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 360 0. olynomial Exterior ngles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 0. rea of a egular olygon: ap where a is the apothem and p is the perimeter. tudent Notes Geometry hapter 8 uadrilaterals KEY age #
how: Ex : Find the sum of the measures of the interior angles of a convex decagon. 0 80 440 Ex : The sum of the measures of the interior angles of a convex polygon is 340 0. 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. 5 80 50 4 x 540 0 00 9 84 x 54 0 9 (x+0) x 00 96 x 84 360 x x 0 96 3 x 66 x a) Ex 5: Find the area of each regular polygon. b) 7(0) 70 6.5 b 8 b 64 4.5 b.75 p 0b 46.6 (6.5)(46.6) 5. 45 3.5 7 a tan 7 3.5 3.5tan 7 a (3.5tan 7)(70) 377.0 Interior angle measure: (0 )80 4 4 0 44 7 tudent Notes Geometry hapter 8 uadrilaterals KEY age #
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
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
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 9 7., 5., 7., 0 9 8 7 6 5 4 3-3 4 5 6 7 8 9 0 - y T x The diagonals of a parallelogram bisect each other, so T is the midpoint of T 9 4 5 :,, 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
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
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
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. 6 5 4 3 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 -3 - - 3 4 5 6 7 8 J - - -3-4 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
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
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
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 #
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 33.94 inches. What can you conclude about the shape? It is a square. tudent Notes Geometry hapter 8 uadrilaterals KEY age #
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
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
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
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
how: Ex: how that XYZW is a trapezoid. y slope YZ =slope XW Y 6 5 4 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 -3 - - 3 4 5 6 7 8 - - -3-4 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 360 65 5 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. 84 360 40 84 m 68 m 40 tudent Notes Geometry hapter 8 uadrilaterals KEY age #7
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) 38. 6 45 tudent Notes Geometry hapter 8 uadrilaterals KEY age #8
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) 0 5 50 Uses rhombus area formula c) 4.5 6 50 Uses triangle area formula E Ex3: Find the area of the trapezoid. h 8sin 60 8 a 8cos 60 4 b 8 4 4 (8sin 60 )(8 6) 83. 8 60 a h 8 tudent Notes Geometry hapter 8 uadrilaterals KEY age #9