GMAT (Subject) / Quantitative - Geometry (Lesson)

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• Vertical angles When two straight lines intersect, the angles opposite each other are equal. They are termed opposite angles or vertical angles (because they share the same vertex).
• Acute angles You can remember this term by thinking of a cute angle - a small, less than 90°, slightly furry angle. An angle is acute if it measures less than 90°.
• Perpendicular Angles 2 lines that intersect at a right angle are perpendicular to each other. It is a relative term, marked by l For example, saying line segment AB is perpendicular to line segment CD can be written as  A ⊥ B
• 3 Rules on Angles All the small angles equal each other. All the big angles equal each other. Any combination of Big angle+Small angle is always 180°.
• Obture angles An angle is obtuse if it is greater than 90° and smaller than 180°. The term obtuse is also used to describe someone who is slow to understand.
• Vertices The term vertex often refers to a figure (i.e., triangle, square etc.), or an angle. A vertex (plural: vertices) is the intersection point of at least two straight lines, or rays (for angles). Pentagon ABCDE has five vertices 
• % Percentile percentile % of girls have a length that is less than or equal to x centimeters. 
• 50 percentile In the graph, this point is a little above the green 50th percentile line, which means that a little more than 50% (but less than 75%) of girls who are 80 centimeters long weigh 10 kg or less. Conversely, this means that a little fewer than 50% (but more than 25%) of girls who are 80 centimeters long weigh more than 10 kilograms
• Number of Diagonals in Polygon 0.5 * (n(n-3))
• Supplementary Angles Two angles are supplementary if they add up to 180° or to a straight angle. However, the angles don't have to be adjacent, as long as they add up to 180°.
• Complementary Angles Two angles are complementary  if they add up to 90° or to a straight angle. However, the angles don't have to be adjacent, as long as they add up to 180°.
• Regular polygon bzw. Equilateral polygon A polygon which has all sides mutually congruent and all angles mutually congruent is called a regular polygon or equilateral polygon. Most polygons in GMAT Geometry problems will be regular.
• Sum of angles for Polygons 180°·(n-2) Triangle - 3 - 180° Quadrilateral - 4 - 360° Pentagon - 5 - 540° Hexagon - 6 - 720° Octagon - 8 - 1080°
• Congruent Congruent means exactly equal in size and shape. Congruent geometric figures are identical.
• Opposing Angles Each of the three angles in a triangle determines the distance between the rays that form the angle. In other words, the angle determines the size of the side opposite to it. Therefore, the largest angle is opposite the largest side, the smallest angle is opposite the smallest side etc. In the same manner, the sides opposite equal angles are also equal, and vice versa.
• Isosceles triangle An isosceles triangle has two equal sides, and consequently two equal angles opposite the equal sides. You need only one angle of an isosceles triangle to figure out all of its angles. For example, if the top angle is 40°, then the other two equal angles can be determined: each of them will equal half of (180-40)=140, or 70°. In an isosceles triangle the height is also the median and the bisector.
• Third side rule Any side in a triangle must be smaller than the sum, and greater than the difference of the other sides a < b+c b < a+c c < a+b
• An obtuse triangle only one obtuse angle in a triangle.
• Types of angles Acute Angle - an angle that is less than 90° Right Angle - an angle that is 90° exactly Obtuse Angle - an angle that is greater than 90° but less than 180° Straight Angle - an angle that is 180° exactly Reflex Angle - an angle that is greater than 180
• Recycled Triangles 3 :  4  : 5  5 : 12 : 13 8 : 15 : 17
• Equilateral triangle An equilateral triangle has three equal sides, and consequently three equal angles. Hence, the angles in an equilateral are 60° each. In an equilateral triangle any height is also the median and the bisector.
• The area of an equilateral triangle is given in the formula Area = a2 * √3 / 4
• Special recycled triangle 1 By drawing a diagonal inside a square you get an isosceles right triangle, namely the 45°:45°:90° triangle. 45 : 45 : 90 traingle is iscosceles
• Special recycled triangle 2 If a is the smallest leg of the triangle, the sides opposite the angles are 30° :   60° : 90°a     : a√3  : 2a
• Are Parallelogram Area=Base×Height
• Is quadrilateral ABCD a parallelogram? (1) BC and AD are parallel. (2) AB and CD are equal in length. statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• Are all sides of quadrilateral ABCD equal (1) The diagonals of ABCD bisect each other at right angles. (2) ABCD is a parallelogram with three equal sides.
• evolution of the quadrilateral T A trapezoid is a quadrilateral with one pair of parallel sides. P A parallelogram is a quadrilateral with opposite equal and parallel sides. R A rectangle has the same properties as a parallelogram, with four right angles. S A square has all the same properties of a rectangle, with all sides equal.
• Rhombus A Rhombus is a special instance of a Parallelogram with all sides equal. The diagonals of a Rhombus are perpendicular to each other, as well as bisect each other and their respective base angles.
• Rectangle All four sides are equal All four sides are parallell Diagonals of a rectangle are equal and bisect each other. Four right angles. perimeter of a rectangle is the sum of its sides, given in the formula Perimeter= 2w+2l.area of a rectangle is given in the formula Area=Length×Width.
• Square All four sides are equal All four sides are parallell Four right angles Diagonals are equal, perpensicular, bisect each other and bisect their respective base angles perimeter of a square is the sum of its sides, given in the formula Perimeter=4s area of a square is given in the formula Area=Side*2
• Circumference Circle The circumference is a special word for the perimeter of a circle.The circumference of a circle is given in the formula Circumference=2πr=π·d.
• The arc arc is a portion of a circle's circumference. arc can be defined by only two points and an indication whether the arc is the major or the minor part of the whole circle. The measure, in angles, of an arc defined by diameter AB is 180
• Measures of arcs "major arc AB" is the greater of the two parts of the circle that are defined by A and B"minor arc AB" is the lesser of the two parts of the circle that are defined by A and B"arc ABC" is the part of the circle on which points A,B,C lie in this orderThe measure of the whole circumference is 360°An arc may be measured in angles, as a fraction of a circle.
• Inscribed circle Inscribed circle is the largest possible circle that can be drawn within a polygon, so that each side of the polygon is tangent to the circle.
• Heigt of triangle The height or altitude, in a triangle is the distance between any vertex and the opposing side.The height or altitude is by definition perpendicular to the opposing side.
• An Arc An arc can be measured as a fraction of a circle, or as a function of the central angle which defines it. The measure of an arc defined by a central angle xº is simply xº. The proportion of the central angle from the entire circle is what matters in determining the measure of an arc.
• Slope Vertical Change / Horizontal Change y2-y1 / x2-x1 A rising line with slope 1 creates a 45° angle with the x-axis. A rising line with slope greater than 45° angle will have a slope > 1 
• Coordinate Plane The coordinate plane (also known as the rectangular coordinate system) is a two-dimensional plane with two perpendicular axes: x and y. The x-axis indicates the horizontal direction while the y-axis indicates the vertical direction of the plane. The coordinate plane is split by the axes into four quadrants. All points in quadrant I have positive x-coordinates and positive y-coordinates, thus (+,+). All points in quadrant II have negative x-coordinates and positive y-coordinates, thus (−,+). All points in quadrant III have negative x-coordinates and negative y-coordinates, thus (−,−). All points in quadrant IV have positive x-coordinates and negative y-coordinates, thus (+,−).
• The Slope The slope is the ratio of the vertical change to the horizontal change between P and R. The vertical change, i.e., the difference between the y coordinates of P and R equals the length of PQ. The horizontal change, i.e., the difference between the x coordinates of P and R equals the length of QR. The slope is the ratio of PQ and QR.
• An inscribed angle in a circle defines an arc. The measure of an arc defined by an inscribed angle xº is 2xº.  when an inscribed angle and a central angle define the same arc, the central angle is double the inscribed angle. Within a given circle, arcs of equal length are arcs of equal degrees. The opposite is true as well: Within a given circle, arcs of equal degrees are arcs of equal length. Equal central angles define equal sections on the circumference of the circle. sector area / circle area = arc length/ circumference
• Line Equations A line equation or linear equation has the form: y=mx+b.x and y are variables standing for the coordinates of any point on the line.m is the slope of the line.b is the y-intercept. The coordinates of the y-intercept are always (0,y), thus for y-intercept x=0. The coordinates of the y-intercept are always (0,y), thus for y-intercept x=0.
• parallel lines same slope. lines that never meet
• Slope of perpendicular the slopes of two lines that are perpendicular to each other are reverse reciprocal if the slope of one line is m, then the slope of the other is -1/m. The slopes of perpendicular lines are reverse and reciprocal. In the above example the slope of one line is 2/5, so the slope of the perpendicular line is the reverse
• Proportion of Arc length / Sector length / etc. The area of the sector, as well as the length of an arc is proportional to the measure of the central angle that defines them. sector area / circle area arc length / circumference xº / 360º
• Sphere Area 4∏r2
• Hemisphere Volume 2/3 * ∏ * r3
• The surface area of a sphere is 4πR2, where R is the radius of the sphere. If the area of the base of a hemisphere is 3, what is the surface area of that hemisphere? The area of the base of a hemisphere is the area of a circle: --> πr2 = 3 --> r2 = 3/π Plug the r2 in the formula for surface area of a sphere: --> 4πR2 = 4π(3/π) = 12 A hemisphere will have half the surface area + area of the base, so the correct answer is 12/2 + 3 = 6 + 3 = 9.

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