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Khan Academy
Geometry
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1
Euclid as the Father of Geometry
by Khan Academy
253,955 views
Learn more: http://www.khanacademy.org/video?v=WqzK3UAXaHs
Euclid and his Elements (and how much Abraham Lincoln liked them) -
2
Language and Notation of Basic Geometry
by Khan Academy
165,716 views
Learn more: http://www.khanacademy.org/video?v=il0EJrY64qE
Understanding basic ideas in geometry and how we represent them with symbols -
3
Lines, Line Segments, and Rays
by Khan Academy
276,324 views
Learn more: http://www.khanacademy.org/video?v=JcqCf762y9w
Difference between lines, line segments, and rays -
4
Language and Notation of the Circle
by Khan Academy
123,050 views
Learn more: http://www.khanacademy.org/video?v=U2W7HPyC0cM
Formal definition of a circle. Tangent and secant lines. Diameters and radii. major and minor arcs -
5
Angle Basics
by Khan Academy
216,919 views
Learn more: http://www.khanacademy.org/video?v=H-de6Tkxej8
Definition of an angle. How to denote an angle using points on the angle (including the vertex) -
6
Measuring Angles in Degrees
by Khan Academy
134,704 views
Learn more: http://www.khanacademy.org/video?v=92aLiyeQj0w
How to measure an angle in degrees -
8
Measuring Angles
by Khan Academy
118,168 views
Learn more: http://www.khanacademy.org/video?v=wJ37GJyViU8
Using the KA virtual protractor to measure angles -
9
Complementary and Supplementary Angles
by Khan Academy
145,000 views
Learn more: http://www.khanacademy.org/video?v=BTnAlNSgNsY
Basics of complementary, supplementary, adjacent and straight angles. Also touching on what it means to be perpendicular -
10
Angles at the intersection of two lines
by Khan Academy
78,884 views
Learn more: http://www.khanacademy.org/video?v=_7aUxFzTG5w
Figuring out angles at the intersection of two lines -
11
Proof-Vertical Angles are Equal
by Khan Academy
71,904 views
Learn more: http://www.khanacademy.org/video?v=wRBMmiNHQaE
Proving that vertical angles are equal -
12
Angles Formed by Parallel Lines and Transversals
by Khan Academy
113,639 views
Learn more: http://www.khanacademy.org/video?v=H-E5rlpCVu4
Parallel lines, transversal lines, corresponding angles -
13
Proof - Sum of Measures of Angles in a Triangle are 180
by Khan Academy
94,299 views
Learn more: http://www.khanacademy.org/video?v=OPG-9IFnJnI
Proof that the sum of the measures of the angles in a triangle are 180 -
14
Triangle Angle Example 1
by Khan Academy
92,954 views
Learn more: http://www.khanacademy.org/video?v=hmj3_zbz2eg
Figuring out angles in a triangle. A little about exterior angles being the sum of the remote interior angles -
15
Triangle Angle Example 2
by Khan Academy
50,482 views
Learn more: http://www.khanacademy.org/video?v=eTwnt4G5xE4
Another example finding angles in triangles -
16
Triangle Angle Example 3
by Khan Academy
43,274 views
Learn more: http://www.khanacademy.org/video?v=iqeGTtyzQ1I
Multiple ways to solve for the angles of multiple triangles -
17
Challenging Triangle Angle Problem
by Khan Academy
39,962 views
Learn more: http://www.khanacademy.org/video?v=95logvV8nXY
Interesting problem finding the sums of particular exterior angles of an irregular pentagon -
18
Proof - Corresponding Angle Equivalence Implies Parallel Lines
by Khan Academy
44,917 views
Learn more: http://www.khanacademy.org/video?v=9_3OxtdqmqE
Proof by contradiction that corresponding angle equivalence implies parallel lines -
19
Finding more angles
by Khan Academy
40,929 views
Learn more: http://www.khanacademy.org/video?v=0gzSreH8nUI
Example of angle hunting! -
20
Sum of Interior Angles of a Polygon
by Khan Academy
99,757 views
Learn more: http://www.khanacademy.org/video?v=qG3HnRccrQU
Showing a generalized way to find the sum of the interior angles of any polygon -
21
Congruent Triangles and SSS
by Khan Academy
117,812 views
Learn more: http://www.khanacademy.org/video?v=CJrVOf_3dN0
What it means for triangles to be congruent -
22
SSS to Show a Radius is Perpendicular to a Chord that it Bisects
by Khan Academy
38,059 views
Learn more: http://www.khanacademy.org/video?v=TgDk06Qayxw
More on the difference between a theorem and axiom. Proving a cool result using SSS -
23
Other Triangle Congruence Postulates
by Khan Academy
56,244 views
Learn more: http://www.khanacademy.org/video?v=8Ld8Csu4sEs
SSS, SAS, ASA and AAS postulates for congruent triangles. Showing AAA is only good for similarity and SSA is good for neither -
24
Two column proof showing segments are perpendicular
by Khan Academy
45,836 views
Learn more: http://www.khanacademy.org/video?v=fSu1LKnhM5Q
Using triangle congruency postulates to show that two intersecting segments are perpendicular -
25
Sum of the exterior angles of convex polygon
by Khan Academy
43,336 views
Learn more: http://www.khanacademy.org/video?v=W9B3VYdC5T8
More elegant way to find the sum of the exterior angles of a convex polygon -
26
Finding Congruent Triangles
by Khan Academy
47,133 views
Learn more: http://www.khanacademy.org/video?v=d5UCZ9hO8X4
Using the SSS, ASA, SAS, and AAS postulates to find congruent triangles -
27
More on why SSA is not a postulate
by Khan Academy
37,088 views
Learn more: http://www.khanacademy.org/video?v=f8svAm237xM
SSA special cases including RSH -
28
Congruent Triangle Proof Example
by Khan Academy
44,920 views
Learn more: http://www.khanacademy.org/video?v=RFesGHsuFZw
Proving that a point is the midpoint via triangle congruency -
29
Congruent Triangle Example 2
by Khan Academy
28,811 views
Learn more: http://www.khanacademy.org/video?v=Xc3oHzKXVh8
Showing that segments have the same length -
30
Congruent legs and base angles of Isosceles Triangles
by Khan Academy
50,890 views
Learn more: http://www.khanacademy.org/video?v=7UISwx2Mr4c
Showing that congruent legs imply equal base angles and vice versa -
31
Equilateral Triangle Sides and Angles Congruent
by Khan Academy
27,650 views
Learn more: http://www.khanacademy.org/video?v=CYNUwiUzlPk
Showing that all of the angles of an equilateral triangle are 60 degrees -
32
Equilateral and Isosceles Example Problems
by Khan Academy
27,388 views
Learn more: http://www.khanacademy.org/video?v=7FTNWE7RTfQ
Three example problems involving isosceles and equilateral triangles. 2 from Art of Problem Solving (by Richard Ruscyk) book -
33
Another Isosceles Example Problem
by Khan Academy
18,341 views
Learn more: http://www.khanacademy.org/video?v=ceDV0QBpcMA
Possible angles for an isosceles triangle -
34
Example involving an isosceles triangle and parallel lines
by Khan Academy
17,748 views
Learn more: http://www.khanacademy.org/video?v=CVKAro3HUxQ
Integrating what we know about isosceles triangles and parallel lines -
35
Figuring out all the angles for congruent triangles example
by Khan Academy
19,619 views
Learn more: http://www.khanacademy.org/video?v=aDCXPdzyS0s
Another example of using congruency to figure out a bunch of angles -
36
Perimeter and Area Basics
by Khan Academy
278,115 views
Learn more: http://www.khanacademy.org/video?v=LoaBd-sPzkU
Definitions of perimeter and area. -
37
Triangle Area Proofs
by Khan Academy
79,075 views
Learn more: http://www.khanacademy.org/video?v=YOYQys52sPs
Proving that the area of any triangle is 1/2 b x h -
38
Interesting Perimeter and Area Problems
by Khan Academy
52,016 views
Learn more: http://www.khanacademy.org/video?v=vWXMDIazHjA
Three example problems involving perimeter and area -
39
Koch Snowflake Fractal
by Khan Academy
81,447 views
Learn more: http://www.khanacademy.org/video?v=azBNsPa1WC4
A shape that has an infinite perimeter but finite area -
40
Area of an Equilateral Triangle
by Khan Academy
57,020 views
Learn more: http://www.khanacademy.org/video?v=UKQ65tiIQ6o
Finding the formula for the area of an equilateral triangle with side s -
41
Area of Koch Snowflake (part 1) - Advanced
by Khan Academy
22,178 views
Learn more: http://www.khanacademy.org/video?v=64bH_27Ehoc
Starting to figure out the area of a Koch Snowflake (which has an infinite perimeter) -
42
Area of Koch Snowflake (part 2) - Advanced
by Khan Academy
14,796 views
Learn more: http://www.khanacademy.org/video?v=vBlR2xNAGmo
Summing an infinite geometric series to finally find the finite area of a Koch Snowflake -
43
Challenging Perimeter Problem
by Khan Academy
28,948 views
Learn more: http://www.khanacademy.org/video?v=1uWZNW5PF-s
Perimeter of rectangle covered by 9 non-overlapping squares. From 200) American Invitational Math Exam -
44
Similar Triangle Basics
by Khan Academy
64,148 views
Learn more: http://www.khanacademy.org/video?v=9ThXDY9Y3oU
Introduction to what it means for triangles to be similar -
45
Similarity Postulates
by Khan Academy
38,633 views
Learn more: http://www.khanacademy.org/video?v=7bO0TmJ6Ba4
Thinking about what we need to know whether two triangles are similar -
46
Similar Triangle Example Problems
by Khan Academy
31,716 views
Learn more: http://www.khanacademy.org/video?v=Ly86lwq_2gc
Multiple examples looking for similarity of triangles -
47
Similarity Example Problems
by Khan Academy
31,960 views
Learn more: http://www.khanacademy.org/video?v=R-6CAr_zEEk
Two example problems involving similarity -
48
Challenging Similarity Problem
by Khan Academy
17,439 views
Learn more: http://www.khanacademy.org/video?v=7aGEvpHaNJ8
Interesting similarity problem where we don't have a lot of information to work with -
49
Similarity example where same side plays different roles
by Khan Academy
18,628 views
Learn more: http://www.khanacademy.org/video?v=Tal_fgREll0
The same side not corresponding to itself in two similar triangles -
50
Finding Area Using Similarity and Congruence
by Khan Academy
18,200 views
Learn more: http://www.khanacademy.org/video?v=EqNzr56h1Ic
Example of using similarity and congruence to find the area of a triangle -
51
Pythagorean Theorem Proof Using Similarity
by Khan Academy
32,053 views
Learn more: http://www.khanacademy.org/video?v=LrS5_l-gk94
Proof of the Pythagorean Theorem using similarity -
52
30-60-90 Triangle Side Ratios Proof
by Khan Academy
48,505 views
Learn more: http://www.khanacademy.org/video?v=SFL4stapeUs
Proving the ratios between the sides of a 30-60-90 triangle -
53
45-45-90 Triangle Side Ratios
by Khan Academy
42,021 views
Learn more: http://www.khanacademy.org/video?v=McINBOFCGH8
Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2) -
54
30-60-90 Triangle Example Problem
by Khan Academy
30,738 views
Learn more: http://www.khanacademy.org/video?v=dgHksfBFbjk
Using what we know about 30-60-0 triangles to solve what at first seems to be a challenging problem -
55
The Golden Ratio
by Khan Academy
226,033 views
An introduction to one of the most amazing ideas/numbers in mathematics
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56
Introduction to angles (old)
by Khan Academy
286,666 views
Learn more: http://www.khanacademy.org/video?v=2439OIVBgPg
What an angle is. Angles in a circle. Complementary and supplementary angles. -
57
Angles (part 2)
by Khan Academy
159,229 views
Learn more: http://www.khanacademy.org/video?v=zrqzG6xKa1A
More on complementary and supplementary angles. Introduction to opposite angles. -
58
Angles (part 3)
by Khan Academy
114,130 views
Learn more: http://www.khanacademy.org/video?v=NLg6hfoKKlE
Angles formed when a transversal intersects parallel lines. -
59
Angles formed between transversals and parallel lines
by Khan Academy
92,747 views
Learn more: http://www.khanacademy.org/video?v=gRKZaojKeP0
Angles of parallel lines -
60
Angles of parallel lines 2
by Khan Academy
70,871 views
Learn more: http://www.khanacademy.org/video?v=0eDwckZOffc
Angles of parallel lines examples -
61
The Angle Game
by Khan Academy
153,430 views
Learn more: http://www.khanacademy.org/video?v=kqU_ymV581c
Using what we know to solve for angles in the Angle Game. -
62
Similar triangles
by Khan Academy
158,035 views
Learn more: http://www.khanacademy.org/video?v=BI-rtfZVXy0
Introduction to similar triangles -
63
Similar triangles (part 2)
by Khan Academy
76,343 views
Learn more: http://www.khanacademy.org/video?v=qO2cTx6DwCA
More on similar triangles -
64
Angle Game (part 2)
by Khan Academy
71,067 views
Learn more: http://www.khanacademy.org/video?v=ko-cYG3d6ec
More examples of the Angle Game. -
65
Acute Right and Obtuse Angles
by Khan Academy
112,115 views
Learn more: http://www.khanacademy.org/video?v=ALhv3Rlydig
The difference between acute, obtuse and right angles -
66
Area and Perimeter
by Khan Academy
333,625 views
Learn more: http://www.khanacademy.org/video?v=kqqmJiJez6o
Area of rectangles and triangles. Perimeter of rectangles. -
67
Circles: Radius, Diameter and Circumference
by Khan Academy
424,111 views
Learn more: http://www.khanacademy.org/video?v=jyLRpr2P0MQ
Understanding the relationship between the radius diameter and circumference of a circle. -
68
Area of a circle
by Khan Academy
117,118 views
Area of a circle and how it relates to radius and diameter
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69
The Pythagorean Theorem
by Khan Academy
261,465 views
Learn more: http://www.khanacademy.org/video?v=AA6RfgP-AHU
Introduction to the Pythagorean Theorem -
70
Pythagorean Theorem II
by Khan Academy
107,757 views
Learn more: http://www.khanacademy.org/video?v=nMhJLn5ives
More Pythagorean Theorem examples. Introduction to 45-45-90 triangles. -
71
45-45-90 Triangles
by Khan Academy
129,326 views
Learn more: http://www.khanacademy.org/video?v=tSHitjFIjd8
Introduction to 45-45-90 Triangles -
72
Intro to 30-60-90 Triangles
by Khan Academy
134,889 views
Learn more: http://www.khanacademy.org/video?v=Qwet4cIpnCM
A few more 45-45-90 examples and an introduction to 30-60-90 triangles. -
73
30-60-90 Triangles II
by Khan Academy
79,578 views
Learn more: http://www.khanacademy.org/video?v=3mLUJSoh6i0
More examples using 30-60-90 triangles. -
74
Solid Geometry Volume
by Khan Academy
160,853 views
Learn more: http://www.khanacademy.org/video?v=ZqzAOZ9pP9Q
Volume of triangular prisms and cubes -
75
Cylinder Volume and Surface Area
by Khan Academy
219,494 views
Learn more: http://www.khanacademy.org/video?v=gL3HxBQyeg0
Finding the volume and surface area of a cylinder -
76
Heron's Formula
by Khan Academy
100,746 views
Learn more: http://www.khanacademy.org/video?v=-YI6UC4qVEY
Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides -
77
Part 1 of Proof of Heron's Formula
by Khan Academy
52,388 views
Learn more: http://www.khanacademy.org/video?v=AreOMvPCxv8
Part 1 of the proof of Heron's Formula -
78
Part 2 of the Proof of Heron's Formula
by Khan Academy
39,758 views
Learn more: http://www.khanacademy.org/video?v=nZu7IZLhJRI
Showing that the expression in part 1 is identical to Heron's Formula -
79
Inscribed and Central Angles
by Khan Academy
113,578 views
Learn more: http://www.khanacademy.org/video?v=MyzGVbCHh5M
Showing that an inscribed angle is half of a central angle that subtends the same arc -
80
Area of Inscribed Equilateral Triangle (some basic trig used)
by Khan Academy
65,042 views
Learn more: http://www.khanacademy.org/video?v=UmiZK6Hgm6c
Problem that requires us to figure out the area of an equilateral triangle inscribed in a circle (A little trigonometry used) -
81
Right Triangles Inscribed in Circles (Proof)
by Khan Academy
77,527 views
Learn more: http://www.khanacademy.org/video?v=b0U1NxbRU4w
Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle -
82
Area of Diagonal Generated Triangles of Rectangle are Equal
by Khan Academy
33,783 views
Learn more: http://www.khanacademy.org/video?v=YTRimTJ5nX4
Area of Diagonal Generated Triangles of Rectangle are Equal -
83
Triangle Medians and Centroids
by Khan Academy
41,736 views
Learn more: http://www.khanacademy.org/video?v=GiGLhXFBtRg
Seeing that the centroid is 2/3 of the way along every median -
84
Triangle Medians and Centroids (2D Proof)
by Khan Academy
20,376 views
Learn more: http://www.khanacademy.org/video?v=Vc09LURoMQ8
Showing that the centroid is 2/3 of the way along a median -
85
Rhombus Diagonals
by Khan Academy
32,999 views
Learn more: http://www.khanacademy.org/video?v=_QTFeOvPcbY
Proof that the diagonals of a rhombus are perpendicular bisectors of each other -
86
2003 AIME II Problem 7
by Khan Academy
20,023 views
Learn more: http://www.khanacademy.org/video?v=jpKjXtywTlQ
Area of rhombus from circumradii of triangles -
87
Perpendicular Radius Bisects Chord
by Khan Academy
46,534 views
Learn more: http://www.khanacademy.org/video?v=q7eF5Ci944U
Simple proof using RSH postulate to show that a radius perpendicular to a chord bisects it -
88
Circumcenter of a Triangle
by Khan Academy
45,420 views
Learn more: http://www.khanacademy.org/video?v=KXZ6w91DioU
Multiple proofs showing that a point is on a perpendicular bisector of a segment if and only if it is equidistant from the endpoints. Using this to establish the circumcenter, circumradius, and circumcircle for a triangle -
89
Circumcenter of a Right Triangle
by Khan Academy
18,339 views
Learn more: http://www.khanacademy.org/video?v=VejCw2NlE60
Showing that the midpoint of the hypotenuse is the circumcenter -
90
Three Points Defining a Circle
by Khan Academy
32,680 views
Learn more: http://www.khanacademy.org/video?v=4_xhiP6g2ow
Showing that three points uniquely define a circle and that the center of a circle is the circumcenter for any triangle that the circle is circumscribed about -
91
Point Line Distance and Angle Bisectors
by Khan Academy
29,397 views
Learn more: http://www.khanacademy.org/video?v=yj4oS-27Q3k
Thinking about the distance between a point and a line. Proof that a point on an angle bisector is equidistant to the sides of the angle and a point equidistant to the sides is on an angle bisector -
92
Incenter and incircles of a triangle
by Khan Academy
25,341 views
Learn more: http://www.khanacademy.org/video?v=21vbBiCVijE
Using angle bisectors to find the incenter and incircle of a triangle -
93
Angle Bisector Theorem Proof
by Khan Academy
32,016 views
Learn more: http://www.khanacademy.org/video?v=TpIBLnRAslI
What the angle bisector theorem is and its proof -
94
Inradius Perimeter and Area
by Khan Academy
16,326 views
Learn more: http://www.khanacademy.org/video?v=5NjgcPri07E
Showing that area is equal to inradius times semiperimeter -
95
Angle Bisector Theorem Examples
by Khan Academy
26,051 views
Learn more: http://www.khanacademy.org/video?v=PlY3e_-9JUA
Using the angle bisector theorem to solve for sides of a triangle -
96
Medians divide into smaller triangles of equal area
by Khan Academy
13,168 views
Learn more: http://www.khanacademy.org/video?v=Xt4sT0fV9Pw
Showing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well -
97
Exploring Medial Triangles
by Khan Academy
9,330 views
Learn more: http://www.khanacademy.org/video?v=Rsi5mzEaCPw
What a medial triangle is and its properties -
98
Proving that the Centroid is 2-3rds along the Median
by Khan Academy
12,136 views
Learn more: http://www.khanacademy.org/video?v=i0VS3eEGjiQ
Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median) -
99
Median Centroid Right Triangle Example
by Khan Academy
12,216 views
Learn more: http://www.khanacademy.org/video?v=k45QTFCHSVs
Example involving properties of medians -
100
Proof - Triangle Altitudes are Concurrent (Orthocenter)
by Khan Academy
20,561 views
Learn more: http://www.khanacademy.org/video?v=aGwT2-RERXY
Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter). -
101
Review of Triangle Properties
by Khan Academy
18,049 views
Learn more: http://www.khanacademy.org/video?v=KUhdMbx5ges
Comparing perpendicular bisectors to angle bisectors to medians to altitudes -
102
Euler Line
by Khan Academy
19,932 views
Learn more: http://www.khanacademy.org/video?v=tUqyJgmGY7k
The magic and mystery of the Euler Line -
103
Euler's Line Proof
by Khan Academy
14,424 views
Learn more: http://www.khanacademy.org/video?v=t_EgAi574sM
Proving the somewhat mystical result that the circumcenter, centroid and orthocenter -
105
Common Orthocenter and Centroid
by Khan Academy
12,728 views
Learn more: http://www.khanacademy.org/video?v=pDZIvyeqX1o
Showing that a triangle with the same point as the orthocenter and centroid is equilateral -
106
Quadrilateral Overview
by Khan Academy
99,306 views
Learn more: http://www.khanacademy.org/video?v=Ka82QC4QvGA
Basics of quadrilaterals including concave, convex ones. Parallelograms, rectangles, rhombi and squares -
107
Proof - Opposite Sides of Parallelogram Congruent
by Khan Academy
33,154 views
Learn more: http://www.khanacademy.org/video?v=LhrGS4-Dd9I
Proving that a figure is a parallelogram if and only if opposite sides are congruent -
108
Proof - Diagonals of a Parallelogram Bisect Each Other
by Khan Academy
35,543 views
Learn more: http://www.khanacademy.org/video?v=TErJ-Yr67BI
Proving that a quadrilateral is a parallelogram if and only if its diagonals bisect each other -
109
Proof - Opposite Angles of Parallelogram Congruent
by Khan Academy
28,286 views
Learn more: http://www.khanacademy.org/video?v=oIV1zM8qlpk
Showing that opposite angles of a parallelogram are congruent -
110
Proof - Rhombus Diagonals are Perpendicular Bisectors
by Khan Academy
27,231 views
Learn more: http://www.khanacademy.org/video?v=GDcVdBAnBdU
Proving that the diagonals of a rhombus are perpendicular -
111
Proof - Rhombus Area Half Product of Diagonal Length
by Khan Academy
16,634 views
Learn more: http://www.khanacademy.org/video?v=3FManXv4mZM
Showing that we can find the area of a rhombus by taking half the product of the lengths of the diagonals -
112
Area of a Parallelogram
by Khan Academy
84,884 views
Learn more: http://www.khanacademy.org/video?v=tFhBAeZVTMw
Showing that the area of a parallelogram is base times height -
113
Problem involving angle derived from square and circle
by Khan Academy
9,813 views
Learn more: http://www.khanacademy.org/video?v=3n0LvI99-KM
Challenging problem to find an angle -
114
Area of a Regular Hexagon
by Khan Academy
45,666 views
Learn more: http://www.khanacademy.org/video?v=LrxZMdQ6tiw
Using what we know about triangles to find the area of a regular hexagon
