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Salvador Amaya 9-5. Polygons. Polygons. 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. Parts of a Polygon. Exterior Angle. Interior angle. Vertex. Diagonal. Side.

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## Polygons

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**Salvador Amaya 9-5**Polygons**Polygons**• 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**Parts of a Polygon**Exterior Angle Interior angle Vertex Diagonal Side**Parts of a Polygon (ctd)**• 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 vs. Convex**• Concave polygon: polygon that has a vertex that points in • Convex polygon: polygon that has all vertices pointing out.**Concave vs. Convex (ctd)**• Concave • Convex**Equilateral vs. Equiangular**• Equilateral: All sides measure the same. • Equiangular: All the angles measure the same. • Equilateral and Equiangular = Regular Polygon**Equilateral vs. Equiangular (ctd)**• Equilateral • Equiangular • Regular**Interior Angles Theorem for Polygons**• The sum of the interior angles of a polygon is (n-2) x 180, being n the number of sides in the polygon.**Interior Angles Theorem for Polygons (ctd)**• What is the sum of interior angles for a nonagon? • (n-2) x 180 • (9-2) x 180 • 7 x 180 • 1260**Interior Angles Theorem for Polygons (ctd)**• 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**Interior Angles Theorem for Polygons (ctd)**• What is the sum of the interior angles for a pentagon? • (n-2) x 180 • (5-2) x 180 • 3 x 180 • 540**Theorems for Parallelograms: Sides**• If a quadrilateral is a parallelogram, then opposite sides are congruent.**Examples**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. If BACD, KHIJ, and FEGH are parallelograms, then…. a c e d b g h f i k j m**Theorems for Parallelograms , converse : Sides**• If opposite sides in a quadrilateral are congruent, then the quadrilateral is a parallelogram.**Examples**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 27 b 22 g h 4 8 f i 22 27 k 4 8 BACD, KHIJ, and FEGH are parallelograms j m**Theorems for Parallelograms: Angles**• If a quadrilateral is a parallelogram, then opposite angles are congruent.**Examples**<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 If BACD, KHIJ, and FEGH are parallelograms, then…. a c e d b g h f i k j m**Theorems for Parallelograms, converse: Angles**• If opposite angles are congruent in a quadrilateral, then the quadrilateral is a parallelogram.**Examples**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 c 128 52 e 36 52 128 d b g 144 h 97 144 f i 83 83 k 36 97 BACD, KHIJ, and FEGH are parallelograms j m**Theorems for Parallelograms: Angles 2**• If a quadrilateral is a parallelogram, then the consecutive angles add up to 180.**Examples**If BACD is a parallelogram, then…. a c <a+<b=180 <b+<d=180 <d+<c=180 <a+<c=180 b d**Theorems for Parallelograms, converse: Angles 2**• If the consecutive angles in a quadrilateral add up to 180, then the quadrilateral is a parallelogram**Examples**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 c 135 65 e 24 65 135 d b g 156 h 103 156 f i 77 77 k 24 103 BACD, KHIJ, and FEGH are parallelograms j m**Theorems for Parallelograms: Diagonals**• If a quadrilateral is a parallelogram, then the diagonals bisect each other.**Examples**If CABD, and GEFH are parallelograms, then…. AL=LD BL=LC EK=KH FK=KG a b 45 f e 48 29 l 32 k 29 45 32 48 g h c d**How to prove that a quadrilateral is a parallelogram**• 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.**Examples, opposite sides**All are parallelograms because both their opposite sides are congruent. 6 2 2 6**Examples, opposite angles**147 33 96 94 94 33 147 96 All are parallelograms because both their opposite angles are congruent.**Examples, consecutive angles**All are parallelograms because their consecutive angles add up to 180. 132 48 48 132 91 99 34 146 146 91 99 34**Examples, diagonals**7 5 7 5 All are quadrilaterals because their diagonals bisect each other.**Examples, pair of sides**All are parallelograms because they have a pair of congruent parallel sides. 45 45**Rhombus**• Rhombus: parallelogram with 4 congruent sides. Its diagonals are perpendicular to each other.**Rhombus Theorems**• If a quadrilateral is a rhombus, then it is a parallelogram.**Examples**Since these are rhombuses because they are equilateral sides, then they are also parallelograms. 28 28 28 28**Rhombus Theorems (ctd)**• If a parallelogram is a rhombus, then its diagonals are perpendicular to each other.**Examples**Since these are rhombuses, then their diagonals are perpendicular to each other**Rhombus Theorems (ctd)**• If a parallelogram is a rhombus, then their diagonals bisect the opposite angles.**Examples**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 a e b c d**Rhombus Theorems (ctd)**• If a pair of consecutive sides is congruent, then the quadrilateral is a rhombus.**Examples**Since these are parallelograms, because they have congruent consecutive sides, then they are rhombuses. 98 98**Rhombus Theorems (ctd)**• If the diagonals in a parallelogram are perpendicular, then it is a rhombus.**Examples**Since the diagonals are perpendicular, then they are rhombuses**Rhombus Theorems (ctd)**• If a diagonal bisects a pair of opposite angles, then it is a rhombus.**Examples**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 b x v c**Rectangle**• Rectangle: parallelogram with 4 right angles . Its diagonals are congruent.**Rectangle Theorems**• If it is a rectangle, then it is a parallelogram.**Examples**Since these are rectangles (4 rt. angles), then they are also parallelograms.**Rectangle Theorems (ctd)**• If it is a rectangle, then the diagonals are congruent.

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