Plane Geometry deals with geometrical entities of dimensions two and less. A Plane along with a point and line is an undefined term in Geometry, its meaning is accepted without a definition. Its meaning is similar to the coordinate plane in Algebra.
Plane Geometry is the subject that deals with axioms, theorems and problems related to Points, Straight lines and two dimensional shapes like Polygons, circles and other plane curves.Greek Mathematician Euclids's book on Geometry "Elements" form the basis for the study of Plane Geometry. Let us now look briefly how the problems are approached and solved in Plane Geometry.

## Plane Geometry Definition

Plane Geometry is the study of entities that a two dimensional space can hold. Systematic reasoning is applied using definitions, Postulates, and theorems to solve problems.

## Plane Geometry Theorems

Some of the Plane geometry theorems which are used as reasoning for supporting statements are listed as follows:

 Name of the Theorem Statement of the Theorem Midpoint Theorem If M is the midpoint of AB, then AM ≅ MB. Ruler Postulate The points on any line or line segment can be paired with real numbers so that, given any points A and B on a line, A corresponds to zero,and B corresponds to a positive real number. Segment Addition Postulate If B is between A and C, then AB + BC = AC. Conversely if AB + BC = AC, then B is between A and C. Segment Congruence Theorem Congruence of segments is reflexive, symmetric and transitive. Protractor Postulate Given $\overrightarrow{AB}$ and a number x between 0 and 180, there is exactly one ray with end point A, extending on either side of $\overrightarrow{AB}$ such that, the measure of the angle formed is xº. Angle addition Postulate If D is in the interior of ∠ABC, them m ∠ABD + m ∠DBC = m ∠ABCConversely if m ∠ABD + m ∠DBC = m ∠ABC, then D is in the interior of ∠ABC Supplement Theorem If two angles form a linear pair then they are supplementary angles. Complement Theorem If the non-common sides of two adjacent angles form a right angle, then the angles are complementary. Angle Congruence Theorem Congruence of angles is reflexive, symmetric and transitive. Vertical Angle Theorem If two angles are vertical, then they are congruent. Corresponding anglesPostulate If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. Alternate Interior anglesTheorem If two parallel lines are cut by a transversal, then each pair of alternate interior anglesis congruent. Consecutive interiorangles Theorem If two parallel lines are cut by a transversal, then each pair of consecutive interior anglesis supplementary. Alternate Exterior angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate exterior anglesis congruent. PerpendicularTransversal Theorem In a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other. Parallel Postulate If there is a line and a point not on the line, then there exists exactly one line throughthe point that is parallel to the given line. Perpendicular BisectorTheorem If a point is equidistant from the end points of a segment, then it lies on the perpendicularbisector of the segment. Conversely a point on the perpendicular bisector of a segment is equidistant from theend points of the segment. Angle Sum Theorem The sum of the measures of the angles of a triangle is 180 degrees. Third angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, then thethird angles of the triangles are congruent. Exterior angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures ofthe two remote interior angles. Side-Side-Side Congruence (SSS)Theorem If the sides of one triangle are congruent to the sides of a second triangle, then thetriangles are congruent. Side-Angle-Side Congruence (SAS)Theorem If two sides and included angle of one triangle are congruent to two sides and includedangle of another triangle, then the two triangles are congruent. Angle-Side-Angle Congruence (ASA)Theorem If two angles and included side of one triangle are congruent to two angles and includedside of another triangle, then the two triangles are congruent. Angle-Angle-Side Congruence (AAS)Theorem If two angles and a nonincluded side of one triangle are congruent to two angles andnonincluded side of another triangle, then the two triangles are congruent. Leg-Leg Congruence (LL)Theorem If the legs of one right triangle are congruent to the legs of another right triangle,then the triangles are congruent. Hypotenuse-Leg-Congruence (HL)Theorem If the Hypotenuse and the leg of one right angle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Hypotenuse-Angle Congruence (HA)Theorem If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent. Leg -Angle congruence(LA) Theorem If one leg and an acute angle of a right triangle are correspondingly congruent to one leg and an acute angle of another right triangle, then the triangles are congruent. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite to those two sides arecongruent. Exterior Angle InequalityTheorem If an angle is an exterior angle of a triangle, then its measure is greater than the either ofits remote interior angles. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the third side. SAS Inequality/HingesTheorem If two sides of a triangle are congruent to the two sides of another triangle, and theincluded angle in one triangle has a greater measure than the included angle in the other,then the third side of the first triangle is longer than the third side of the second triangle. SSS Inequality Theorem If two sides of a triangle are congruent to two sides of another triangle, and the third sidein one triangle is longer than the third side in the other, then the angle opposite to thethird side in the first triangle is greater than the corresponding angle in the second triangle. Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of another triangle then the triangles are similar. Side-Side-Side (SSS) Similarity If the measures of corresponding sides of two triangles are proportional, then the twotriangles are similar. Side-Angle-Side (SSS) Similarity. If the measures of two sides of a triangle are proportional to the measures of the two sides of another triangle, and the included angles are congruent, then the two triangles are congruent. Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these two sides into segments of proportional lengths. Converse of the TriangleProportionality Theorem If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. Mid-segment Theorem A Midsegment of a triangle is parallel to one side of a triangle and its length is one-half the length of that side. Angle Bisector Theorem An angle bisector in a triangle separates the opposite sides into segments in the sameratio as the other two sides. Pythagorean Theorem In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. Converse of the Pythagorean Theorem If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. Interior Angle Sum Theorem The sum of the measures of the interior angles of a convex polygon of n sides = 180(n -2) degrees. Exterior angle SumTheorem The sum of the measures of the exterior angles, one at each vertex of a convex polygon is360 degrees.

## Plane Geometry Proofs

The Proofs in Plane geometry are of two types, Direct and Indirect proofs.In Direct Proofs deductive reasoning and four main methods are applied.

Two Column Proof:
This type is also called the formal Proof. This type of proofs contain statements and the reasoning supporting them in two columns. The statements are validated by reasons step-wise till the statement required to be proved is reached.
Given: $\overrightarrow{QS}$ bisects $\angle{TQR}$ and $\overrightarrow{SQ}$ bisects $\angle{TSR}$.

Prove: $\angle{TQS}$ $\cong$ $\angle{RQS}$

 Statements Reasons 1. $\overrightarrow{QS}$ bisects $\angle{TQR}$. 1. given 2. $\angle{TQS}$ $\cong$ $\angle{RQS}$ 2. definition of angle bisector 3. $\overline{QS}$ $\cong$ $\overline{QS}$ 3. Reflexive property 4. $\overrightarrow{SQ}$ bisects $\angle{TSR}$. 4. given 5. $\angle{TSQ}$ $\cong$ $\angle{RSQ}$ 5. definition of angle bisector 6. $\Delta{TQS}$ $\cong$ $\Delta{RQS}$ 6. ASA postulate

Paragraph Proof:
It is an informal proof written in the form of a paragraph which explains why a conjecture is true.

Flow Proof:
Flow proofs use boxes and connecting arrows for proving statements. The statements are written in boxes and are connected by arrows using logical order. The Postulates, properties and theorems supporting the statements are
written beside the boxes.

Coordinate Proof:
This method makes use of coordinate geometry formulas to prove statements.

Indirect Proof:
This type of proofs are also called Proofs by contradiction. The statement to be proved is initially assumed as false. Logical reasoning is then used to contradict theorems or postulates or one of the assumptions. The contradiction thus arrived deems the assumption as false.

## Plane Geometry Formulas

There are a number of formulas in Plane Geometry to calculate measures like area, perimeter, length, height and angle which are dealt with in pages devoted for these concepts.

Formulas in coordinate geometry provide an algebraic approach to prove statements. The common formulas used in proofs are as follows:
Distance formula:
The distance between two points with coordinates (x1, y1) and (x2, y2)
d = $\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$

Section Formula:
The coordinate of a point which divides the segment joining (x1, y1) and (x2, y2) internally in the ratio m : n
$(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n})$

For external division the formula is
$(\frac{mx_{2}-nx_{1}}{m+n},\frac{my_{2}-ny_{1}}{m+n})$.

Mid-point Formula:
The midpoint of the segment joining (x1, y1) and (x2, y2) is
$(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

## Plane Geometry Problems

### Solved Examples

Question 1: Find the angle measure marked as x.

Solution:

Draw a line parallel to the given parallel lines through the vertex of the angle x, separating x into two angles as 1 and 2.

m ∠1 = 48º                                Corresponding angles Postulate
m ∠2 = 180 -124 = 56º              Consecutive interior angles Theorem
x = m ∠1 + m ∠2                        Angle Addition Property
= 48 + 56 = 104º.

Question 2: Write a Paragraph proof

 Given : ∠D ≅ ∠F            GE bisects ∠DEF.Prove: DG ≅ FG.

Solution:

As GE bisects ∠ DEF, ∠DEG ∠FEG. Side GE is common to both the triangles DEG and FEG. It is also given ∠D ≅ ∠F.
Hence by AAS criterion of congruence ΔDEG ≅ ΔFEG. Thus DGFG by CPCTC.

Question 3: The coordinates of the vertices of parallelogram ABCD is given as A(2, 5) , B(3, 3), C(-2, -3) and D(-3, -1).
Find the coordinates of the intersection of the diagonals of the parallelogram.

Solution:

It is a property of parallelograms, that diagonals bisect each other.
Hence to find the intersection of the diagonals, it is sufficient to find the midpoint of any one diagonal.

Midpoint of diagonal AC is $(\frac{2-2}{2},\frac{5-3}{2})$ = (0, 1)

It can also be verified that the midpoint of diagonal BD is also (0, 1).
Hence the diagonals intersect at (0, 1).

## Plane Geometry Practice

### Practice Problems

Question 1: BD is the bisector of ∠ABC. m ∠ABD = 2x + 25 and m ∠CBD = x + 45.

Find x and also m ∠ABC.                 (Answer: 20, 130º)
Question 2: Find the Perimeter and area of the given triangle rounded to one decimal value.

(Answer: 44.3 cm, 60 cm2 )

Question 3: ΔABC is isosceles with ABAC.  AD is the median on BC.
Write a two column proof to show that ΔABD ≅  ΔACD.
Question 4: Write a paragraph proof.

 Given: SR || PQ.           ∠P ≅ ∠RProve: ΔPQS ≅ ΔRSQ

Question 5: Triangle ABC with vertices A(-4, 2), B(-2, 0) and C(-4, -2) is transformed to ΔDEF with vertices
D(4, 2), E(2, 0) and F(4, -2). Graph and identify the transformation. Also verify that it is a
congruence transformation.