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Transcript

REVIEW PACKET POLYGONS Properties of a parallelogram: use these for algebra problems and proofs o Opposite sides parallel To prove using coordinate: find slopes of opposite sides and show that the slopes are equal o Opposite sides are equal o Opposite angles are equal o Consecutive angles are supplementary o Diagonals bisect each other o One set of opposite sides are equal and parallel Properties of a Rectangle: use these for algebra problems and proofs o Opposite sides parallel To prove using coordinate: find slopes of opposite sides and show that the slopes are equal o Opposite sides are equal o Opposite angles are equal o Consecutive angles are supplementary o Diagonals bisect each other o 4 right angles to prove using coordinate: find slope of all sides, show opposite sides have = slopes and that adjacent sides have negative reciprocal slopes. o Congruent diagonals Properties of a Rhombus: use these for algebra problems and proofs o Opposite sides parallel To prove using coordinate: find slopes of opposite sides and show that the slopes are equal o Opposite sides are equal o Opposite angles are equal o Consecutive angles are supplementary o Diagonals bisect each other o Diagonals are perpendicular o 4 equal sides use distance formula 4 times to show all sides are = o Diagonals bisect opposite angles Properties of a Square: use these for algebra problems and proofs o Opposite sides parallel To prove using coordinate: find slopes of opposite sides and show that the slopes are equal o Opposite sides are equal o Opposite angles are equal o Consecutive angles are supplementary o Diagonals bisect each other o Diagonals are perpendicular o 4 equal sides use distance formula 4 times to show all sides are = REVIEW PACKET o Diagonals bisect opposite angles o 4 right angles o Congruent diagonals Properties of a trapezoid: use these for algebra problems and proofs o Only one pair of opposite sides parallel o Isosceles trapezoid: 2 = legs, 2 = base angles Congruent diagonals Logic Inverse: negate both statements o Example: if it rains, then I bring an umbrella. Inverse: If I do not bring an umbrella then I it did not rain. Converse: Switch the order o Example: Converse: If I bring an umbrella then it rains Contra positive: Switch and negate o Example: If I do not bring an umbrella then it is not raining. o LOGICALLY EQUIVALENT Transformations Isometry: size doesn’t change Reflection: notation: rx-axis. Does not preserve orientation. Opposite or reverse or indirect isometry o rx-axis : Reflect over the x-axis: switch the sign of the y value Example: (7,-3) goes to (7,3) o ry-axis: Reflect over the y-axis: switch the sign of the x value Example: (7,-3) goes to (-7,-3) o ry=x: Reflect over the line y=x: flip flop the x and y value. Example: (7,-3) goes to (-3,7) o rorigin: Reflection through the origin: switch the sign of both x and y Example: (7,-3) goes to (-7,3) Rotation: Spin your paper counterclockwise. Notation: R90. Is a direct isometry o R90: moves one box to the left (counterclockwise) Rule: (x,y) goes to (-y,x) Example: (7,-3) goes to (3,7) o R180: moves two boxes to the left (counterclockwise) Rule: (x,y) goes to (-x,-y) same as Reflection through the origin Example: (7,-3) goes to (-7,3) o R270: moves three boxes to the left (counterclockwise) Rule: (x,y) goes to (y,-x) Dilation: Changes size. Notation Dk , Is not an isometry o k is called a scale factor o multiply both the x and y value of a point by k Example: D2 on the point (7,-1) becomes (14,-2) Translation: Is a slide. Only the location of the points changes. Is a direct isometry. REVIEW PACKET o Notation: T(a,b), where you add a to the x value and b to the y value o You add values to the x and y coordinates. Example: T(-2,5), on the point (1,-3). New point: (-2+1, 5+-3) = (-1,2) Composition of Transformations: Read from RIGHT to LEFT o Example: ry axis T 2, 4 (7,3) First do the Translation: (7+-2, -3+4) = (5,1) Now Reflect the NEW POINT over the y-axis. (5,1) goes to (-5,1) Helpful hints: o If a question says which of the following is not an isometry: look for the one with Dilation o If a question asks which of the following does not preserve orientation or is an indirect isometry look for the answer with some sort of line reflection. (this does not include reflection through the origin since the origin is not a line) CIRCLES Equation of a circle: ( x h) 2 ( y k ) 2 r 2 o (h,k) is the center of the circle. r is the radius. o When given the equation, flip the sign of the numbers to find the center and square root to find the radius. Example: ( x 2) 2 ( y 3) 2 25 . Center is (2,-3) and radius is 5 TRIANGLES Sum of interior angles in a triangle is 180 Triangle Inequality: the sum of any two sides of a triangle needs to be longer than the third side. B o RULE: AB+BC> AC BC+AC> AB AB+AC>BC A C Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. o RULE: a = b + c c b a Types of Triangles: o Isosceles Triangle: 2 = angles across from 2 = sides o Equilateral Triangle: 3 = sides, 3 = angles (all 60) o Scalene Triangle: all 3 sides are NOT = REVIEW PACKET o Obtuse Triangle: a triangle with ONE obtuse angle and 2 acute angles o Acute Triangle: ALL THREE angles are acute. Pythagorean Theorem o a 2 b 2 c 2 , where c has to be the longest side (the hypotenuse) o only works for a right triangle The Longest Side is across from the largest angle. The exterior angle of a triangle is always greater than either of the two nonadjacent interior angles ANGLES Supplementary Angles: 2 angles whose sum in 180 Complementary Angles: 2 angles whose sum is 90 Acute Angle: an angle less than 90 Obtuse Angle: an angle greater than 90 Linear Pair: 2 angles that form a line PARALLEL LINES CUT BY A TRANSVERSAL Alternate interior angles: equal: 4 & 6 and 5&3 Corresponding angles: equal: 2 & 5, 4 & 7, 1 & 6, and 3 & 8 A Same side interior: supplementary: 4 & 5, and 3 & 6 Alternate Exterior angles: equal: 2 & 7 and 1 & 8 Same side Exterior angles: supplementary: 2&7 and 1 & 8 C t 2 4 5 7 1 3 6 8 TO USE FOR PROOFS Alternate interior angles on parallel lines are equal o Highlight the parallel lines and the transversal and look for the “Z” Corresponding angles on parallel lines are equal o Highlight the parallel lines and the transversal and look for the “F” PROOFS B 3 types of proofs: Congruent Triangle Only, CPCTC o Congruent Triangle only: Prove statement looks like: ABCDEF Last reason will be: SAS, ASA, AAS, SSS or HL NO ASS OR SSA HL only works for right triangle o CPCTC: When proving parts, like angles or segments. Prove statement may look like: AB CD or ABC DEF CPCTC COMES AFTER PROVING TWO TRIANGLES ARE CONGRUENT The keyword from the statement helps you find the reason for the next statement D