposed on the other by a dilation and a sequence of isometries.

• AA, SAS

Circles

Circle: center, radius

Points on circle have distance r from the center point O.

Unit Circle: radius length 1.

Arc: a connected subset of the points on circle.

Chord: line segment connecting 2 points on circle.

Central Angle: vertex is center of circle, rays intersect in 2 different points. (pie slice)

Inscribed Angles vertex is on circle, rays intersect circle in 2 points.

Inscribed: verities on circle.

For an inscribed square, all four points of a square are on the circle.

Tangent Line: Line that intersects circle at only 1 point.

Radian: measure of the central angle in a unit circle with arc length of 1.

Inscribed Angle Theorem: An inscribed angle is half of a centeral angle that subtends the same arc.

Corollary: Any two inscribed angles have the same arc on the circle are congruent.

Power of the Point Theorem 1: If \(\overline{AB}\) and \(\overline{CD}\) are chords of circle intersecting in x inside a circle. Then \(Ax\cdot xB=Cx\cdot xD\)

Power of the Point Theorem 2: LEt P be a point outside a given circle. Suppose we draw two rays from the point P: one ray intersects the circle at points A and B (in that order), and the other intersects the circle at the points C and D (in that order). Then \(PA\cdot PB=PC\cdot PD\)

Isometries and Symmetries

The set of isometries with composition is a group:

0. Closure

1. Associativity

2. Identity

3. Inverses

Symmetry A symmetry is an isometry that sends a geometric figure to itself.

• 6 symmetries of an equilateral triangle

• 8 symmetries of a square

• 2n symmetries of a regular polygon

Taxicab Geometry

Euclidean Distance: \(d_E(A,B)=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}\)

• \(\pi\approx 3.14\)

Taxicab Distance: \(d_T=|x_B-x_A|+|y_B-y_A|\)

• \(\pi = 4\)

Isometries for taxicab (traingles):

• translations

• rotations by \(90^\circ k\) where k is an integer

• combinations

Spherical Geometry

No parallel may be drawn through a point not on a given line.

Equation of a Sphere: \(S^2=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=\rho^2\}\)

• great circles are straight lines (equator and longitudes)

Distance of a Sphere: \(d_s(A,B)=\rho\cdot\text{arc cos}(\frac{A\cdot B}{\rho ^2})\)

Triangle Angle Measurements

• Can have three right angles

• All three angles added together will be greater than \(180^\circ\)

Area of a Sphere: \(\rho^2\cdot E\) (where the excess \(E=\alpha+\beta+\gamma-180^\circ\))

Consider the surface area of a sphere to be \(4\pi\rho^2\), then the area of a triangle on a sphere will be a proportion of that.

Hyperbolic Geometry

More than one parallel may be drawn through a point not on a given line.

Inversions about a circle

• preserve angles (conformal)

Cross Ratio: Given four distinct points (A,B,C,D) in the plane, the cross ratio is define \((A,B;C,D)=\frac{AC\cdot BD}{BC\cdot AD}\)