Formal Geometry UNIT 6 - Quadrilaterals

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1 Formal Geometry UNIT 6 - Quadrilaterals 14-Jan 15-Jan 16-Jan 17-Jan 18-Jan Day 1 Day Day 4 Kites and Day 3 Polygon Basics Trapezoids Proving Parallelograms Day 5 Homefun: Parallelograms Pg #1 19, 1 3, 7 9, 30 33, 47, 50, Homefun: Pg # 8 11, 16 even, 4 7, 41 44, 65, 66, 71, 75 Homefun: Day 3 Worksheet Homefun: Day 4 Worksheet Early Unit Review 1-Jan -Jan 3-Jan 4-Jan 5-Jan Day 1 4 Day 6 Day 7 Day 8 MLK JR Day Mini Celebration of Knowledge Rhombuses and Rectangles Homefun: Day 6 Worksheet Squares Homefun: Day 7 Worksheet Quadrilateral Family Tree Homefun: Day 8 Worksheet 8-Jan 9-Jan 30-Jan 31-Jan 1-Feb Quick Mini Cele on Properties Day 9 Coordinate Proofs Homefun: Day 9 Worksheet Day 10 Proof Workshop Part I Homefun: Day 10 Worksheet Day 11 Proof Workshop Part II Homefun: Day 11 Worksheet Unit Review Celebration of Knowledge

2 The basics on POLYGONS Day 1 Polygons Basics Objectives: SWBAT identify, name and describe polygons. SWBAT use the sum of the measures of the interior angles of a quadrilateral. 3 or more sides formed by non-collinear segments Each side intersects with two other sides, one at each endpoint Concave Convex Regular Polygons Has dips No Dips All sides and Angles are Congruent Naming Polygons Named by the number of sides Polygon # of Sides # of Triangles Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon Sum of the Interior Angle Measures n-gon n (n-) (n )180

3 Interior Angles of Polygon Theorem n 180 N= number of sides Exterior Angles of Polygon Theorem Exterior or external angles add up to 360s degrees Find the value of x Find the angle of a reagular 14 - gon n 180 n 14 (14 ) n 180 n 5 (5 ) If the sides are, then angles are (given the diagram s shape and convex) n 180 n 7 (7 ) x 160 x (4x 3) 101 (4x 3) 5x x x 349 x x 900 x 15.57

4 You are given the measure of each interior angle of a regular n-gon. Find the number of sides. Regular means all the angles are congruent mangle of a regular polygon n 180 n m n 180 n n 180 n n 180 n m n 180 n n 180 n n 180 n m n 180 n n 180 n n 180 n Cross Mulitply Cross Mulitply Cross Mulitply 108n n n180n n 180n 7n 360 n 5 10n n n180n n 180n 60n 360 n 6 140n n n180n n 180n 40n 360 n 9 You are given the number of sides of a regular polygon. Find the measure of each exterior angle external n 360 external external n 360 external external n 360 external 8 45

5 10. Find the value of x. Sum of external x 3x x x 150 x 5

6 Day Trapezoids and Kites Objectives: SWBAT use properties of trapezoids and kites. Trapezoids A Quad with 1 pair of parallel sides, bases, legs, and two pair of base angles Isosceles Trapezoid leg base base leg A trapezoid with congruent legs 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 AD CB 1. PQRS is an isosceles trapezoid. Find m P: 110 Find m Q: 110 Find m R: mp mq 360 x x x 110

7 . ABCD is an trapezoid. Find m R: Angles R and V are consecutive Interior (SUPP) mr mv 180 mr mr147 Find m N: Angles N and E are consecutive Interior (SUPP) mn me 180 mn mn Determine if the following trapezoid is an isosceles trapezoid. Given: BC = 3, CD = 4, FG = 5, CD EG, BF GE If it is an isosceles trapezoid then BD FG BCD is a right triangle so a b c BC CD BD Midsegment of a Trapezoid x 5 x x 5 YES IT IS AN ISOSCLES TRAPEZOID x All three segments are parallel BC MN AD Midsegment is the average length of the two bases 1 MN ( AD BC )

8 4. M and N are the midpoints on PS and QR, respectively. Find the length of the midsegment. 1 QU PT SR M and N are the midpoints on PS and QR, respectively. Find its bases. 1 NR MS PQ 1 x 1 x 4 x 6 1 x1 x x1 x1 x 11 MS x 4 MS 114 MS 15 PQ x 6 PQ 116 PQ 5 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.

9 Sides of a Kite Theorem A quadrilateral with exactly two consecutive congruent sides. 6. 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 MN 10 MN MQ MQ 10 NP NR RP NP 6 NP 40 NP 10 NP PQ PQ 10 Perimeter MN NP PQ QM Perimeter Perimeter 04 10

10 7. Find a. 8. 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 10 x 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 18

11 10. 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.

12 Day 3 Properties of Parallelograms Objectives: SWBAT use properties of parallelograms in real-life situations. 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 180 mp mq 180 mq mr 180 mr ms 180 Diagonals of a Parallelogram Theorem If a quadrilateral is a parallelogram, then diagonals bisect each other

13 1. FGHJ is a parallelogram. Find the unknown lengths. a. JH = 5 Since it is a parallelogram, then the opposite sides are congruent, b. FH = 3 Since it is a parallelogram, the diagonals bisect each other. Given the following are all parallelograms.. Find the angle measures. a. m R Since it is a parallelogram, the opposite angles are congruent. mr mp mp70 b. m Q Since it is a parallelogram, the consecutive (same side) angles are supplementary. mr mq mq 180 mq110 c. m S Since it is a parallelogram, the consecutive (same side) angles are supplementary. mr ms ms 180 ms Find the value of x and y. Q R 3x x 60 x 0 3y10 y 6 y 16 y 8 3y 10 P 3x 10 S y 6

14 4. Solve for the following variables. Opposite Angles are Congruent z 70 Alternate interior angles are congruent y degrees in a triangle x y x x x 9 5. Solve for the variables. y10 3y y 10 z 70 x x y x 9 x 3

15 6. Find the following segments. 3x3y1 3y13x 3y13x 3 y 7 x You will have to use substitution to solve the following example Diagonals of parallelograms bisect each other 1 x 4y 6y x 4 y 4y 1 x y x y 7 x 1 x 7 x x 1 x 14 x x 1 4x14 x 7 x 14 x 4 y 7 x y 74 y 3 AE x 4y AE AC AE EC BE ED BD AE AE 0 BD = AE = AC = 7. Given LJHG GK x 7x KJ 4 6x Find the value of GK x 7x 4 6x x GK KJ x 4 0 x x x 7 0 x 6 0 x 7 x 6 GK x 7x 7 7(7) GK 0 GK x 7x 6 7( 6) 36 4 GK 78

16 Day 4 Proving Quadrilaterals are Parallelograms Definition of Parallelogram Objectives: SWBAT prove that a quadrilateral is a parallelogram. Converse Opposite Side Parallelogram Theorem Converse Opposite Angle Parallelogram Theorem Converse Consecutive Angles Parallelogram Theorem Converse Diagonals of a Parallelogram Theorem Parallel and Congruent Parallelogram Theorem

17 Examples: Determine if the following quadrilaterals are parallelograms. If so why or why not? Solve for the following variables so that the quadrilateral is a parallelogram

18 10. Which of the following is not always true of parallelogram ABCD? A. AB BC, DC BC B. AB DC, BC AD C. m A + m B = 180 D. AB + BC = AD + DC 1. A wooden frame has screws at A, B, C, and D so that the sides of it can be pressed to change the angles occurring at each vertex. AB CD and AB CD, even when the angles change. Why is the frame always a parallelogram? A. The angles always stay the same, so ABCD is a parallelogram. B. All sides are congruent, so ABCD is a parallelogram. C. One pair of opposite sides is congruent and parallel, so ABCD is a parallelogram. D. One pair of opposite sides is congruent, so ABCD is a parallelogram.

19 Day 6 Rhombuses and Rectangles Objectives: SWBAT use properties of sides and angles of rhombuses and rectangles SWBAT use properties of diagonals of rhombuses and rectangles Review: 5 Parts of a Parallelogram: Rhombus Rectangle True or false, if false explain why it is false. 1. A rectangle is a parallelogram.. A parallelogram is always a rhombus. 3. A rhombus is always a Rectangle. 4. A rhombus is always a Kite. 5. A rhombus is always a parallelogram.

20 Matching: Which of the following quadrilaterals has the given property? 6. All sides are congruent. A. Parallelogram 7. All angles are congruent. B. Rectangle 8. The diagonals are congruent. C. Rhombus 9. Opposite angles are congruent. D. Quadrilateral 10. Sum of Interior angles equals 360 degrees. 11. What is the value of x and y in the rectangle to the right? 1.

21 13. Use the rhombus PRYZ, to find the measurements of the following given that A) PY PK = 4y + 1 RK = 3x - 1 KY = 7y 14 KZ = x + 6 B) RZ C) m YKZ D) YZ 14. Based on the figure below, which statements are true? I. The figure is a rectangle II. The figure is a parallelogram III. 6x 4 = 9x + 3 IV. 9x + 3 = 10x V. x = 8 VI. The longest side has a length of 60.

22 Day 7 Squares Objectives: SWBAT use properties of sides and angles of Squares SWBAT use properties of diagonals of Squares Review: 5 Parts of a Parallelogram: Square Identify each figure as a quadrilateral, parallelogram, rhombus, rectangle, trapezoid, kite, square or none of the above.

23 1. Given Square ABCD, find the following. AD = CD = BD = m DCB = m BAD = m CBD =. Given that the figure to the right is a square, find the length of a side. x 14 x The quadrilateral at the below is a square. Solve for x and y. y y 4 16 y x= y= 5x Given Square SQUR, find the following. RE 16 mm. EQ = m SEQ = EU = m SQU = SU = m UEQ = RU = m SQE =

24 Day 8 Quadrilateral Family Tree Objectives: SWBAT identify special quadrilaterals.

25 Shape Description of Sides Description of Angles Interesting Information Quad Trapezoid Isosceles Trapezoid Kite Parallelogram Rectangle Rhombus Square

26 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 1 pair of opp. sides Diagonals are Diagonals are Diagonals bisect each other Quadrilateral ABCD has at least one pair of opposite sides congruent. Draw the kinds of quadrilaterals meet this condition (5).

27 Day 9 COORDINATE QUADS Objectives: SWBAT identify types of quads using a coordinate plane. Characteristic Definition Formula Congruent Perpendicular Parallel Midsegment / Midpoint Coordinate Proof 1. Use the coordinate plane, and the Distance Formula to show that KLMN is a Rhombus. K(, 5), L(-, 3), M(, 1), N(6, 3)

28 . Use slope or the distance formula to determine the most precise name for the figure A( 1, 4), B(1, 1), C(4, 1), D(, ). A. Kite B. Rhombus C. Trapezoid D. Square 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(, 1) B. A( 1, ) C. A(, 3) D. A( 3, )

29 4. Given a Trapezoid ( 3,4), B( 5, ), C(5, ), and D(3,4). Find the following a) Is the trapezoid Isosceles? b) What are coordinates of the midsegment for the trapezoid? c) What is the length of the midsegment?

30 Classifying Quadrilaterals ZERO ONE TWO YES! NO NO YES! YES! NO

31 Day 10 Proof Workshop Part 1 Objectives: SWBAT do Proofs involving Quadrilaterals 1. Given: Diagram at the right Prove: ABCD is a parallelogram Statements 1) 1) Given ) ) Given 3) ABCD is a parallelogram 3) Reasons. Given: PG GU ; PU bisects GPL Prove: Quad GULP is a trapezoid Statements 1) PG GU 1) ) PU bisects GPL ) Reasons 3) 3) Definition of Isosceles Triangle 4) GPU UPL 4) 5) 5) Substitution 6) GULP is a trapezoid 6)

32 3. Given: ABCD is a parallelogram; DE FB Prove: 1 Statements 1) ABCD is a parallelogram 1) ) ) Reasons 3) 3) Opposite Angle Parallelogram Theorem 4) DE FB 4) 5) DEA BFC 5) 6) 1 6) 4. Given: DE AC ; BF AC ; AE FC ; DE FB Prove: ABCD is a parallelogram Statements Reasons 1) DE AC 1) ) BF AC ) 3) 3) Definition of Perpendicular 4) DEA BFC 4) 5) AE FC 5) 6) DE FB 6) 7) 7) SAS 8) AD BC 8) 9) 9) CPCTC 10) AD BC 10) 11) ABCD is a parallelogram 11)

33 5. Given: ABCD is a rectangle M is the midpoint of AB Prove: DM CM Statements Reasons 1) 1) ) AD BC ) 3) M is the midpoint of AB 3) 4) MA MB 4) 5) DAM CBM 5) 6) DAM CBM 6) 7) DM CM 7) 6. Given: HS SB ; RS SO ; HR HO Prove: RHOB is a rhombus Statements Reasons 1) HS SB 1) ) RS SO ) 3) HR HO 3) 4) RHOB is a parallelogram 4) 5) HR OB RB HO 6) RHOB is a rhombus 6) 5)

34 7. Given: ABCD is a kite; AB AD ; BC DC Prove: B D Statements Reasons 1) AB AD 1) ) BC DC ) 3) AC AC 3) 4) ABC ADC 4) 5) B D 5)

Quadrilaterals. Polygons Basics

Quadrilaterals. Polygons Basics 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