CHAPTER 6. SECTION 6-1 Angles of Polygons POLYGON INTERIOR ANGLE SUM

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1 HPTER 6 Quadrilaterals SETION 6-1 ngles of Polygons POLYGON INTERIOR NGLE SUM iagonal - a line segment that connects two nonconsecutive vertices. Polygon interior angle sum theorem (6.1) - The sum of the measures of the interior angles of an n sided convex polygon is (n-)

2 EXMPLE 1 Find the sum of the measures of the interior angles of a convex nonagon n = 9 ( n )180 ( 9 )180 ( 7) EXMPLE The alien overlords that conquered us in chapter 5 have constructed new cities in which the humans live. They have everything we used to have, but the architecture is a bit odd. The human malls, for example, has 5 hallways that come together to form a food court that is a regular pentagon. Find the measure of one interior angle in that pentagon.! n = 5 ( n )180 ( 5 )180 ( 3) EXMPLE 3 Find the number of sides of a regular polygon that has an interior angle measure of 150 ( n )180 = 150n 180n 360 = 150n 30n = 360 n = 1

3 POLYGON EXTERIOR NGLE SUM Polygon Exterior ngle Sum Theorem (6.) - The sum of the exterior angles of a polygon, with one angle at each vertex is EXMPLE 4 Find x 5x + 4x 6 + 5x 5 + 4x x 1 + x x + 5 = 360 4x 6 5x x 1 = 360 5x 5 4x + 3 5x x + 3 6x 1 SETION 6- Parallelograms 31x = 37 x = 1

4 SIES N NGLES OF PRLLELOGRMS Parallelogram - a quadrilateral where both pairs of opposite sides are parallel. Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent Theorem If a quadrilateral is a parallelogram then its opposite angles are congruent Theorem If a quadrilateral is a parallelogram then its consecutive angles are supplementary Theorem If a parallelogram has one right angle then it has 4 right angles,, m + m = 180 m + m = 180 m + m = 180 m + m = 180 EXMPLE 1 In m = 3, = 80 and = 15 find: =15 m =148 m = IGONLS OF PRLLELOGRMS Theorem If a quadrilateral is a parallelogram then the diagonals bisect each other Theorem If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles X X X X X

5 EXMPLE Find the value of each indicated variable in the parallelogram: r = 4r = 18 s t r = 9 ( t) X = X 8s = 7s + 3 s = 3 4r 7s + 3 m = m t = 18 t = 9 8s 18 X EXMPLE 3 Given the coordinates of the parallelogram, what are the coordinates of the intersection of the diagonals? M(-3,0), N(-1,3), P(5, 4) and R(3,1) What do we know about the diagonals of a parallelogram? What is the point of intersection of the diagonals on each diagonal? What are the names of the diagonals? ( 1, ) M N R P EXMPLE 4 Given:!, & are diagonals that intersect at P Prove: & bisect each other Statement Reason Given Opp. sides of a parallelogram are congruent P P vert. angle thm P! ef. of parallelogram P P lt. int. angle thm P P S P P, P P PT P is the midpoint of & ef. of a midpoint & bisect each other ef of segment bisector

6 SETION 6-3 Tests for Parallelograms ONITIONS FOR PRLLELOGRMS Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram Theorem If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram EXMPLE 1 Is it a parallelogram? Justify your answer NO 55 NO YES YES YES NO

7 EXMPLE The end of the 198 tari 600 video game Raiders of the Lost rk shows Indy on a platform. The closer you are to the ark, the more of the game you completed, like a percent complete meter. Explain why consecutive angles on the scissor lift platform will always be supplementary EXMPLE They are all parallelograms, and consecutive angles are supplementary EXMPLE 3 Find x and y to make the quadrilateral a parallelogram 4x 1 3( y +1) 4y 3( x + ) 4x 1 = 3( x + ) 4x 1 = 3x + 6 x 1 = 6 x = 7 3 y +1 ( ) = 4y 3y + 3 = 4y y + 3 = y = 5 y = 5

8 SUMMING IT LL UP To show that a quadrilateral is a parallelogram: Show that both pairs of opposite sides are parallel Show that both pairs of opposite sides are congruent Show that both pares of opposite angles are congruent Show that the diagonals bisect one another Show that a pair of opposite sides is both parallel and congruent EXMPLE 4 What is the easiest way to show that QRST is a parallelogram given Q(-1, 3), R(3, 1), S(, -3) and T(-, -1) Use the slope formula to show that opposite sides are parallel m QR = 1 m ST = 1 m RS = 4 m TQ = 4 You could, of course, always put coordinate geometry problems on a graph as part of your work. EXMPLE 5 Write a coordinate proof for the following: If a parallelogram has one right angle, then it has 4 right angles. We need to create a parallelogram, but we need to make sure we have one right angle. The easiest way to do that is to use horizontal and vertical lines, because they don t take much effort to put on the graph. oordinate proofs are general cases, they don t use any actual numbers other than zero in the coordinates.

9 EXMPLE 5 Write a coordinate proof for the following: If a parallelogram has one right angle, then it has 4 right angles. 0 a = undef m = =0 b 0 a 0 m = = undef (0,a) b b a a m = =0 0 b (0,0),,, m = (b,0) (b,a) SETION 6-4 Rectangles PROPERTIES OF RETNGLES Rectangle - a parallelogram with 4 right angles. y definition, a rectangle has the following properties: ll four angles are right angles Opposite Sides are parallel and congruent Opposite angles are congruent onsecutive angles are supplementary iagonals bisect each other Theorem If a parallelogram is a rectangle, then the diagonals are congruent

10 PROPERTIES OF RETNGLES Rectangle - a parallelogram with 4 right angles. y definition, a rectangle has the following properties: ll four angles are right angles Opposite Sides are parallel and congruent Opposite angles are congruent onsecutive angles are supplementary iagonals bisect each other Theorem If a parallelogram is a rectangle, then the diagonals are congruent EXMPLE 1 Find: = = 6 m E = 50 m E = 5 First, we know opposite sides of a rectangle are congruent. Second, if the diagonals bisect each other and are congruent to each other that effectively makes 4 isosceles triangles, which makes finding angle measures much easier E Third, we know we have right angles, which means we have a bunch of complementary angles that we can use to find our remaining missing angles EXMPLE WXYZ is a rectangle. find WX if ZY = x+3 and WX = x+4 W Y P X Z WX = ZY x + 4 = x + 3 x + 4 = 3 x = 1 x = 1 WX = x + 4 WX = 1 ( ) + 4 WX = 5

11 PROVING PRLLELOGRMS RE RETNGLES Just like you can prove a quadrilateral is a parallelogram, you can prove a parallelogram is a rectangle. Theorem If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. E Given! if then! is a rectangle EXMPLE 3 Given: is a rectangle Prove: Honestly, there are so many different ways to make this one work, I didn t even bother to make a proof, we are just going to work it out by hand. EXMPLE 4 Given the quadrilateral W(-, 4), X(5, 5), Y(6, -) and Z(-1, -3) determine YES if it is a rectangle.

12 SETION 6-5 Rhombi and Squares PROPERTIES OF RHOMI N SQURES Rhombus - a parallelogram with 4 congruent sides Theorem If a parallelogram is a rhombus then its diagonals are perpendicular Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles E E E, E E E E, E E EXMPLE 1 Find the m given the rhombus E 39.5 Option 1 Use the fact that diagonals of a rhombus are angle bisectors to say that m = 39.5 and m = 79. Since a rhombus is a parallelogram and consecutive angles are supplementary m = 101 Option Use the fact that diagonals are perpendicular in a rhombus making m E = 90 and use the triangle angle sum thm to find m = Since diagonals are angle bisectors that means m = 101. rhombus is a parallelogram so opposite angles are congruent and m = 101

13 PRLLELOGRMS Square - a parallelogram with 4 congruent sides and 4 right angles. square has both the properties of a rectangle and a rhombus. PROVING QURILTERLS RE RHOMI OR SQURES Theorem If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus Theorem If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus Theorem If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus Theorem If a quadrilateral is both a rectangle and a rhombus, then it is a square EXMPLE Given:!, 1, 6 Prove: is a rhombus E 5 6 Statement! 1 5 bisects & is a rhombus Given Reason ef. of a parallelogram lt. int. thm ef. of bisector If a diagonal of a parallelogram bisects opposite angles the parallelogram is a rhombus

14 EXMPLE 3 lassify the quadrilateral made where the paths of equal width intersect RHOMUS EXMPLE 4 Given (-, -1), (-1, 3), (3, ) and (, -) determine if is a rhombus, rectangle or square. EXMPLE 4 Given (-, -1), (-1, 3), (3, ) and (, -) determine if is a rhombus, rectangle or square. We know it is a parallelogram, so opposite sides will be congruent and parallel First we need to see if we have a right angle, if we do, it will eliminate rhombus 3 1 m = ( ) 1 ( ) m = 4 1 m = 4 m = ( ) m = 1 4

15 EXMPLE 4 Given (-, -1), (-1, 3), (3, ) and (, -) determine if is a rhombus, rectangle or square. 3 3 ( 1) m = m = 3 ( 1) 1 ( ) 1 4 m = 4 m = m = 4 1 Now, we know it is either a rectangle or square, so we need to check and see if non opposite sides are congruent. EXMPLE 4 Given (-, -1), (-1, 3), (3, ) and (, -) determine if is a rhombus, rectangle or square. = ( 1 ( )) + ( 3 ( 1)) = (1) + ( 4 ) = = ( 4 ) + ( 1) = 17 SETION 6-6 Trapezoids and Kites = 17 ( 3 ( 1)) + ( 3) = = Square

16 TRPEZOIS Trapezoid - a quadrilateral with exactly one pair of parallel sides ase(s) - the parallel side(s) of a trapezoid! Legs - the nonparallel sides of a trapezoid & are bases Isosceles trapezoid - a trapezoid with congruent legs & are legs in an isosceles trapezoid E TRPEZOIS Theorem If a trapezoid is isosceles then each pair of base angles is E congruent. Theorem 6. - If a trapezoid has one pair of congruent base angles, then it is, an isosceles trapezoid Theorem trapezoid is isosceles if and only if its diagonals are congruent EXMPLE 1 K Given the isosceles trapezoid, find: m MJK =117 JK = L J M

17 EXMPLE Given (-, 5), (-3, 1), (6, 1), (3, 5) verify it is a trapezoid and determine if it is an isosceles trapezoid Start with the idea that in a trapezoid, the bases are parallel, so check the slope of each side and determine which sides are the bases m = 4 m = 0 m = 4 3 m = 0 Now that we know it is a trapezoid, see if =, making it isosceles = = = 1+16 = 17 ( 3 ( ) ) + ( 1 5) ( 1) + ( 4) = ( 3 6) + ( 5 1) = ( 3) + ( 4) = = 5 = 5 It is NOT isosceles MISEGMENTS Midesgment - a line segment that connects the midpoints of the legs of a trapezoid Trapezoid midsegment theorem (6.4) - The midsegment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases X X X, Y! XY! XY = Y + EXMPLE 3 Given the trapezoid, if = 1 and = find XY X Y + XY = 1 + XY = XY = 34 XY = 17

18 PROPERTIES OF KITES Kite - a quadrilateral with exactly two pairs of consecutive congruent sides. Opposite sides of a kite are neither congruent nor parallel Theorem If a quadrilateral is a kite, then its diagonals are perpendicular Theorem If a quadrilateral is a kite then exactly one pair of opposite angles is congruent E, EXMPLE 4 Find the perimeter of the kite E + E = E + E = = = = = 100 = = 10 = = = = 8 10 = = E P =

19 FINL EXMPLE What is given (-1, 4), (, 6), (3, 3) and (0, 1)?