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CHAPTER 8 QUADRILATERALS By Margo Pennington and Ashlynn Kolarik SECTION 1 A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Diagonals of polygons form triangles which can be useful in calculating the sum of interior angles. Theorem 8.1- Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex n-gon is (n-2)*180◦ Corollary to Theorem 8.1- Interior Angles of a Quadrilateral: The sum of the measures of the exterior angles of a quadrilateral is 360◦ Example 1- Find the sum of the interior angle measures A pentagon has 5 sides. Use the polygon interior angles theorem to find the sum of interior angles. (5-2)*180◦ =540◦ TRY THIS! Find the sum of the measures of the interior angles of a convex heptagon. SOLUTION (n-2)180 n=7 5*180 =900◦ CONTINUED Example 2: Find the number of sides given the sum of interior angles. The sum of the measures of the interior angles of a convex polygon is 720◦. (n-2)*180◦=720◦ (n-2)=4 N=6, the polygon is a hexagon. Example 3: Find the value of x shown in the diagram. x + 68 + 126 + 106 = 360 x + 300 = 360 x = 60 Theorem 8.2-Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360◦ TRY THIS! Find the value of x. ∠1=2x° ∠2=72° ∠3=19° ∠4=5x-2° ∠5=31° SOLUTION 2x + 5x + 72 + 19 + 31- 2 = 360 7x + 120 = 360 7x = 240 x ≈ 34.3 SECTION 2 A parallelogram is a quadrilateral with both pairs of opposite sides parallel Theorem 8.3: If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem 8.4: If a quadrilateral is a parallelogram, then its opposite angles are congruent. Example 1: Find the values of x and y. x-7 x-7=23 y◦ x=30 90° y=90 23 Theorem 8.5: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary Theorem 8.6: If a quadrilateral is a parallelogram, then its diagonals bisect each other. TRY THIS! Find the values of x and y. AB=y+12 BC=-4x+11 CD=2y-1 AD=2x-3 SOLUTION y + 12 = 2y - 1 y = 13 2x – 3 = 4x + 11 x = -7 SECTION 3 Theorem 8.7: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 8.8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Example 1: Solve a real-world problem. A carpenter wants to build a table that is a parallelogram. Find the value of x and y that make both pairs of opposite angles congruent. x+1=101 x+1 80 x=100 y=80 101 y TRY THIS! Find the values of x and y that make the quad a p-gram. SOLUTION 5y + 15 = 110 5y = 95 y = 19 2x + 40 = 70 2x = 30 x = 15 CONTINUED TRY THIS! Find the value of x. SOLUTION x + 40 = 2x + 18 x = 22 CONTINUED Section Wrap- Up~ Proving that A Quadrilateral is a Parallelogram: 1.) Opposite sides are parallel. 2.) Opposite sides are congruent. 3.) Opposite angles are congruent. 4.) One pair of opposite sides are congruent and parallel. 5.) Diagonals bisect each other. SECTION 4 A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle. CONTINUED Comparing and Contrasting Parallelograms: TRY THIS! Classify the Quadrilaterals a.) b.) SOLUTION a.) Rhombus because all sides are congruent. b.) Square because all sides are congruent and all angles are 90°. CONTINUED Theorem 8.11: A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem 8.12: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 8.13: A parallelogram is a rectangle if and only if its diagonals are congruent. Example 2: Solve a real world problem An inspector wants to confirm that a certain lap pool is a rectangle. Can he assume based on the measurements that it is one? 88 92 92 88 e He cannot assume this is a rectangle because the angles should each be ninety degrees. PROPERTIES OF QUADRILATERALS Rectangle Property All sides congruent Rhombus Square X X Diagonals bisect each other X X X X Both pairs opp. Sides congruent X X X X Diagonals are Congruent X X X X Both pairs of opp. Sides parallel X X X X X X Diagonals are perpendicular 1 pair of opp. Sides are parallel. X X X X 1 pair opp. angles are congruent X X X X X X X All angles congruent Kite Trapezoid X X

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