Theorem in Plane Geometry A list of theorems with some common terminologies

Theorems in Plane Geometry

1. Points and Straight Lines 6. Pythagoras’ Theorem 2. Parallel Lines 7. Mid-point theorem, Intercept theorem 3. Triangles and Polygons and Equal ratios theorem 4. Congruence and Similarity 8. Special lines in triangles 5. Quadrilaterals 9. Circles and Tangents

1. Points and Straight Lines

A C

O

D B A O B A O B

If AOB and COD are st. lines, If AOB is a st. line, then If , then then AOB is a st. line ( )*

* No one use this nowadays.

2. Parallel Lines

i. Angles related to parallel lines

A B A B A B

C D C D C D

If AB//CD, then If AB//CD, then If AB // CD, then

ii. Test for parallel lines

A B A B A B

C D C D C D

If , then AB // CD If , then AB // CD If , then AB // CD

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3. Triangles and Polygons

i. Sum of angles in a triangle/ polygon

A A

B C B C

Interior angle sum = Exterior angle sum =

ii. Isosceles Triangles

A A A

Definition:

An isosceles triangle is a triangle with 2 B C B C B C sides equal. D A If If If , then the then then followings are equivalent: (i) (ii) bisects B C (iii)

iii. Equilateral Triangles Some terminologies:

1. Supplementary angles A Definition: 2 angles and are supplement to each other if . In this case and An equilateral triangle are called supplementary angles. 2. Complementary angles is a triangle with all 3 B C 2 angles and are complement to each sides equal. other if . In this case and are called complementary angles. A If is , then

B C

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4. Congruence and Similarity

i. Test for congruent triangles

ii. Tests for similar triangles

B B B

Y Y C A C A C A

Y

Z X Z X

Z X

iii. When we are given a pair of congruent / similar triangles, what we can know are: Congruent Similar

B B B B

C A C A C A C A

Y Y Y Y

Z X Z X Z X Z X

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5. Quadrilaterals

Definition: Parallelograms are quadrilaterals with two pairs of opposite sides parallel.

i. Properties of parallelograms

A D A D A D

O O O

B C B C B C

If is a , If is a , If is a , then and then then AC and BD bisect each other and ( and )

ii. Tests for a parallelogram

A D A D A D A D

O

B C B C B C B C

If and , If , If and , If and , then is a . then is a . then is a . then is a .

iii. Other types of parallelogram 1. Rhombus 2. Rectangle 3. Square Rhombus is a parallelogram Rectangle is a parallelogram Square is a rectangle with 2 with two adjacent sides with one angle equal to a right adjacent sides equal. equal. angle.

Properties: Properties: Properties: (i) its 4 sides are equal (i) its 4 angles are equal. (i) it has all properties of (ii) the 2 diagonals are (4 right angles) rhombus and rectangles. perpendicular to each (ii) its diagonals are equal (ii) its diagonals make angles other. of with the sides

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6. Pythagoras’ Theorem

If If then then

7. Mid-point Theorem, Intercept theorem and Equal ratios theorem

A A B A

H K C D H K

E F B C B C

If and , If If then ; and , and ,

1 then then and 2

K H A A A A

H K H K

B C B C B C B C H K

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8. Special lines in triangles Circumcenter A  The intersection of 3 perpendicular bisectors P  AG=BG=CG  Position of G: G B Acute inside the triangle. C Right-angled on the hypotenuse . Q Obtuse outside the triangle.

Perpendicular bisector Circumcenter Incenter A A  Intersection of 3 angle bisectors.  Position of G: G Always inside the triangle.

B P C *Coordinate of G

Angle bisector In-center Centroid A A  Intersection of 3 medians. AG BG CG 2     F GD GE GF 1 G E  Area of the six small triangles are the B same. M D C *Coordinate of G

Median Centroid Orthocenter A A  Intersection of 3 altitudes.  Position of G: Acute inside the triangle. G Right-angled on the right angle. B Obtuse outside the triangle. H C

Altitude Orthocenter

*It is assumed that the coordinate of the vertices , and are , and respectively, and that , and are the sides opposing to the

vertices , and respectively.

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9. Circles and Tangents a. Chords of a Circle

Some terminologies: 1. Circle: A circle is a collection of point such that the distance between each of the points O O and a particular point O is a constant. 2. Center, radius and Diameter: The point ‘O’ is called the center. A M B A B M Radius is any line segment connecting the center O and a point at the If , If , circumference. then then Diameter is any line segment with ends at the circumference and passing through the center O. 3. Chord: A chord is a line segment AB, where A and B are on the circumference. D D 4. Arc (i) Major arc N N Major arc is an arc that subtends an angle greater than at the O O center. C C (ii) Minor arc Minor arc is an arc that subtends an A B A B M M angle less than at the center. If , If , 5. Segment A segment is a region inside circle such then then that it is bounded by a chord AB and a corresponding arc AB. The arc can be a minor arc AB or a major arc AB.

6. Sector A segment is a region inside circle such

Major arc that it is bounded by an arc CD and two radius OC and OD. The arc can be a minor Center arc CD or a major arc CD. 7. Tangent O B Tangent is any line such that it cuts the circle at only one point. A 8. Secant Diameter Radius Minor arc Secant is any line such that it cuts the circle at two distinct points. Circle Arc Segment B B A A O O chord AB

C D Tangent Secant Sector OCD Chord Sector and Segment Tangent and secant

8 b. Angles in a Circle

O O O O

In above cases,

C

A B O

A B

If AB is a diameter, then If AB is a chord, then

c. Angles, Arcs and Chords

D

A A A

D D C

C B B C B   AB a AB a    b  b CD CD

9 d. Cyclic Quadrilateral i. Properties of a Cyclic Quadrilateral

A A

D D

B C B C

If ABCD is a cyclic quad., If ABCD is a cyclic quad., then then

ii. Tests for Concyclic Points

A A A

D D D

B C B C B C

If , If , If , then A,B,C,D concyclic then A,B,C,D concyclic then A,B,C,D concyclic

Some terminologies: 1. Collinear 3 points are collinear if there is a line passing through all 3 points. 2. Concurrent 3 lines are concurrent if they all pass through a common point. 3. Concyclic 4 points are concyclic if there is a circle passing through all 4 points.

collinear concurrent concyclic

10 e. Tangent

P

O O O T

P T Q P T Q Q

If is the tangent at , If , If and are tangents, then: then then is the tangent at (i) (ii) (iii)

C C

B B

P A Q P A

If is the tangent at A, then If , then is the tangent at .

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