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GEOMETRY: RULES OF SHAPES

Rules of different shapes.
by

Srikripa Krishna Prasad

on 20 December 2011

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Transcript of GEOMETRY: RULES OF SHAPES

Rules of Shapes The circle
-No sides, no angles
-Infinite number of lines of symmetry
-Line from the center to any point on the circumference of the circle is called the radius
-A line that completely crosses the circle and passes through the center is called the diameter; the diameter is twice the radius
-A line that joins any two points on the circumference of the circle is called a chord; the diameter is the longest chord of the circle
-Any section of the circumference that lies between the chord is called the arc
-a line that touches a circle at just one point is a tangent
-the formula to find the area of a circle is pi times the radius squared (pi is the ratio of the circumference of a circle to its diameter)
-the formula to find the circumference of a circle is 2 times pi times the radius The square
-4 equal sides
-4 equal angles
-all angles right (90 degrees)
-all angles add up to 360 degrees
-all angles equidistant
-2 pairs of parallel lines (two lines on the same plane
that extend forever and never meet)
-all sides perpendicular (lines that intersect to form right angles)
-2 diagonals
-4 lines of symmetry
-formula for area of a square is side times side
-formula for perimeter of a square is 4a, where a is the side
The Rectangle
-4 sides
-opposite sides equal
-4 right angles
-all sides perpendicular to each other
-2 lines of symmetry
-2 diagonals
-2 pairs of parallel sides
-formula for area of a rectangle is
length times width
-formula for perimeter is 2l + 2w
-therefore, the square is a rectangle
The Triangle
-three sides and angles
-in a triangle, the sum of any two sides have to be greater
than the length of the third side
-scalene, isosceles, equilateral
-scalene: all sides and angles unequal
-isosceles: two sides and angles equal
-equilateral: all sides and angles equal
-an equilateral triangle is always an isosceles triangle
but an isosceles is never an equilateral
-right angled, obtuse angle, acute angle
-acute angle triangle: three acute angles present in triangle
-obtuse angle: at least one obtuse angle present in triangle
-right angle: one right angle present in triangle
and two acute angles
-isosceles triangles have one line of symmetry
-equilaterals have three lines of symmetry
-scalene triangles have no lines of symmetry
-all angles in a triangle have to equal 180 degrees
-in a triangle, when a side of the triangle is produced,
the exterior angle so formed is equal to the sum of the
two opposite interior angles
a b c The formula to find the hypotenuse of a triangle (the side opposite the right angle) is a²+b²=c² The parallelogram
-four sides
-opposite sides and angles equal
-two pairs of parallel sides
-two obtuse angles, two acute angles
-adjacent angles are supplementary
-all angles have to equal 360 degrees
-formula for area of a parallelogram is base times height
-formula for perimeter is 2l plus 2w
-two diagonals that bisect each other
-two lines of symmetry The Trapezoid
-4 sides
-Two obtuse angles, two acute angles
-all angles equal to 360 degrees
-one pair of parallel sides
-perimeter = Side A plus Side B plus Side C plus Side D
-area of a trapezoid is 1/2 times the
sum of the two bases times the height (1/2 x [b1+b2] x h)
-two diagonals
-one line of symmetry Quadrilaterals a b c d The Kite
-it is a quadrilateral (4 sides)
-two lines of symmetry
-two diagonals that bisect at
90 degree angles (diagonals are perpendicular)
-adjacent sides are equal
-one set of congruent angles
-all angles have to equal 360 degrees
-perimeter of a kite is sides a+b+c+d
-area of a kite is length of
diagonal a x length of diagonal b divided by 2
The Rhombus
-4 equal sides
-two pairs of parallel sides
-2 lines of symmetry
-2 diagonals
-opposite angles equal
-all angles equal 360 degrees
-perimeter is side a+b+c+d
-area of a rhombus is base x height
-therefore, any square is a rhombus Transversals and Angles in Shapes -there are transversals(line intersecting two or more given lines in a plane at different points) in any shape that has one or more pairs of parallel sides
-for example, taking the parallelogram below,
if you extend one pair of parallel sides, you would
get two transversals in between them Here are some more examples: -Ex. 130 degrees x y x + y = 130 degrees -if you draw a diagonal in any quadrilateral, you create a transversal -Here are some examples
The Rosetta Stone of Shapes -one of the most useful tools in history was the Rosetta Stone
-it was a stone tablet that had three scripts written on it in different languages
-it allowed for the decoding for many languages
-similarly, when talking about shapes, a Rosetta stone could be the Equilateral Triangle
-since all sides and angles in an equilateral triangle are equal, they can help decode the angles of other shapes
Example: -this trapezoid is made of three equilateral triangles
-therefore, the trapezoid's angles can easily be found, using the triangles' angle measures -in fact, any shape or angle can be a Rosetta Stone, as long as it aids you in finding the missing angles in other shapes or figures Ex. 50 degrees "Rosetta Stone." So the opposite angle would be 50 degrees, while the other two angles would be 130 degrees each
Angle Properties of Transversals and Parallel Lines
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 When two parallel lines are crossed by another line, called the transversal, the angles that are on the same position as another and are matching are corresponding angles. Ex. <1 and <9
<2 and <10
<3 and <11
<4 and <12 Vertical angles are the angles that are opposite each other when two angles cross. They are called vertical because they share the same vertex.
Ex. <1 and <4
<2 and <3
<5 and <8
<6 and <7 Transversal: a line that intersects two different lines at different points Parallel lines: lines that extend forever in two directions and never meet Supplementary angles: angles that add up to 180 degrees
Ex. <1 and <2
<2 and <4
<9 and <14 Complementary angles: angles that add up to 90 degrees Alternate interior angles: Two angles that lie between the parallel lines and are on oppposite sides of the transversal
Ex. <3 and <10
<4 and <9 Alternate exterior angles: angles that lie outside of the parallel lines on opposite sides of the trasnversal.
Ex. <1 and <12
<2 and <11 Consecutive interior angles: Two angles between the parallel lines on the same side of the transversal
Ex. <3 and <9
<4 and <10 Consecutive exterior angles: two angles that lie outside of the parallel lines on the same sides of the trasnversal
Ex. <1 and <11
<2 and <12 Other Polygons:

The Hexagon:
-6 equal sides
-6 equal angles
-angles have to add up to 720 degrees
-each angle is 120 degrees
-9 diagonals
-perimeter is the sum of the
lengths of all six sides
-area: divide into four
triangles, find area of each
triangle, add them up
The Pentagon
-5 equal sides and angles
-all angles equal 540 degrees
-each angle is 108 degrees
-5 diagonals
-no parallel sides
-perimeter is the sum of the
lengths of all five sides
area: divide pentagon into
triangles, find area of triangles,
add them up The Octagon:
-8 equal sides and angles
-135 degree angles
-angles have to add up to 1080 degrees
-20 diagonals
-perimeter is the sum of the length of all 8 sides
-area: divide into 6 triangles, find the area of each, add them up
The Heptagon:
-7 equal sides and angles
-each angle equals,128 degrees (approx)
-sum of interior angles is 900 degrees
-14 diagonals
-perimter is the sum of the length of
all the sides
-area: divide into 5 triangles,
find the area of each, add them up -9 equal sides and angles
-interior angles are 140 degrees
-sum of interior angles is 1260 degrees
-27 diagonals
-perimeter is the sum of the lengths of all sides
area: divide into 7 triangles, find the area of each, add them up -10 equal sides and angles
-interior angles are 144 degrees
-sum of interior angles are 1440 degrees
-35 diagonals
-perimeter is the sum of the length of all the sides
-area: divide into 8 triangles, find the area of each, add them up Now, let's try some problems 1. A B C The height of Triangle ABC is 4 cm and the base is 3 cm. Find the hypotenuse. 4 cm 3 cm 2. Height/Altitude Base Hypotenuse Legs 145° 57° Find angles R and S using what you know about transversals and angles 3. 52 ° 40° x Find the value of x. Answers

1. Solution:
a²+b²=c²
=3²+4²=c²
=9+16=c²
=√25
=5
Therefore, c= 5 cm. 2. Solution:
57° 145° Since there is a transversal running through PQ and RT, it means angle Q and angle S should be the same. Therefore, angle A and angle B added should equal 57°. Now, angle U is 145°. The lines RU, UV and RT form an incomplete parallelogram. So, angle S should be 180°-145°, which equals 35°. Now all that remains is to do is find the other half of angle R, which is 57°-35°. This equals 22°. P Q S T R U v P Q R T S U V 3. 22° 35° x=88° A B C D E 52 ° 40° x A B C D E AB II DC
Therefore angle b = angle D
So, if angle D is 52°, and angle C is 40°, then x=180°-(52°+40°)
x=88° Base 1 Base 2
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