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Geometry EOC Study guide Benchmarks: Be able to find the converse, inverse, and contrapositive of a statement. Determine if two propositions are logically equivalent Find the lengths and midpoints of line segments in two-dimensional coordinate system Use coordinate geometry to find slopes of parallel lines, perpendicular lines, and equations of lines. Examples: If today is Sunday, then tomorrow is Monday If tomorrow is Monday , then today is Sunday is the _________________ of the conditional statement If today is not Sunday, then tomorrow is not Monday is the __________________ of the conditional statement It today is not Monday, then tomorrow is not Sunday is the ___________________ of the conditional statement. These two statements are logically equivalent: If Lisa is in France, then she is in Europe If Lisa is not in Europe, then she is not in France The second statement is the _______________________ of the first. Midpoint: the point on a line segment that is the same distance from both endpoints. The midpoint ______________ a line segment. Midpoint Formula: Slope formula: Slope intercept form of a line: __________________________ lines have the same slope. ___________________________lines have opposite reciprocal slopes. To determine the slope of the line 2y = 3x -6 you need to isolate the ___ variable. The slope of the line 2y = 3x – 6 is ____ The line -6x + 4y = 15 is ______________ Identify and use the relationships between special pairs of angles formed by parallel lines and transversals. to the line 2y = 3x – 6 The line -10x – 15y = 4 is ______________ to the line 2y = 3x - 6 Corresponding angles (draw a picture Alternate interior angles (draw a picture) Alternate exterior angles(draw a picture) Identify and describe convex, concave, regular, and irregular polygons. The converse of the parallel line conjecture states if two lines are cut by a transversal to form pairs of congruent _________________ angles, congruent ____________________________ angles, or congruent _________________________ Angles, then the lines are _____________________. Convex polygons: No diagonals will fall outside the polygon Concave polygons: At least one diagonal is outside the polygon Regular polygons: All angles and sides are congruent Irregular polygon: Does not have all angles or all sides congruent Determine the measures of interior and exterior angles of polygons, justifying the method used. The sum of the exterior angles of ANY polygon is always _________ degrees Formula to find the sum of the interior angles in a polygon: Formula to find the measure of one angle in a regular polygon: Use properties of congruent and similar polygons to solve mathematical or real work problems. Apply transformations to polygons to determine congruence, similarity, and symmetry. Know the images formed by translations, reflections, and rotations are congruent to the original shape. Remember that before you can solve a math problem using proportions or set any parts of polygons equal to each other you need to first prove the two figures congruent or similar. Triangle congruence shortcuts: SSS SAS ASA AAS Triangle similarity shortcuts: AA SSS SAS Dilations of polygons show ____________________ , not _____________________. A dilation either enlarges or shrinks a polygon by a scale factor that changes all values of every coordinate of all vertices Translation is done by sliding a figure in a straight line either up or down. (figures will remain congruent b/c size or shape does not change) When a figure moves to the right by adding a constant number the only values changing are the ____ values When a figure moves down by subtracting a constant number the only values changing are the ____ values. Reflections produce a figures mirror image (figures will also remain congruent) Reflection about y-axis (x,y) > (-x,y) Reflection about x-axis (x,y) > (x,-y) Reflection about the line y=x (x,y) > (y,x) Know the difference between area and perimeter of polygons Rotations are when all points in the original figure rotate an identical number of degrees about a fixed point. (figures will remain congruent) My suggestion to keep rotations as simple as possible is the use easier points that fall on the x or y axis since we know that each quadrant on the coordinate plane is a 90˚ angle 90 ˚ rotation counterclockwise (x,y) > (-y,x) 90˚ rotation clockwise (x,y) > (y,-x) Area formulas: Rectangle/Square/Parallelograms: Triangles: Rhombuses/kites: Trapezoids: Circles: Circumference of a circle: Regular polygons: Determine how changes in dimensions affect the perimeter and area of common To determine how two figures compare in volume or surface area, pick a number to geometric figures. start with and find the surface area or volume. Then change the original number however the problem is telling you to and find the new SA or volume and compare both answers. Example: Kendra has a compost box that has a shape of a cube. She wants to increase the size of the box by extending every edge of the box by half of its original length. After the box is increased in size, what % of the volume of the new box is the old box? Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite. Quadrilaterals: polygon with 4 sides Trapezoid: a quadrilateral with _____ pair of parallel sides The ____________________ angles are supplementary A trapezoid is a/n _____________________ trapezoid if the pair of ___________________________ sides are _________________. Base angles of a/n ____________________ trapezoid are ________________________. Diagonals of a/n ______________________ trapezoid are _________________________. Kites: a quadrilateral with two pairs of consecutive sides congruent Non vertex angles are __________________ Diagonals are perpendicular The diagonal connecting the vertex angles is the perpendicular ___________________of the other diagonal. The vertex angles are bisected by a Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals. diagonal. Parallelograms: a quadrilateral with 2 pair of parallel sides. Opposite angles are ____________________. Consecutive angles are ___________________. Opposite sides are _____________________. Diagonals of a parallelogram ________________ each other. Rectangle: (subset of a parallelogram) have all angles congruent. (four 90˚ angles) All 5 properties of parallelograms apply to rectangles in addition to the fact that diagonals of a rectangle not only bisect each other but are also ____________________. Rhombus: (subset of a parallelogram) have all sides congruent. All 5 properties of parallelograms also apply to rhombuses in addition to the fact that diagonals of a rhombus are perpendicular to each other as well as bisect each other. Diagonals of a rhombus also bisect the angles of the rhombus. Square: A regular parallelogram (the baby of rectangle and a rhombus) The diagonals of a square are _____________________, _____________________, and __________________ each other. A square has all properties of parallelograms, rectangles, and rhombuses. When in the coordinate plane you need to use the distance formula to determine if all sides are congruent. (Remember that if the lines are not vertical or horizontal you cannot count the squares!!!!) You also need to find slopes of opposite sides to determine if they are parallel and slopes of consecutive sides to determine if they are perpendicular. Remember parallel lines have the Prove theorems involving quadrilaterals Classify, construct and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter. same slope Perpendicular lines have opposite reciprocal slopes. See page 298 and review a view examples Right triangle: Triangle with a 90˚ angle. The side opposite the right angle is called the __________________________ The other two sides are called ________ You can only use sine, cosine, and tangent as well as the Pythagorean Theorem in right triangles!!!!!! Remember not to assume a triangle is right if it is not stated in the problem, if no angles are drawn Acute: a triangle with all angles less than 90˚ Obtuse: one angle in a triangle is greater than 90˚ Remember that there can only be one angle greater than 90˚ in a triangle Scalene: no sides are congruent Isosceles: two sides congruent Equilateral: all sides congruent Equiangular: all angles congruent Equilateral triangles are regular polygons whose three angles each measure 60˚ Remember that the height of a triangle has to be perpendicular to its base. In an equilateral or isosceles triangle you need to find the height by bisecting the vertex angle which then becomes the perpendicular bisector of the base. Altitude: the perpendicular segment from a vertex to its opposite side The _______________________is the point of concurrency for the altitudes Medians: the segment that connects a vertex to the midpoint of its opposite side The _____________________ is the point of concurrency of the medians. Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles (CPCTC) Apply the inequality theorems Angle bisectors: the segment that cuts an angle in half The ____________________ is the point of concurrency that connects the angles bisectors Perpendicular bisectors: the line that bisects and is perpendicular to each side of the triangle The ___________________________ is the point of concurrency of the perpendicular bisectors. Remember that BEFORE you can prove parts congruent by CPCTC you first HAVE to prove triangles congruent by SSS, SAS, ASA, or AAS. Triangle inequality conjecture: the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Side-Angle inequality conjecture: in a triangle if one side is longer than another side, then the angle opposite the longer side larger than the angle opposite the shorter side. Prove and apply the Pythagorean theorem and its converse Triangle Exterior Angle conjecture: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Pythagorean theorem: Converse of the Pythagorean theorem: If the lengths of the three sides of a triangle satisfy the Pythagorean theorem equation then the triangle is a right triangle. If a2 + b2 > c2 then the triangle is acute If a2 + b2 < c2 then the triangle is obtuse State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle Use special right triangles (30˚-60˚-90˚ and 45˚-45˚-45˚) to solve problems. Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent, and concentric circles. In a right triangle, the altitude drawn to the hypotenuse is the geometric mean of the two pieces of the hypotenuse Geometry mean: when you multiply two numbers and find the square root of their product. Mean/Average: when you add two or more numbers and divide by the total numbers in the set. 30˚-60˚-90˚ The short leg is _____ the hypotenuse The long leg is ________ The hypotenuse is _____ the short leg 45˚-45˚-90˚ The hypotenuse is the short leg multiplied by ______. Circumference: The perimeter of a circle defined by the equation __________ or _________ Radius: The distance from the center of a circle to a point on the circle. The radius in of a circle does not change no matter where you move the point on the circle. The radius is ________ the diameter Arc: Two points on a circle and the continuous part of the circle between them. Arc length: The portion of the circumference of the circle described by an arc, measured in units of length. the formula for arc length is: The formula for a sector is: Chord: A line segment whose endpoints lie on the circle Secant: Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents) A line the intersects the circle in two points Tangent: A line that touches a point on a circle only once. a tangent to a circle is __________________ to the radius drawn to the point of tangency Tangent segments to a circle from a point outside the circle are _____________________. Concentric Circles: Circles that share the same center The area between concentric circles is called a/n _________________ To find the area of the annulus you need to subtract the area of the __________ circle from the area of the __________ circle. Central angles are _______________ the measure of the intercepted arc Inscribed angles are _____________ the measure of the intercepted arc. List all conjectures and equations for intersection of secants, chords, and tangents: (chapter 6 pg. 310) Given the center and the radius, find the equation of a circle in the coordinate plane, or given the equation of a circle in center radius form, state the center and the radius of the circle (chapter 5 pg. 504) Describe the relationships between the faces, edges, and vertices of polyhedral (chapter 10) Explain and use formulas for lateral area, surface area, and volume of solids. List all formulas for surface area: List all formulas for volume: Determine how changes in dimensions affect the surface area and volume of common geometric solids Define and use trigonometric ratios (sine, cosine, tangent, cotangent, secant, and cosecant) in term of right triangles.