1、Quadrilaterals,A quadrilateral 四邊形 is a polygon of four sides.,Bases 底 altitude 高,opposite angles,Parallelogram平行四邊形,A rhombus 菱形 is an equilateral parallelogram A square 正方形 is an equilateral and equiangular parallelogram. Also, it is an equilateral rectangle, or an equiangular rhombus. The diagona

2、ls cut the corner angles in half , and they cross at right angles. A kite 箏形 has two pairs of adjacent sides that are equal. A kite has one pair of equal opposite angles. The diagonals of a kite cross at right angles.,Trapezoid梯形,An isosceles trapezoid 等腰梯形 is a trapezoid having two equal nonparalle

3、l sides (legs 腰). The base angles of an isosceles trapezoid equal. The diagonals of an isosceles trapezoid equal. Right-angled trapezoid直角梯形,Proving quadrilaterals to be parallelograms Geometry Idea 8,Properties of parallelogram Both pairs of opposite sides are parallel One pair of opposite sides bo

4、th congruent and parallel Both pairs of opposite sides are congruent Both pairs of opposite angles congruent Diagonals bisect each other How to prove a quadrilateral is a parallelograms?,A counter example,Another counter example,Demonstrating the need to consider all information given Geometry idea

5、9,Three equal circles A,B, and C meet at a common point P and intersect each other (pair wise) at points K, L, and M. To prove,Parallelepiped 平行六面體,The three points of intersection of the pairs of circles lie on a circle equal in size to the other three circles,Where is the center of the fourth circ

6、le,Midline of a triangle geometry idea 10,quadrilateral,midpoint,side,diagonals,Inside parallelogram,vertex,Midline KL is parallel to FH and has half the length of FH,Invariants in geometry,What are invariant when we distort this figure into several differently shaped quadrilaterals ABCD? Why? Under

7、 what conditions that the inner quadrilateral is a a rhombus, a rectangle, a square?,Inner quadrilateral,opposite sides,Length of the median of a trapezoid,Median: the segment that joins the midpoints of the nonparallel sides of the trapezoid Median JL is parallel to the bases The length of the medi

8、an of a trapezoid is the average of the lengths of the bases,Geometry Idea 11,Pythagorean Theorem,The Converse of the Pythagorean Theorem For any three positive numbers a, b, and c such that a2+b2=c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between th

9、e sides of lengths a and b.,In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). The square of the hypotenuse of a right

10、triangle is equal to the sum of the squares on the other two sides.,Geometry Idea 12,Proof using similar triangles,We draw the altitude from point C, and call D its intersection with the side AB. The new triangle ACD is similar to our triangle ABC, because they both have a right angle (by definition

11、 of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well.,Proof by rearrangement趙爽勾股圓方圖,劉徽對(duì)勾股定理的證明,the sum of equals are equal,Begin the class by asking the class the question: What do Pythagoras, Euclid, and President James A. Garfiel

12、d have in common? Whats the answer?,A proof given by the Pythagoreans,A proof given by the American President,b,a,a,b,c,c,We can prove it is a trapezoid,A proof given by Euclid,share the same base and altitude triangle must be congruent to triangle ,Congruent Triangles,Triangles that have the same s

13、ize and shape. In congruent triangles, all 3 pairs of corresponding angles have the same measure, and all 3 pairs of corresponding sides have the same measure.,Side-Angle-Side (SAS),If two sides and the included angle of a triangle are congruent, respectively, to two sides and the included angle of

14、another triangle, then the two triangles are congruent.,Side-Side-Side (SSS),If three sides of a triangle are congruent, respectively, to the three sides of another triangle, then the two triangles are congruent.,Angle-Angle-Side (AAS),If two angles and a non-included side of one triangle are congru

15、ent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.,Angle-Side-Angle (ASA),If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.,

16、Comparing areas of similar polygons,相似比 the ratio of similitude (the ratio of any two corresponding linear parts) 相似比的平方 the square of the ratio of similitude,Geometry Idea 24,Area1=Area2+Area3,Area1:Area2:Area3=c2:a2:b2 Area1=kc2, Area2=ka2, Area3=kb2 c2=a2+b2 Area1=Area2+Area3,The fundamental term

17、s,Point: a point has position only. It has no size, no length, width, or thickness. Line: a line has length but has no width or thickness. It may be straight, curved, or a combination of these. It could be thought that it is generated by a moving point. Straight line, curved line, broken line One an

18、d only one straight line can be drawn through any two points. Two points determine a straight line. A straight line is unlimited in extent and may be extended in either direction indefinitely.,Surface: a surface has length and width but no thickness. A plane surface or a plane is a surface such that

19、 a straight line connecting any two of its points lies entirely in it. Three noncollinear points determine a plane. Planar figures平面圖形 curved surface Solid: a solid is an enclosed portion of space bounded by plane or curved surfaces. A solid has length, width, and thickness. Spatial figures空間圖形,Line

20、 segment: it joins two points and is a part of a line. An interior point of a segment is any point of the segment between its endpoints. The length of a line segment is the distance between its endpoints. Equal segments are segments having the same measure. They may be shown by crossing them with the same number of strokes(一劃） a single stroke equal sides = sides having equal measures, equal angles If a line divides a segment into two equa