Lines & angles

Congruent全等的

Length

Intersection

Bisect平分

在GMAT里只是平分角

Vertices頂點

Obtuse angle鈍角

Perpendicular

Complementary angles互補為90度

Height/altitude高Polygon

三維圖形

Width

Acute angles銳角

Opposite angles對頂角Vertical angles

Straight angle平角180度

Supplementary angles互補為180度

不需要相鄰

Polygons(N-2)*180角度

所有角所有邊相等Regular polygon

Equilateral polygon

Perimeter周長

閉合長度

The number of diagonals in a polygon對角線的個數：n(n-3)/2

n是邊長

對角線并不一定相等比如正六邊形

Parallel linesA transversal intersects parallel linesAll the small angles equal each other

All the big angles equal each other

Any combination of Big angle + Small angle is always 180

反之亦然

Triangles

base(底邊)

Bisector in a triangle三角形有三條角平分線

Median in a triangle三角形有三條邊平分線

Height/altitude 高三角形有三條高

Perimeter of a triangle所有邊的和

Impossible triangles and the third side rule三角形兩邊之和大于第三邊最小的兩邊的平方和小于第三條邊的平方和，必為鈍角三角形

三角形兩邊之差小于第三邊最小的兩邊的平方和大于第三條邊的平方和，必為銳角三角形b-c

The angles in a triangle sum up to 180

Obtuse

Equilateral三個角60度

三條邊相等

角平分線也是中線

面積底*高/2

如果三角形三邊長為a,b,c,

The largest side is opposite the largest angle大邊對大角

小邊對小角

同角度對的邊也相等

若兩個三角形相似，則這兩個三角形的面積比等于相似比的平方

三角形的一個外角等于其不相鄰的兩個內角之和

IsoscelesBase angles兩條腰

Vertex angle頂角角平分線是中線

Right trianglesHypotenuse直角三角形的弦

Pythagorean theorem勾股定理

3:4:5

5:12:13

8:15:17

7:24:25

Circles

Circumference圓周長

Concentric circles同心圓

Radius, diameter, chordChord弦

RadiusThe distance from the center of the circle to its boundary lineDiameterA chord that passes through the center

Longest chord

Tangent lineTangentA line outside the circle that touches the outer boundary at a single point

A radius drawn to the point of tangency is always perpendicular to the tangent

Circumference圓周長

周長是直徑的3倍多點

Inscribed or peripheral anglesAn angle with the vertex on the edge of the circle.

Its rays 射線 are along chords of the circle

Inscribe circlesThe largest possible circle that can be drawn within a polygon

Each side of the polygon is tangent to the circle

Circumscribed circles外接圓A circumcircle passes through all the vertices of a polygon

所有的三角形和正多邊形都有外接圓

Central anglesOriginates from the center and is formed by two radii

Area

Sector 扇形像pizza塊

Quadrant四分之一圓周

SegmentA portion of the circle’s area defined by a chordSemicircleA segment defined by a diameter

Measuring arcsA portion of a circle’s circumference

Major part

Minor part

The measure of an Arc is equal to the central angle that defines it

The measure of an arc is double the inscribed angle that defines itAll the inscribed angles that define the same arc of 2x are equal to each other, and equal to x

An angle with its vertex on the edge of a circle is half the measure of the arc defined by its rays

解釋圓心角：頂點在圓心上且兩條邊是圓的弦的角成為圓心角

圓周角：頂點位于圓周上且兩條邊是圓的弦的角為圓周角

圓中同一段弧所對的圓心角是圓周角的兩倍

An inscribed angle defining a diameter is a right angleAll inscribed angles lying on a diameter equal 90The proportion of arc/circumference = sector/area=central angle/360

Arcs of equal length are arcs of equal degreesEqual central angles define equal arcs

Equal inscribed angles define equal arcsMethod for solving questions involving arcs and anglesCopy the figure to your noteboard

Indicate the measure of every arc on the circle

Then solve for the answer to the question

Quadrilaterals

四邊形

Parallelogram 平行四邊形Opposite sides that are equal and parallel

Adjacent angles sum up to 180

area=base * height

The diagonals bisect each other, but are not necessarily equal

性質對邊互相平行且相等

對角相等，四個內角和=360

若平行四邊形的一個內角等于90，那么他的所有內角都=90

RhombusA special instance of a parallelogram with all sides equal

The diagonals are perpendicular to each other, as well as bisect each other and their respective base angles

性質菱形的面積等于兩條對角線乘積的一半

對角線平分四個內角

RectanglePerimeter2w+2l

Arealength*widthDiagonalsEqual and bisect each other

4個直角的平行四邊形

性質周長一定時，正方形面積最大

面積一定時，正方形周長最小

SquarePerimeter4s

AreaSide * sideDiagonals相等、垂直，相互平分，平分底角

4個直角，四條邊相等的平行四邊形

性質每條對角線長度的平方的一半=正方形的面積

Evolution

總結

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