Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review

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Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon

1 Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon - a polygon with n-sides. Diagonal a segment joining two non-adjacent vertices of a polygon. Regular Polygon A polygon where all sides and all angles are congruent. Convex Polygon has no diagonal with points outside the polygon Concave Polygon has at least one diagonal with points outside the polygon Type Classify Classifying a polygons by sides: 3 sides - 6 sides - 9 sides - 4 sides - 7 sides - 10 sides - 5 sides - 8 sides - n sides - Naming a polygon - A polygon is named by starting at any vertex and going clockwise or counterclockwise to the next consecutive vertex. This polygon can be named two different ways starting at point M: 1) Polygon M 2) Polygon M 1

2 6-0 Interior and Exterior Angles of Polygons How many triangles can be drawn in this square by drawing in diagonals (diagonals cannot intersect) *The sum of the interior angles of a triangle is 180 The sum of the interior angles of a square is = 360 Investigate the number of triangles that can be drawn in the following shapes with diagonals: Triangles Triangles Triangles Sum = 180* = Sum = 180* = Sum = 180* = *There are always 2 less triangles than sides in every convex polygon. Polygon-Angle Sum Theorem The sum of the measures of the interior angles of a convex n-gon is Triangle = (3 2) * 180 Pentagon = = 1 * 180 = = 180 = Finding the missing angle value This shape at right is a pentagon. The interior angle sum is (5 2)*180 = 540 The sum of the angles currently is (437 + x) Substituting we have: (437 + x) = 540 Solving for x gives the answer x = 103 2

3 What is the sum of the measures of the exterior angles of an equilateral triangle and of a rectangle? Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles of any polygon is 360 ***Special rules for REGULAR polygons*** All of the interior angles are congruent ***After you find the measure of the interior angles, divide that number by the # of vertices to find the measure of each vertex angle. All of the exterior angles are congruent Divide 360 by the number of vertices to find the measure of each exterior angle., *n is the number of sides of the polygon. To find each interior angle of a regular polygon then divide by the number of sides. *n is the number of sides of the polygon What is the exterior angle of a regular polygons? Hexagon = Pentagon = = = What is the measeure of the interior angles of the regular polygons? Octagon = Dodecagon = = = = 135 = 3

4 6-1 Classifying Quadrilaterals Parallelogram Rectangle Square Kite Trapezoiod Isosceles Trapezoid Rhombus 4

5 6-2 Properties of Parallelograms All properties on this page are true when a quadrilateral is a parallelogram Parallelogram: Is a quadrilateral with both pairs of opposite sides are parallel. Parallelogram Opposite Side Congruence Theorem: If a quadrilateral is a parallelogram, then its opposite sides are congruent. Parallelogram Consecutive Interior Angle Theorem: If a quadrilateral is a parallelogram, its consecutive angles are supplementary. Parallelogram Opposite Angle Congruence Theorem: If a quadrilateral is a parallelogram, its opposite angles are congruent. Parallelogram Diagonal Bisector Theorem: If a quadrilateral is a parallelogram, then its diagonals bisect each other. Congruent Segments Theorem For Parallel Lines. If three (or more) parallel lines make congruent segments on one transversal, then they make congruent segments on all transversals. 5

6 6-3 Proving a quadrilateral is a parallelogram Converse to Parallelogram Opposite Side Congruence Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Converse to Parallelogram Consecutive Interior Angles Theorem: If an angle of a quadrilateral is supplementary to both consecutive angles, then the quadrilateral is a parallelogram. Converse to Parallelogram Opposite Angle Congruence Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Converse to Parallelogram Diagonal Bisector Theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. Parallelogram Congruent and Parallel Congruence Theorem: If one pair of opposite sides of a quadrilateral is both congruent and parallel, the quadrilateral is a parallelogram. Proving a Quadrilateral is a Parallelogram 6

7 Write a developmental proof for each of the following. 1. Given: AB DC and AD BC Prove: ABCD is a Parallelogram A B D C 2. Given: and AB DC Prove: ABCD is a Parallelogram A B D C 7

8 3. Given: A B 180 and B C 180 Prove: ABCD is a Parallelogram A B D C 4. Given: AE EC and BE ED Prove: ABCD is a Parallelogram A E B D C 8

9 5. Given: DAB BCD and ABC CDA Prove: ABCD is a Parallelogram A B D C 9

10 6-4 Special Classes of Parallelograms All Parallelogram rules apply as well as each shapes unique rules. Rhombus: Is a parallelogram with all four sides congruent. Rhombus Perpendicular Diagonal Theorem: A parallelogram is a rhombus if and only if its diagonals are Perpendicular. Rhombus Diagonals Bisect Vertex Angle Theorem: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Rectangle: Is a parallelogram with four right angles. Rectangle Diagonal Congruence Theorem: A parallelogram is a rectangle if and only if its diagonals of are congruent. Square: Is a rectangle that is also a rhombus. 10

11 Developmental Proofs Given: JKLM is a rhombus Prove: J K Z M L Given: JKLM is a parallelogram at Z Prove: JKLM is a rhombus J Z K M L 11

12 6-5 Trapezoids and Kites 1. Trapezoid: Is a quadrilateral with exactly one pair of parallel sides. 2. Isosceles Trapezoid: If the legs of the trapezoid are congruent. 3. Trapezoid Base Angles Congruence Theorem: If a trapezoid is isosceles, the each pair of base angles is congruent. 4. Trapezoid Base Angles Congruent Theorem Converse: If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. 5. Trapezoid Congruent Diagonals Theorem: A trapezoid is isosceles if and only if its diagonals are congruent. 6. Trapezoid Midsegment Theorem: 1) The midsegment of a trapezoid is half the sum of the bases 2) The midsegment of a trapezoid is parallel to the bases of the trapezoid. 7. Kite: Is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. 8. Kite Perpendicular Diagonals Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular. 9. Kite One Opposite Congruent Angle Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. 12

13 Given: WHAT is a kite, with Prove: Developmental Proofs W H S A T 13

Geometry/Trigonometry Unit 5: Polygon Notes Period:

Geometry/Trigonometry Unit 5: Polygon Notes Name: Date: Period: # (1) Page 270 271 #8 14 Even, #15 20, #27-32 (2) Page 276 1 10, #11 25 Odd (3) Page 276 277 #12 30 Even (4) Page 283 #1-14 All (5) Page