Geometry Tutorial -grade 9 - Sat Geometry Standard

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Parallel Lines ➢ Two lines which would never intersect each other no matter how much you extend them will be parallel to each other.

➢ Distance is always the perpendicular distance between two lines or between point and a line. The distance between two parallel lines remains constant at every point. ➢ A line intersecting the given set of parallel lines is called the transversal. ➢

Region bounded between the lines is called interior region while the remaining area outside the lines is called exterior region. s t L a b c

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* When considering properties of pair of angles for parallel lines, please take care of the transversal you are considering and make sure angles you are considering are made with the transversal. ➢ pair of angles which are on same side of transversal, with one angle in the interior and other angle in the exterior form a pair of corresponding angles. Example: a,e; c,g; b,f ; d,h. For given parallel lines and a transversal, all pair

www.gurukul24x7.com of corresponding angles are equal. ➢ Pair of angles in the interior region and on opposite side of the transversal are called alternate interior angles. Example: c,f; d,e;. For given parallel lines and a transversal, all pair of alternate interior angles are equal. ➢ Similar to alternate interior angles, when pair of angles in the exterior region and on opposite side of the transversal,they are called alternate exterior angles. Example: a,h; b,g;. For given parallel lines and a transversal, all pair of alternate exterior angles are equal. ➢ Pair of angles on same side of the transversal and in the interior region are called co-interior angles. Co-interior angles between parallel lines are supplementary. This is also the Euclid's fifth postulate or parallel postulate, which states that if the sum of co-interior angles between a pair of lines is less than 180 degrees then the two lines when extended will meet at some point. ➢ Playfair's Axiom – Exactly one line can be drawn through a point not on a given line and parallel to that line. ➢ Whenever two lines intersect, the pair of opposite angles formed are called vertically opposite angles and they are equal. m n b a f l c e d figure 2 p In figure 2, a and e are formed between two same intersecting lines i.e. line m and line l and they are opposite to each other, hence vertically opposite angles so angle a = angle e. Angle d is between lines p and m while angle b is between lines m and n. Since they are not formed between same 2 lines, so they are not vertically opposite.

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Angles of a Polygon ➢ Number of diagonals in a polygon: (Number of vertices in the polygon-3) X (Number of vertices /2) Reasoning:Each vertex can make a diagonal with N-number of adjacent vertices= N-3 So total diagonals made by polygon with N vertices (or N sides) =N X (N-3). But in this way each diagonal is counted twice, from A to B an then B to A,so we divide N X (N-3) by 2 to get correct number of diagonals.

➢ Internal and External angles of polygon:Angle made between two adjacent sides of a polygon is called an internal angle. For N sided polygon there would be N internal angles. Angle made in the exterior region by extending a side of the polygon such that it lies between that line (side) and a side adjacent to it is called an exterior angle.

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Side AB is extended such that angle x lies between AD (or AB extended) and side BC (adjacent to AB), it is in the exterior region of the polygon and is the exterior angle. Angle ABC is the interior angle. The adjacent interior and exterior angles form a linear pair i.e. angle ABC + angle CBD=180 (they are supplementary).

➢ Convex and Concave Polygon: If any interior angle of a polygon is a reflex angle (greater than 180 and less than 360) then it is called a concave polygon. If all interior angles are less than 180 deg then it is called a convex polygon. In convex polygon all diagonals lie in the region enclosed by the polygon but in concave at-least one diagonal lies outside the region enclosed by the polygon. This document covers

www.gurukul24x7.com only convex polygons

Acute interior angle reflex interior angle

Equi means equal so ➢ equiangular polygon- all angles are equal. ➢ Equilateral polygon- all sides of the polygon are equal ➢ Regular polygon- it is both equilateral and equiangular

Number of sides in the polygon

Name of the polygon

3

Triangle (Tri for 3)

4

Quadrilateral (Quad means 4). Example:square, rectangle

5

Pentagon (Pent means 5)

6

Hexagon (Hex means 6)

7

Heptagon (hept means 7)

8

Octagon (oct means 8)

9

Nonagon

10

Decagon (dec means10)

www.gurukul24x7.com 11

Undecagon (1 + dec and uni is used for 1)

12

Dodecagon (2+dec)

*highlighted ones are less commonly used.

Angles and the triangle ➢ Sum of interior angles of a triangle is 180 deg. This is called angle sum property of the triangle or the ➢ Corollary ; A triangle can have only one obtuse or right angle. (corollary means the statement follows readily from a previous statement, in this case the angle sum property of triangle). ➢ Corollary: If two angles of a triangle are congruent to two angles of another triangle, then the third pair of angles are also congruent. ➢ In equilateral triangle, all angles are 60 degrees. ➢ Exterior angle theorem:Exterior angle to any angle of a triangle is formed by extending either arm of that angle outward such that the angle lies between the extended side of the triangle and the other arm. The sum of interior angle and it's exterior angle is 180 degrees. Exterior angle is equal to the sum of opposite interior angles or the non-adjacent interior angles, this is the exterior angle theorem.

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www.gurukul24x7.com angle ACD is between BD (BC extended to BD) and other arm of angle C i.e. AC. Similarly angle BCE is between AE (AC extended to AE) and other arm of angle C i.e. BC. So angle ACD and BCE are exterior angle (exterior to angle C). Angle ACB=180-(Angle A + angle B) (Angle sum property of triangle) -------Equation 1 Angle ACD=180 – angle ACB (Linear pair) -------Equation 2 Substituting value of angle ACB in equation 2 Angle ACD=180-(180-(angle A + angle B)) =180 – 180 + angle A +angle B Hence angle ACD= angle A + angle B ---this proves exterior angle theorem. ➢ Sum of interior angles of a polygon: A 'N' sided polygon can be divided into N-2 separate triangles because if we draw triangles from a single vertex of a polygon then it can form diagonals with N-3 vertices. Each diagonal except the last one i.e. (each of the N-4 diagonals) will generate one triangle and one non-triangular polygon (with number of sides >3) but the last diagonal will generate 2 triangles. So total number of triangles formed = (N-3-1) + 2=N-2 triangles. => Sum of interior angles of a polygon = 180 X number of triangles into which the 'N' sided polygon can be separated = 180 X (N-2) *=> means (this implies)

AN OCTAGON Number of triangles = 8 -2=6! Last diagonal gives 2 triangles.

➢ Sum of all exterior angles of the polygon = 360 degrees. Reasoning: interior angle + its exterior angle=180 => For all angles of N sided polygon: (sum of all interior angles) +(sum of all exterior angles) = 180 X N

www.gurukul24x7.com But sum of all interior angles = 180(N-2) => 180(N-2) + sum of all exterior angles = 180N =>sum of all exterior angles= 360 Please note that in the above 2 results for sum of all interior angles and sum of all exterior angles, all the interior angles( or all exterior angles) need not be equal. In case all the interior angles of the polygon are equal (of course all the exterior angles will also be equal), then for N sided polygon, each exterior angle =360 /N. Hence each interior angle will be equal to :[180 -(360/N)] OR (360/N). So both expressions mean the same!

1 180 N −2 =180N