adjacent angle_GRE/GMAT 数学之平面几何

Lines & angles

Congruent全等的

Length

Intersection

Bisect平分

在GMAT里只是平分角

Vertices顶点

Obtuse angle钝角

Perpendicular

Complementary angles互补为90度

Height/altitude高Polygon

三维图形

Width

Acute angles锐角

Opposite angles对顶角Vertical angles

Straight angle平角180度

Supplementary angles互补为180度

不需要相邻

Polygons(N-2)*180角度

所有角所有边相等Regular polygon

Equilateral polygon

Perimeter周长

闭合长度

The number of diagonals in a polygon对角线的个数：n(n-3)/2

n是边长

对角线并不一定相等比如正六边形

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(底边)

Bisector in a triangle三角形有三条角平分线

Median in a triangle三角形有三条边平分线

Height/altitude 高三角形有三条高

Perimeter of a triangle所有边的和

Impossible triangles and the third side rule三角形两边之和大于第三边最小的两边的平方和小于第三条边的平方和，必为钝角三角形

三角形两边之差小于第三边最小的两边的平方和大于第三条边的平方和，必为锐角三角形b-c

The angles in a triangle sum up to 180

Obtuse

Equilateral三个角60度

三条边相等

角平分线也是中线

面积底*高/2

如果三角形三边长为a,b,c,

The largest side is opposite the largest angle大边对大角

小边对小角

同角度对的边也相等

若两个三角形相似，则这两个三角形的面积比等于相似比的平方

三角形的一个外角等于其不相邻的两个内角之和

IsoscelesBase angles两条腰

Vertex angle顶角角平分线是中线

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圆周长

Concentric circles同心圆

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圆周长

周长是直径的3倍多点

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

Its rays 射线 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外接圆A circumcircle passes through all the vertices of a polygon

所有的三角形和正多边形都有外接圆

Central anglesOriginates from the center and is formed by two radii

Area

Sector 扇形像pizza块

Quadrant四分之一圆周

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

解释圆心角：顶点在圆心上且两条边是圆的弦的角成为圆心角

圆周角：顶点位于圆周上且两条边是圆的弦的角为圆周角

圆中同一段弧所对的圆心角是圆周角的两倍

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

四边形

Parallelogram 平行四边形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

性质对边互相平行且相等

对角相等，四个内角和=360

若平行四边形的一个内角等于90，那么他的所有内角都=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

性质菱形的面积等于两条对角线乘积的一半

对角线平分四个内角

RectanglePerimeter2w+2l

Arealength*widthDiagonalsEqual and bisect each other

4个直角的平行四边形

性质周长一定时，正方形面积最大

面积一定时，正方形周长最小

SquarePerimeter4s

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

4个直角，四条边相等的平行四边形

性质每条对角线长度的平方的一半=正方形的面积

Evolution

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