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Salvador Amaya 9-5 A polygon is a closed figure that has no curved sides. Therefore, it needs to have at least three sides. Not a polygon Yes, a polygon Yes, a polygon Exterior Angle Interior angle Diagonal Side Vertex Interior angle: angle in the inside of a polygon Exterior angle: angle in the outside of a polygon Vertex: point where two lines meet Diagonal: segment that connects two vertices of a polygon Side: segment in a polygon Concave polygon: polygon that has a vertex that points in Convex polygon: polygon that has all vertices pointing out. Concave Convex Equilateral: All sides measure the same. Equiangular: All the angles measure the same. Equilateral and Equiangular = Regular Polygon Equilateral Equiangular Regular The sum of the interior angles of a polygon is (n-2) x 180, being n the number of sides in the polygon. What is the sum of interior angles for a nonagon? (n-2) x 180 (9-2) x 180 7 x 180 1260 What is the measure of an angle in a regular hexagon? [(n-2) x 180] / n [(6-2) x 180] / 6 (4x 180) / 6 720 / 6 120 What is the sum of the interior angles for a pentagon? (n-2) x 180 (5-2) x 180 3 x 180 540 If a quadrilateral is a parallelogram, then opposite sides are congruent. If BACD, KHIJ, and FEGH are parallelograms, then…. a c e d b AC is cong. To BD, AB is cong. to CD, EF is cong. to GM, EG is cong. to FM, HI is cong. to KJ, HK is cong. to IJ. g h i f k j m If opposite sides in a quadrilateral are congruent, then the quadrilateral is a parallelogram. If AC is cong. To BD, AB is cong. to CD, EF is cong. to GM, EG is cong. to FM, HI is cong. to KJ, HK is cong. to IJ, then…. a c 68 e 13 13 68 d b 27 22 g h 4 k 8 i f 22 27 8 4 j BACD, KHIJ, and FEGH are parallelograms m If a quadrilateral is a parallelogram, then opposite angles are congruent. If BACD, KHIJ, and FEGH are parallelograms, then…. a c e <a is cong. to <d, <c is cong. to <b, <e is cong. to <m, <f is cong. to < g, <h is cong. to <j, <k is cong. to <i d b g h i f k j m If opposite angles are congruent in a quadrilateral, then the quadrilateral is a parallelogram. If <a is cong. to <d, <c is cong. to <b, <e is cong. to <m, <f is cong. to < g, <h is cong. to <j, <k is cong. to <i, then…. a 128 52 c e 36 52 128 d b 144 h 97 k 83 83 97 j i f 144 BACD, KHIJ, and FEGH are parallelograms 36 m g If a quadrilateral is a parallelogram, then the consecutive angles add up to 180. If BACD is a parallelogram, then…. a c <a+<b=180 <b+<d=180 <d+<c=180 <a+<c=180 b d If the consecutive angles in a quadrilateral add up to 180, then the quadrilateral is a parallelogram If <a+<b=180, <a+<c=180, <b+<c=180, <d+<c=180, <e+<g=180, <e+<f=180, <g+<m=180, <m+<f=180, <h+<i=180, <h+<k=180, <k+<j=180, <j+<i=180, then…. a 135 65 c e 24 65 135 d b 156 h 103 k 77 77 103 j i f 156 BACD, KHIJ, and FEGH are parallelograms 24 m g If a quadrilateral is a parallelogram, then the diagonals bisect each other. If CABD, and GEFH are parallelograms, then…. a AL=LD BL=LC EK=KH FK=KG b 45 f e 48 29 l 32 k 48 45 32 g c d 29 h Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. Its consecutive angles are supplementary. Its diagonals bisect each other. It has a pair of congruent parallel sides. Both pairs of opposite sides are parallel. All are parallelograms because both their opposite sides are congruent. 6 2 2 6 147 96 33 94 94 33 96 All are parallelograms because both their opposite angles are congruent. 147 All are parallelograms because their consecutive angles add up to 180. 132 48 48 132 91 99 91 99 34 146 146 34 7 5 7 5 All are quadrilaterals because their diagonals bisect each other. All are parallelograms because they have a pair of congruent parallel sides. 45 45 Rhombus: parallelogram with 4 congruent sides. Its diagonals are perpendicular to each other. If a quadrilateral is a rhombus, then it is a parallelogram. Since these are rhombuses because they are equilateral sides, then they are also parallelograms. 28 28 28 28 If a parallelogram is a rhombus, then its diagonals are perpendicular to each other. Since these are rhombuses, then their diagonals are perpendicular to each other If a parallelogram is a rhombus, then their diagonals bisect the opposite angles. a Since this is a rhombus, then… <BAE is cong. to <CAE <EBA is cong. to <EBD <BDE is cong. to <CDE <DCE is cong. to <ACE e b c d If a pair of consecutive sides is congruent, then the quadrilateral is a rhombus. Since these are parallelograms, because they have congruent consecutive sides, then they are rhombuses. 98 98 If the diagonals in a parallelogram are perpendicular, then it is a rhombus. Since the diagonals are perpendicular, then they are rhombuses If a diagonal bisects a pair of opposite angles, then it is a rhombus. Since…. <ZXB is cong. to <CXB, then CXZV is a rhombus <XZB is cong. to <VZB, then CXZV is a rhombus <ZVB is cong. to <CVB, then CXZV is a rhombus <XCB is cong. to <VCB, then CXZV is a rhombus z x b v c Rectangle: parallelogram with 4 right angles . Its diagonals are congruent. If it is a rectangle, then it is a parallelogram. Since these are rectangles (4 rt. angles), then they are also parallelograms. If it is a rectangle, then the diagonals are congruent. a c z b w q d AD is cong. to CB x ZV is cong. to CX e c v Since these are rectangles, the diagonals are congruent. QR is cong. to EW r If a quadrilateral has a right angle, then it is a rectangle. They are all rectangles because they each have a right angle. If the diagonals in a parallelogram are congruent, then it is a rectangle. p o f Since diagonals are congruent, these are rectangles i PU= 15 IO= 15 h FS=1 GH=1 u IH=98 JK=98 g s l k j h Square: parallelogram with 4 sides that are congruent and angles are congruent. The diagonals are congruent and perpendicular. It is both a rhombus and a rectangle. 2 pairs of congruent consecutive sides. Perpendicular diagonals. 1 pair of congruent angles. If it is a kite, then diagonals are perpendicular. Since the diagonals in the quadrilaterals are perpendicular, they are kites. If it is a kite, then there is 1 pair of opposite congruent angles. 103 103 110 110 One pair of parallel sides. Base- nonparallel side Leg- parallel side. Base angles- angles whose common side is a base. Isosceles trapezoid- has 2 congruent legs If it is an isosceles trapezoid, then the 2 pairs of base angles are congruent. Since these are isosceles triangles, their base angles are congruent. If there are 2 pairs of congruent angles, base angles, it is an isosceles triangle. 70 110 110 70 Since the base angles are congruent, these are isosceles trapezoids. 83 83 97 97 An isosceles trapezoid has to have congruent diagonals. a g b AD is cong. to BC f v b j n c d m VM is congruent to NB h These are isosceles trapezoids because their diagonals are congruent. FJ is cong. to GH The midsegment in a trapezoid is parallel to both bases and it is half as long as the sum of the two other bases. (b1+b2)/2 What is the midsegment of this trapezoid? (b1+b2)/2 10 10+35/2 45/2 22.5 35 What is the measure of the missing base? 67-21= 46 46+67= 113 21 67 What is the midsegment of this trapezoid? (b1+b2)/2 135 135+11/2 146/2 73 11 _____(0-10 pts.) Describe what a polygon is. Include a discussion about the parts of a polygon. Also compare and contrast a convex with a concave polygon. Compare and contrast equilateral and equiangular. Give 3 examples of each. _____(0-10 pts.) Explain the Interior angles theorem for polygons. Give at least 3 examples. _____(0-10 pts.) Describe the 4 theorems of parallelograms and their converse and explain how they are used. Give at least 3 examples of each. _____(0-10 pts.) Describe how to prove that a quadrilateral is a parallelogram. Give at least 3 examples of each. _____(0-10 pts.) Compare and contrast a rhombus with a square with a rectangle. Describe the rhombus, square and rectangle theorems. Give at least 3 examples of each. _____(0-10 pts.) Describe a trapezoid. Explain the trapezoidal theorems. Give at least 3 examples of each. _____(0-10 pts.) Describe a kite. Explain the kite theorems. Give at least 3 examples of each.