Lines and Angles角度
1) lines & line segments
2) angles & degrees
3) 180 degree in a straight angle, and 90 degree in a right angle
4) angle bisectors; perpendicular bisectors
5) two angles on a line are supplementary
6) vertical angles are congruent
7) angles formed by a transversal intersecting a pair of parallel lines
Vertex 角的顶点
90 degree in a right angle
Perpendicular 垂直的 : if two lines or segments meet at a right angle
Congruent 全等
Bisectors 二等分
Angle bisector
Midpoint
Perpendicular bisector: a line bisect a segment and also perpendicular to it
Supplementary
Vertical angles对顶角: vertical angles are congruent
Transversal 横断的：穿过两条平行线的直线
Triangles - part 1
1) the sum of the three angles in any triangle must be 180 degree
2) two angles of a triangle must be acute锐角；the third could be acute, right直角, or abtuse钝角
3) biggest angle opposite longest side; smallest angle opposite shortest side
4) the sum of any two sides of a triangle is greater than the third side (triangle inequality theorem)
5) any side of a triangle must be greater than the difference of the other two sides and less than the sum of the other two sides
Vocabulary:
Sides
Vertex/Vertices
Assumptions & Estimation
On the GRE, we can assume that a line that looks straight in fact is straight, but we cannot assume virtually anything else purely from the diagram
We must rely on the facts and relations specified in the problem text or in special symbols in the diagram
Geometry Strategies - part 1
1) always draw a diagram, and label it with what you are told and what you can deduce
2) you may have to extend a line or introduce a new line
3) you may have to assign variables to lengths or angles and do algebra
4) in diagrams, remember to ‘look big’ and ‘look small’
Triangles - part 2
1) isosceles等腰三角形: equal bases, and opposite angles equal
2）equilateral 等边三角形：all equal sides, all 60 degree angles
3) area = 0.5bh; any side can be the base, and the altitude is perpendicular to this side
4) the altitude, perpendicular bisector, line from vertex to opposite midpoint, and angle bisector are four completely different lines in most triangles, but the line of symmetry in an isosceles triangle plays all four of those roles
Right Triangles
1) right triangles have one 90 degree angle and two acute angles
2) right triangles have one hypotenuse and two les
3) Pythagorean Theorem: a^2 + b^2 = c^2 (only for right triangles)
4) Pythagorean triplets:{3,4,5}, {5,12,13}, {8,15,17} and {7,24,25}
5) if the sides given are larger, divide down by the GCF, do the computations in the smaller triangle, then scale back up.
Similar Triangles
1) similar figures = same shape, different size
2) angles in similar figures are equal
3) we can prove two triangles are similar if they simply share two angles
4) sides in similar figures are proportional
5) the scale factor, k, is the factor by which all lengths in the smaller figure were multiplies to arrive at the lengths in the larger figure
6) if all the lengths are multiplied by k, then area is multiplied by k^2
Special Right Triangles
1) one special triangle, the isosceles right triangle, has angles of 45-45-90 and sides of 1-1-sqrt(2)
2) the other results from dividing an equilateral triangle in half, and has angles of 30-60-90 and sides of 1-sqrt(3)-2
3) we can use these patterns and proportional reasoning to solve a variety of problems
Quadrilaterals
Trapezoid 梯形
Parallelogram 平行四边形
Rectangle 长方形
Rhombus 菱形
Square 正方形的
1）all quadrilaterals: sum of angles = 360 degree
2) ‘big four’ parallelogram properties: parallel opposite sides; equal opposite side; equal opposite angles; diagonal bisect each other
3) Rhombus = 4 equal sides + ‘big four’
4) Rectangle = all 90 degree angles + ‘big four’
5) Square = a rectangle and a rhombus
6) Trapezoid = exactly one pair of parallel sides
7) Symmetrical trapezoid = equal legs; equal angles on each side; equal diagonals
Area of Quadrilaterals
1) square: A = s^2
2) rectangle, rhombus, parallelogram: A = bh
3) trapezoid: A = ((b_1+b_2)/2)h
4) for ‘slanty’ shapes, think about subdividing into rectangles and right triangles
5) expect to find the Pythagorean theorem involved in anything involving a slant倾斜
Polygons 多边形
不闭合，交叉，有弧线 都不是 多边形 (GRE 只考convex的多边形）
1） 3-sides = triangle; 4-sides = quadrilateral; 5-sides = pentagon; 6-sides = hexagon 8-sides = octagon
2) a segment from one vertex to a non-adjacent vertex is a diagonal
3) the sum of the angle in an n-sided polygon equals (n-2)*180
Regular Polygons
1. Regular polygons have all equal sides and all equal angles
2. We can find the sum of the angles using the (n-2)*180 formula, and divide by n to find the measure of each individual angle
Circles
1) all radii(radius半径的复数) of a circle are the same length
2) Chord弦 = both endpoints on the circle
3) diameter = chord through the center; this is longest possible chord in a circle
4) c = pi*d = 2*pi*r
5) arc弧 = a piece of the curve of a circle, denoted on the test by three points
6) A = pi*r^2
7) the #1 circle strategy: find radius first, and use radius to find everything else
Circle Properties
1) if two sides of a triangle are radii, the triangle is isosceles
2) a central angle has the same measure as the arc it intercepts
3) equal length chords intercept equal arcs
4) an inscribed内接 angle has half the measure of the arc is intercepts
5) an angle inscribed in a semicircle is 90 degree
6) two inscribed angles intersecting the same chord on the same side are equal
7) a tangent line is perpendicular to a radius at the point of tangency
Circles, Arcs, and Sectors
Vocabulary: circular sector
We find arc-length and areas of a sector by setting up part-to-whole proportions
arc-length/(2*pi*r) = angle/360
area of sector/(pi*r^2) = angle/360
Volume and Surface Area
Total surface area (total S.A.)
1) Cube立方体(a special case of rectangular solids): V = s^3, S.A. = 6s^2
2) Rectangular solid长方体: V = hwd, S.A. = 2hw+2hd+2wd
3) Cylinder圆柱体: V = pi*(r^2)*h, S.A. = 2pi*r^2 + 2pi*rh
2D - area of shapes 3D - volume of shapes
Face-diagonal; Space diagonal
Sphere球体：
Every point on the surface is equidistance from the center
The sphere is circular in every direction
Scale Factor and Scaling
1) between two similar figures, every length is multiplied by a constant ratio known as a scale factor
2) the ratio of areas equals k^2
3) the ratio of volumes equals k^3
Units of Measurement
1) change units with unit conversions, which are often given in the text of the problem
2) for areas, we have to square the unit-conversion fraction
3) for volume, we have to cube the unit-conversion fraction
