CHAPTER 8 QUADRILATERALS

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1 HTE 8 UILTEL In this chapter we address three ig IE: ) Using angle relationships in polygons. ) Using properties of parallelograms. 3) lassifying quadrilaterals by the properties. ection: Essential uestion 8 Find ngle Measures in olygons How do you find a missing angle measure in a convex polygon? Warm Up: Key Vocab: iagonal segment that joins two nonconsecutive vertices of a polygon. diagonals Theorems: olynomial Interior ngles Theorem: The sum of the measures of a convex n-gon is ( n ) 80 orollary - Interior ngles of a uadrilateral: The sum of the measures of the interior angles of a quadrilateral is olynomial Exterior ngles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is rea of a egular olygon: ap where a is the apothem and p is the perimeter. tudent Notes Geometry hapter 8 uadrilaterals KEY age #

2 how: Ex : Find the sum of the measures of the interior angles of a convex decagon Ex : The sum of the measures of the interior angles of a convex polygon is lassify the polygon by the number of sides. 340 n 80 3 n 5 n The polygon is an 5-gon Ex 3: Find the value of x in each of the diagrams shown below x x (x+0) x x x x x 66 x a) Ex 5: Find the area of each regular polygon. b) 7(0) b 8 b b.75 p 0b 46.6 (6.5)(46.6) a tan tan 7 a (3.5tan 7)(70) Interior angle measure: (0 ) tudent Notes Geometry hapter 8 uadrilaterals KEY age #

3 ection: Essential uestion 8 Use roperties of arallelograms How do you find angle and side measures in a parallelogram? Warm Up: Key Vocab: arallelogram quadrilateral with OTH pairs of opposite sides parallel. Theorems: a quadrilateral is a parallelogram, its opposite sides are congruent. and tudent Notes Geometry hapter 8 uadrilaterals KEY age #3

4 a quadrilateral is a parallelogram, its opposite angles are congruent. and. a quadrilateral is a parallelogram, its consecutive pairs of angles are supplementary. x y 80. x y y x a quadrilateral is a parallelogram, its diagonals bisect each other. M M and M M. M how: Ex: Find the values of x and y. G y-8 7 H x 7 y 8 36 F x 36 K tudent Notes Geometry hapter 8 uadrilaterals KEY age #4

5 Ex: Find the indicated measure. a) NM = b) KM = 4 c) m JML = 70 d) m KML = 40 Ex3: The diagonals of parallelogram intersect at point T. What are the coordinates of point T? 9., , 5., 7., y T x The diagonals of a parallelogram bisect each other, so T is the midpoint of T :,, losure: What are the properties of a parallelogram? parallelogram s opposite sides are parallel and congruent. Its opposite pairs of angles are congruent. Its consecutive pairs of angles are supplementary. Its diagonals bisect each. tudent Notes Geometry hapter 8 uadrilaterals KEY age #5

6 ection: Essential uestion 8 3 how that a uadrilateral is a arallelogram How can you prove that a quadrilateral is a parallelogram? Warm Up: Theorems: both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. and both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram. and. tudent Notes Geometry hapter 8 uadrilaterals KEY age #6

7 one pair of opposite sides of a quadrilateral is congruent N parallel, and O and, the quadrilateral is a parallelogram. M how: Ex: The figure shows part of a stair railing. Explain how you know the support bars M and N are parallel. ince M N and MN, MN is a parallelogram. Therefore, M N Ex: For what value of x is quadrilateral TU a parallelogram? 8x 3 4x 4x 3 x 8 4x T 8x-3 U tudent Notes Geometry hapter 8 uadrilaterals KEY age #7

8 Ex3: uppose you place two straight narrow strips of paper of equal length on top of two lines of a sheet of notebook paper. you draw a segment to join their left ends and a segment to join their right ends, will the resulting figure be a parallelogram? Explain. Yes, since the segments are congruent N the lines on the notebook paper are parallel, we can use the theorem that says one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram Ex4: how that FGHJ is a parallelogram. Option : how OTH pair of opposite sides congruent y G H Option : how one pair of opposite sides congruent N parallel (have the same slope.) Option 3: how OTH pair of opposite sides parallel. F x J For example: FJ=GH= 5 m m FJ GH FGHJ losure: How do you prove that a quadrilateral is a parallelogram? how that the quadrilateral has. both pair of opposite sides parallel.. both pair of opposite sides congruent. 3. one pair of opposites sides parallel N congruent. 4. both pair of opposite angles congruent. 5. diagonals that bisect each other. tudent Notes Geometry hapter 8 uadrilaterals KEY age #8

9 ection: Essential uestion 8 4 roperties of hombuses, ectangles, and quares What are the properties of parallelograms that have all sides or all angles congruent? Warm Up: Key Vocab: hombus parallelogram with four congruent sides E F ectangle parallelogram with four right angles H G m E m F m g m H 90 I J quare parallelogram with four congruent sides N four right angles L K IJ JK KL LI tudent Notes Geometry hapter 8 uadrilaterals KEY age #9

10 Theorems: hombus orollary quadrilateral is a rhombus IFF it has four congruent sides., quad is a rhombus. quad is a rhombus,. ectangle orollary quadrilateral is a rectangle IFF if it has four right angles. E F m E m F m g m H 90, quad is a rectangle, H G quad is a rectangle. m E m F m g m H 90. quare orollary quadrilateral is a square IFF it is a rhombus N a rectangle. I J IJ JK KL LI N m I m J m K m L 90, quad is a rectangle, L K quad is a square. IJ JK KL LI N m I m J m K m L 90. tudent Notes Geometry hapter 8 uadrilaterals KEY age #0

11 Theorems: parallelogram is a rhombus IFF its diagonals are perpendicular., is a rhombus. is a rhombus,. parallelogram is a rhombus IFF each diagonal bisects a pair of opposite angles. bisects N and bisects N, is a rhombus. is a rhombus, bisects N and bisects N. parallelogram is a rectangle IFF its diagonals are congruent. E F Z H G EG HF, EFGH is a rectangle, EFGH is a rectangle. EG HF. tudent Notes Geometry hapter 8 uadrilaterals KEY age #

12 how: Ex: For any rectangle, decide whether the statement is always, sometimes or never true. a.) lways; ll rectangle are parallelograms and opposite sides of a parallelogram are congruent. b.) ometimes; provided that the rectangle is a square. ut not all rectangles are squares. Ex: lassify the special quadrilateral. Explain your reasoning. It is a rhombus. Its is a parallelogram because opposite angles are congruent. ince a pair of adjacent sides are congruent, all four side are congruent. Ex3: You are building a case with glass shelves for collectibles. 4 in 4 in 4 in 4 in a.) Given the shelf measurements in the diagram, can you assume that the shelf is a square? Explain. No, it has four congruent sides so it is a rhombus. However, we do not know whether the angles are right angles. b.) You measure the diagonals and find they are both inches. What can you conclude about the shape? It is a square. tudent Notes Geometry hapter 8 uadrilaterals KEY age #

13 Ex4: ketch a square EFGH. List everything that you know about it. o Opposite sides are parallel o ll sides are congruent o ll angles are congruent right angles o The diagonals are congruent and perpendicular and they bisect each other o Each diagonal bisects a pair of opposite angles. losure: omplete the Venn diagram for the properties that are LWY true. ectangle hombus tudent Notes Geometry hapter 8 uadrilaterals KEY age #3

14 ection: Essential uestion 8 5 Use roperties of Trapezoids and Kites What are the main properties of trapezoids and kites? Warm Up: Key Vocab: Trapezoid quadrilateral with exactly one pair of parallel sides. ases The parallel sides of a trapezoid. ase ase ngles Either pair of angles whose common side is a base of a trapezoid. leg ase angles ase angles leg Legs The nonparallel sides of a trapezoid. ase E F Isosceles Trapezoid trapezoid with congruent legs. H G EH FG Midsegment of a Trapezoid segment that connects the midpoints of the legs of a trapezoid. M M is the midsegment tudent Notes Geometry hapter 8 uadrilaterals KEY age #4

15 Kite quadrilateral that has two pairs of consecutive congruent sides, but in which opposite sides are NOT congruent. and Theorems: trapezoid is isosceles IFF its base angles are congruent. E F EH FG, H H G O E F, G H G N E F EH FG. trapezoid is isosceles IFF its diagonals are congruent. E F HF EG, H G trapefgh is isosceles. trapefgh is isosceles, HF EG. tudent Notes Geometry hapter 8 uadrilaterals KEY age #5

16 midsegment is drawn in a trapezoid, Midsegment Theorem for Trapezoids it is parallel to each base N its length is one half the sum of the lengths of the bases. M is the midsegment of a trap M, N M M M a quadrilateral is a kite, its diagonals are perpendicular. kite a quadrilateral is a kite, exactly one pair of opposite angles is congruent. kite tudent Notes Geometry hapter 8 uadrilaterals KEY age #6

17 how: Ex: how that XYZW is a trapezoid. y slope YZ =slope XW Y slope XY 3 and slope ZW 4 X 3 Z Therefore, YZ XW and XY ZW. ince exactly one pair of sides are parallel, XYZW is a trapezoid. x W Ex: The top of the table in the diagram is an isosceles trapezoid. Find m N, m O, and m. M m m M 65 m N m O Ex3: In the diagram, HK is the midsegment of the trapezoid EFG. Find HK. 6 cm E 6 8 HK cm H K G 8 cm F Ex4: Find m in the kite shown m 68 m 40 tudent Notes Geometry hapter 8 uadrilaterals KEY age #7

18 ection: 8 7 rea of pecial uadrilaterals Essential uestion How can you find areas of special quadrilaterals? Warm Up: Formulas: rea of a arallelogram bh rea of a hombus dd rea of a Trapezoid ( ) hb b how: Ex: Find the area of the parallelogram. 9 h 6sin 45 (6sin 45 )(9) tudent Notes Geometry hapter 8 uadrilaterals KEY age #8

19 Ex: In the rhombus, 0 and 5. The area can be found in more than one way. Fill in the blanks for each formula, then compute the area..5 a).5 50 Uses parallelogram area formula b) Uses rhombus area formula c) Uses triangle area formula E Ex3: Find the area of the trapezoid. h 8sin 60 8 a 8cos 60 4 b (8sin 60 )(8 6) a h 8 tudent Notes Geometry hapter 8 uadrilaterals KEY age #9