Quadrilaterals. Polygons Basics

Transcription

1 Name: Quadrilaterals Polygons Basics Date: Objectives: SWBAT identify, name and describe polygons. SWBAT use the sum of the measures of the interior angles of a quadrilateral. A. The basics on POLYGONS 3 or more sides formed by non-collinear segments Each side intersects with two other sides, one at each endpoint B. Identifying Polygons State whether the following figures are polygons are not Yes yes yes No No (not a segment) (can t have a two collinear sides) C. Naming Polygons (named by the number of sides they have) # of Sides Type of polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon Need to Memorize these!! 7 Heptagon 8 Octagon 9 Nonagon 0 Decagon Hendecagon Dodecagon N n-gon

2 Convex vs. Concave Convex Has no dips Convex Concave Concave- Has dips in the sides Examples: Identify the polygon and state whether it is convex or concave... Concave Octagon Convex Pentagon D. Regular Polygons If it is equiangular and equilateral Examples: Decide whether the polygon is regular No Yes No

3 Interior Angles of Polygon Theorem Number of degrees in the interior angles of a polygon = 80( n-) N is the number of sides Quadrilaterals A quad will always have 360 degrees. P Examples x x 80( n ) 50 3x 80(4 ) 50 3x 360 3x 0 x 70 S x x R 70 Q. Given the following diagram, find m<jkl 4x 3 7x 8 4x 3 5x 5 5x 80( n ) x 8 80(4 ) x m JKL 4x 3 x x x x 8 5x 5 5x

4 Properties of Parallelograms Objectives: SWBAT use properties of parallelograms in real-life situations. A. Parallelogram and its Properties Parallelogram~ A quadrilateral where both opposite sides are parallel to each other. Opposite Side Parallelogram Theorem If a quadrilateral is a parallelogram, then opposite sides are congruent Opposite Angles Parallelogram Theorem If a quadrilateral is a parallelogram, then the opposite angles are congruent Consecutive Angles Parallelogram Theorem If a quadrilateral is a parallelogram, then the Consecutive angles are supplementary mp ms 80 mp mq 80 mq mr 80 mr ms 80 Diagonals of a Parallelogram Theorem If a quadrilateral is a parallelogram, then diagonals bisect each other Examples:. FGHJ is a parallelogram. Find the unknown lengths. a. JH 5 F G b. JK K 3 J H Since it is a parallelogram, then the opposite sides are congruent, and the diagonals bisect each other. If FG=5, then the opposite side will be the same so JH=5. JH 5 Since the diagonals bisect each other, if KG = 3 then JK = 3. JK 3

5 . Find the angle measures. a. m R b. m Q Q R P 70 S Since it is a parallelogram, the opposite angles are congruent. mr mp mp70 Since it is a parallelogram, the consecutive (same side) angles are supplementary. mr mq mq 80 mq0 3. Find the value of x and y. Q R 3y 0 P 3x 0 Since it is a parallelogram, the opposite sides are congruent. S y 6 QP RS 3y0 y 6 y 6 y 8 Since it is a parallelogram, then the consecutive angles are supplementary. mp ms 80 3x x 60 x 0 4. Gracie, Kenny, Teresa, and Billy Bob live at the four corners of a block shaped like a parallelogram. Gracie lives 3 miles away from Kenny. How far apart do Teresa and Billy Bob live from each other? 3 miles because opposite sides of parallelograms are congruent

6 Proving Quadrilaterals are Parallelograms Objectives: SWBAT prove that a quadrilateral is a parallelogram. Definition of a Parallelogram If both pairs of opposite sides are parallel to the corresponding opposite side in a quadrilateral, then it is a parallelogram. Converse Opposite Side Parallelogram Theorem If both pairs of opposite sides are congruent to the corresponding opposite side in a quadrilateral, then it is a parallelogram. Converse Opposite Angle Parallelogram Theorem If both pairs of opposite angles are congruent to the corresponding opposite angle in a quadrilateral, then it is a parallelogram. Converse Consecutive Angles Parallelogram Theorem If one angle of a quadrilateral is supplementary (add up to 80) to both of the angles on its sides, then it is a parallelogram. Converse Diagonals of a Parallelogram Theorem If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Parallel and Congruent Parallelogram Theorem B C If one pair of opposite sides are congruent and parallel then it is a parallelogram A D D

7 Examples: Determine if the following quadrilaterals are parallelograms. If so why or why not?.. 3. Yes since it has two pairs of opposite No, you only have Yes, since pair of Congruent sides, then it satisfies the one pair of parallel sides of sides are Converse Opposite Side Parallelogram sides. parallel and congruent Theorem then it satisfies the Parallel and Congruent Parallelogram Theorem Yes since it has an angle that is supplementary, to both of the other angles it neighbors, then it satisfies the Converse Consecutive Angle of a Parallelogram Theorem No, you can not mix And match theorems to Prove that a quadrilateral is a Parallelogram. Yes, since both of the opposite angles are congruent to their corresponding opposite angles, then it satisfies the Converse Opposite Angles of a Parallelogram theorem.

8 Solve for the following variables so that the quadrilateral is a parallelogram Oppo. Sides Are 7x 5x 6 x 8 x 9 Consec utive Angles Are Supplementary 7 3y 80 3y 08 y 36 Diagonals Bi sec t Each other 5 30x x Oppo. Angles Are 3y 4x x 08 4x x 7 Diagonals Bi sec t Each other 0 4y y 6

9 Trapezoids and Kites Objectives: SWBAT use properties of trapezoids and kites. A. Trapezoids pair of parallel sides, bases, legs, and two pair of base angles Isosceles Trapezoid leg base base leg Legs are congruent B. Trapezoid Theorems Isosceles Trapezoid Angle Theorem If a trapezoid is isosceles, then the base angles are congruent A B D C Converse of the Isosceles Trapezoid Angle Theorem If a pair of base angles are congruent, then it is Isosceles trapezoid Isosceles Trapezoid Diagonals Theorem A trapezoid is isosceles if and only if the diagonals are congruent Examples: PQRS is an isosceles trapezoid. BD AC AE BE DE CE S Find m P: mp mq 360 x Find m Q: 0 0 x x 0 Find m R: 70 P Q R

10 Midsegment of a Trapezoid C All three segments are parallel BC MN AD MN ( AD BC ) D 3. M and N are the midpoints on PS and QR, respectively. Find the length of the midsegment. P Q MN AD BC M N 8 S 8 R A riser is designed to elevate a speaker. The riser consists of 4 trapezoidal sections that can be stacked one on top of the other to produce trapezoids of varying heights. If only the bottom two stages are used, what is the width of the top resulting riser? Middle Bottom Top Top of Bottom Two Risers Bottom New Top

11 KITE ~ A quadrilateral with exactly two consecutive congruent sides. Kite s Perpendicular Diagonals Theorem If a quadrilateral is a kite, then the diagonals are perpendicular Kite Opposite Angle Theorem If a quadrilateral is a kite, then exactly one pair of Opposite angles are congruent. Examples:. MQNP is a kite (so the diagonals are perpendicular). Find MN, NQ, NP, PQ, and the Perimeter If it is a kite, then the diagonals are congruent (so we can use Pythagorean theor3em) MN RN MR MN MN NP NR RP NP 6 NP 40 MN 0 NP 0 Perimeter MN NP PQ QM MN MQ NP PQ MQ 0 Perimeter PQ 0 Perimeter Find a. 3. Find x Kite is a quad, so internal angles add up to 360 a a a 80 Kites have one pair of Congruent Angles x x x x 0 x 05

12 4. Given Kite ABCD, find x, y, and z. AC BD z 90 BDC is isosceles BC DC CBD CDB y 35 BAD is isosceles BA DA BAC DAC x 8 The diagonal RB of kite RHBW forms an equilateral triangle with two of the sides, and m < BWR = 40. Draw and label a diagram showing the diagonals, and the measures of all the angles.

13 Rhombuses and Rectangles Objectives: SWBAT use properties of sides and angles of rhombuses and rectangles SWBAT use properties of diagonals of rhombuses and rectangles A. Properties Rhombus~ A parallelogram with four congruent sides. Diagonals are perpendicular and bisect each angle Rectangle~ A parallelogram with four right angles. Diagonals are all congruent. True or false, if false explain why it is false.. A rectangle is a parallelogram. True. A parallelogram is always a rhombus. False, but a rhombus is always a parallelogram. 3. A rhombus is always a Rectangle. False, it they have different properties. 4. A rhombus is always a Kite. False, Kites Congruent Sides are consecutive. 5. A rhombus is always a parallelogram. True Matching: Which of the following quadrilaterals has the given property?. All sides are congruent. C A. Parallelogram. All angles are congruent. B B. Rectangle 3. The diagonals are congruent. B C. Rhombus 4. Opposite angles are congruent. A, B, C D. Quadrilateral 5. Sum of Interior angles equals 360 degrees. A,B,C,D

14 What is the value of x and y in the rectangle to the right? Oppo. Sides Are PS QR 4x 0 x 5 Corner Angles Are mpsr mpsr mpsq mqsr y x y y y y 5 Answer is A

15 Use the rhombus PRYZ, to find the measurements of the following given that PK = 4y + RK = 3x - KY = 7y 4 KZ = x + 6 A) PY 4y 7 y4 3y 4 5 3y 5 y 4y PK 4 5 PK 0 PK PK PY PY PY PK 4 B) RZ 3x x 6 x 6 x 7 3x RK 3 7 RK RK 0 RK RZ RZ RZ RK 0 40 C) m YKZ Diagonals are perpendicular, therefore m YKZ 90 D) YZ KY KZ YZ 0 YZ YZ 9 YZ YZ YZ

16 4. Based on the figure below, which statements are true? I. II. III. IV. The figure is a rectangle (False, not all diagonals are congruent) The figure is a parallelogram (True opposite sides are congruent 6x 4 = 9x + 3 (False, it is not a rectangle so not all diagonals are congruent) 9x + 3 = 0x (True, it is a parallelogram so diagonals bisect each other). V. x = 8 (False, x = 5) VI. The longest side has a length of 60. True, (5)=60 Since it is a Parellelogram 9x 3 0x 9 x 9 x 3 x x 5 Since it is a Parellelogram 6x 4 5x 5 x 5 x x x 5 If is is a Rectangle, all Diagonals are congruent 9x x x x Not A Rectangle

17 Squares Objectives: SWBAT use properties of sides and angles of Squares SWBAT use properties of diagonals of Squares A. Properties Square~ a parallelogram with four congruent sides and four right angles. Examples: Identify each figure as a quadrilateral, parallelogram, rhombus, rectangle, trapezoid, kite, square or none of the above. Square Rectangle Quadrilateral Parallelogram Kite Trapezoid

18 . Given that the figure to the right is a square, find the length of a side. All Sides Are x4 x5 x 9 Sides 44 x 4 x 5. The quadrilateral at the below is a square. Solve for x and y. All Sides Are y 6 y 3y 8 y 6 Sides 0 Since the Sides 0 5x 0 0 5x 0 x 4 y y 4 6 y 5x 0 x= y= 3. Given Square SQUR, find the following. EQ = 6 mm. m SEQ = 90 degrees EU = 6 mm. m SQU = 90 degrees SU = 3 mm. m UEQ = 90 degrees RU = 6 mm. m SQE = 45 degrees RE 6 mm.

19 Quadrilateral Family Tree Objectives: SWBAT identify special quadrilaterals. The Family Tree Quadrilateral (Interior Angles Add up to 360) Kite Parallelogram Trapezoid Rectangle Rhombus Isosceles Trapezoid Square

20 Shape Description of Sides Description of Angles Interesting Information Quad 4 non congruent nor parallel sides No equal angles All angles add up to 360 degrees Trapezoid set of parallel sides (no congruent sides) No equal angles but angles by legs are supp. Nevada is close to a right trapezoid Isosceles Trapezoid Kite Parallelogram Rectangle Rhombus Square pair of parallel sides and pair of congruent sides (not same) No parallel sides pairs of neighboring congruent sides pairs of congruent and parallel sides pairs of congruent, parallel and perpendicular sides pairs of parallel sides and all 4 sides are congruent pairs of parallel and perpendicular sides and all 4 sides are congruent Base angles are congruent and angles by legs are supp. set of congruent angles. Opposite angles are congruent and neighboring angles are supp. Diagonals are congruent Diagonals are perpendicular. Diagonals Bisect Each other 4 Right angles Diagonals are congruent. Opposite angles are congruent and neighboring angles are supp. It is a parallelogram. Diagonals are perpendicular. It is a parallelogram. 4 Right angles Diagonals are congruent, an perpendicular. It is a Rectangle and Rhombus

21 Fill in the table. Put an X in the box if the shape always has the property. Property Parallelogram Rectangle Rhombus Square Trapezoid Both pairs of opp. sides Exactly pair of opp. sides Diagonals are Diagonals are Diagonals bisect each other X X X X X X X X X X X X X 3. Quadrilateral ABCD has at least one pair of opposite sides congruent. Draw the kinds of quadrilaterals meet this condition (5). Parallelogram Square Rhombus Rectangle Isosceles Trapezoid

22 COORDINATE QUADS Objectives: SWBAT identify types of quads using a coordinate plane. Characteristic Definition Formula Congruent Equal Sides / Angles d x x y y Perpendicular Opposite Reciprocal Slope Negative Reciprocal Slope y y m x x Parallel Same Slope y y m x x Midsegment / Midpoint Bisecting a segment or is the middle x x, y y Coordinate Proof ~ Using algebra and formulas on a coordinate plane to prove a statement.. Use the coordinate plane, and the Distance Formula to show that KLMN is a Rhombus. K(, 5), L(-, 3), M(, ), N(6, 3) KL LM MN NK,5,3,3,, 6,3 6,3, Since all sides are congruent, the quadrilateral is a rhombus

23 Use slope or the distance formula to determine the most precise name for the figure: A(, 4), B(, ), C(4, ), D(, ). Graph it first, that will give you some insight into the shape, and allow you to eliminate obviously wrong answers. A. Kite B. Rhombus C. Trapezoid D. Square Now we know that it is a kite or rhombus use the slope to see if the opposite sides are parallel or not. BC, 4, m 4 m 3 AD, 4, 4 m m 3 AB, 4, 4 m m 3 CD 4,, m 4 3 m Since BC AD and AB CD it must be B 3. Given points B ( 3,3), C(3, 4), and D(4, ). Which of the following points must be point A in order for the quadrilateral ABCD to be a parallelogram? A. A(, ) B. A(, ) C. A(, 3) D. A( 3, ) Since it is a parallelogram, it has the same slope as CD which is m = rise = 6 run. So start at point B, move the same slope of m = 6, which lands you at A(, 3), so the answer is C.

24 4. Given a Trapezoid ( 3,4), B( 5, ), C(5, ), and D(3,4). Find the following a) Is the trapezoid Isosceles? AB 3, 4 5, , 4 5, DC Since AB DC then Yes b) What are coordinates of the midsegment for the trapezoid? 3, 4 5, 3 5 4, 8, 4, AB 3, 4 5, 4 3 5, 8, DC 4, c) What is the length of the midsegment? Midsegment 4, 4,

25 Classifying Quadrilaterals YOU HAVE A KITE ZERO YOU HAVE A TRAPEZOID ONE How many pairs of parallel sides? TWO ARE THE 4 SIDES EQUAL? YES! NO YOU HAVE A RHOMBUS NO YES! ARE THE 4 CORNERS 90 DEGREES? YES! ARE THE 4 CORNERS 90 DEGREES? NO YOU HAVE A SQUARE YOU HAVE A RECTANGLE YOU HAVE A PARALLELOGRAM

26 Distance between a Line and a point Objectives: SWBAT find the distance between a line and a point Shortest distance between two points is always a straight line that is perpendicular to both points. Steps for Success: ) Find the equation perpendicular to the given line through the point given. ) Find the point where the two lines intersection using substitution. 3) Use the distance formula to find the distance between the given point and Just watch an example: Find the distance between the line y = x + 5, and the point (7, ) Step Step Step 3 m point (7, ) y y m x x y 7 y x x 7 5 y x 5 y x y x5 Since they both equal y 5 x x 5 Multiply everything by to cancel out the fraction 5 x x 5 x 5 4x0 x x 5 5x x x now plug the x value back int o one of the equations and solve for y y x5 y 5 y 5 y 3 Point,3