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Warm – up 10/14 • Use the same sheet for warm-ups • Answer the following – Come up with your own definition of a quadrilateral (Remember a quad is a polygon) – What is the internal angle sum of a quadrilateral? Chapter 5 Quadrilaterals • Apply the definition of a parallelogram • Prove that certain quadrilaterals are parallelograms • Apply the theorems and definitions about the special quadrilaterals 5-1 Properties of Parallelograms Objectives • Apply the definition of a parallelogram • List the other properties of a parallelogram through new theorems Quadrilaterals • Any 4 sided figure He’s Back…. • Parallelograms are special types of quadrilaterals with unique properties • If you know you have a parallelogram, then you can prove that these unique properties exist… • With each property we learn, say the following to yourself.. – “If a quadrilateral is a parallelogram, then _____________.” Definition of a Parallelogram ( ) What do you thinkisthe definition is based thepairs If a quadrilateral a parallelogram, then on both diagram? of opposite sides are parallel. ABCD A B Partners: What do we know about the angles of a parallelogram b/c it has parallel sides? D C Naming a Parallelogram Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise. ABCD A D B C A D B C • The fact that we know opposite sides are parallel, we can deduce addition properties through theorems… Theorem Opposite sides of a parallelogram are congruent. A D B C What would be our plan for solving this theorem? Theorem Opposite angles of a at parallelogram What did we discuss the beginningareofcongruent. the lesson about s-s int. angles? A D B C Theorem The diagonals of a parallelogram bisect each other. A D B C What is another name for AC and BD? Parallelograms: What we now know… If a quad is a parallelogram, then… • From the definition.. 1. Both pairs of opposite sides are parallel • From theorems… 1. 2. 3. 4. Consecutive angles are supplementary Both pairs of opposite sides are congruent Both pairs of opposite angles are congruent The diagonals of a parallelogram bisect each other True or False • Every parallelogram is a quadrilateral True or False • Every quadrilateral is a parallelogram True or False • All angles of a parallelogram are congruent True or False • All sides of a parallelogram are congruent True or False • In ABCD, if m A = 50, then m C = 130. Hint draw a picture True or False • In ABCD, AC and BD bisect each other Hint draw a picture White Board Practice Given ABCD • Draw the parallelogram with the diagonals intersecting at E – Use different tick marks to show all the segments that are congruent White Board Practice A D B C White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. 6 R x = 80 y = 100 a=6 b=9 xº S yº 9 b 80º U a T White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. R S xº yº 9 12 a b U 45º 35º T x = 100 y = 45 a = 12 b=9 White Board Groups • Given this parallelogram with the diagonals drawn. x=5 y=6 5-2:Ways to Prove a Quad is a Parallelogram Objectives • Learn about ways to prove a quadrilateral is a parallelogram What we already know… • If a quad is a parallelogram, then… – 5 properties • What we are going to learn.. – What if we don’t know if a quad is a parallelogram, how can we prove that it is one? Its Friday night, you and your “quad” friends try to get into the parallelogram club… “If my quad has __(insert any of the statements)_, then it is a parallelogram. 1. 2. 3. 4. 5. both pairs of opposite sides parallel both pairs of opposite sides congruent both pairs of opposite angles congruent diagonals that bisect each other one pair of opposite sides are both congruent and parallel Make chart in notes w/ diagram Use the Definition of a Parallelogram • If both pairs of opposite sides of a quadrilateral are parallel then… • the quadrilateral is a parallelogram. A B The definition is a biconditional, so it can be used either way. D C Theorem • If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. • Show that the diagonals bisect each other. A B X D C Theorem • If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. • Show that both pairs of opposite angles are congruent. A D B C Theorem • If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. • Show that both pairs of opposite sides are congruent. A D B C Theorem • Show that one pair of opposite sides are both congruent and parallel. Partners: Draw in an AUX line to create 2 triangles… Use those triangles to prove find the properties you need to prove you have a parallelogram A D B C The diagonals of a quadrilateral _____________ bisect each other A. B. C. D. Sometimes Always Never I don’t know If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram A) B) C) D) Sometimes Always Never I don’t know If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is ___________ a parallelogram A. B. C. D. Sometimes Always Never I don’t know If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is __________ a parallelogram A.) B.) C.) D.) Sometimes Always Never I don’t know Whiteboards • Open book to page 173 –Answer the following… • #2 • #3 • #6 • #9 5-3 Theorems Involving Parallel Lines Objectives • Apply the theorems about parallel lines and triangles Theorem (skip) If two lines are parallel, then all points on one line are equidistant from the other line. Demo: 6 volunteers How do we measure the distance from a point to a line? m What does equidistant mean? n Theorem (noodle theorem) If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. A D B E C F Theorem (skip) A line that contains the midpoint of one side 1 of a triangle and is parallel to a another side 2 passes through the midpoint of the third side 3 . A 3 X Y 1 B 2 C Theorem A segment that joins the midpoints of two sides 1 2 of a triangle is parallel to the third side and its length is half the length of the third side. 3 A If BC is 12 then XY =? 2 X Y 1 3 B C White Board Practice • Given: R, S, and T are midpoint of the sides of ABC AB BC AC ST RT RS 12 14 B 18 R 15 10 22 S 10 9 7.8 A T C White Board Practice • Given: R, S, and T are midpoint of the sides of ABC AB BC AC ST RT RS 12 14 18 6 7 B 9 R 20 15 22 10 10 18 15.6 5 S 7.5 11 9 7.8 A T C B • ST is parallel to what side? R S • BC is parallel to what side? A T C White Board Practice 1. BD = 3x – 2; AB = 11 2. AC = 12x ; BD = 2x +40 R A S T U B C D 5.4 Special Parallelograms Objectives • Apply the definitions and identify the special properties of a rectangle, rhombus and square. QUADRILATERALS parallelogram Rhombus Rectangle Square Parallelograms: What we now know… If a quad is a parallelogram, then… • From the definition.. 1. Both pairs of opposite sides are parallel • From theorems… 1. 2. 3. 4. Consecutive angles are supplementary Both pairs of opposite sides are congruent Both pairs of opposite angles are congruent The diagonals of a parallelogram bisect each other Rectangle By definition, it is a quadrilateral with four right angles. R S V T Rhombus By definition, it is a quadrilateral with four congruent sides. B C A D Square By definition, it is a quadrilateral with four right angles and four congruent sides. B C What do you notice about the definition compared to the previous two? The square is the most specific type of quadrilateral. A D Theorem The diagonals of a rectangle are congruent. WY XZ What can we conclude about the W smaller segments that make up the diagonals? Z P X Y Finding the special properties of a Rhombus Apply the properties of a parallelogram to find 2 special properties that apply to the Rhombus. Hint: both properties involve angles. Theorem The diagonals of a rhombus are perpendicular. K X J L M What does the definition of perpendicular lines tell us? Theorem Each diagonal of a rhombus bisects the opposite angles. K X J L M X M Z Y Theorem The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. X M Z Y Discuss special angle relationships!! White Board Practice • Quadrilateral ABCD is a rhombus Find the measure of each angle A 1. ACD 2. DEC E 3. EDC 4. ABC B 62º D C White Board Practice • Quadrilateral ABCD is a rhombus Find the measure of each angle A 1. ACD = 62 2. DEC = 90 E 3. EDC = 28 4. ABC = 56 B 62º D C White Board Practice • Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m PON = M 29º 2. m PMO = L 3. PL = 4. MO = P N 12 O White Board Practice • Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m PON = 90 M 29º 2. m PMO = 61 L 3. PL = 12 4. MO = 24 P N 12 O White Board Practice • ABC is a right ; M is the midpoint of AB 1. If AM = 7, then MB = ____, AB = ____, and CM = _____ . A 4 2. mL1 = 40, find the rest. M 5 3 C 2 1 B A. B. C. D. Always Sometimes Never I don’t know • A square is ____________ a rhombus A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a parallelogram ____________ bisect the angles of the parallelogram. A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a rhombus are ___________ congruent. A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a rhombus ___________ bisect each other. 5.5 Trapezoids Objectives • Apply the definitions and learn the properties of a trapezoid and an isosceles trapezoid. Trapezoid A quadrilateral with exactly one pair of parallel sides. Trap. ABCD B C How does this definition differ from that of a parallelogram? A D Anatomy Of a Trapezoid • The bases are the parallel sides Base R S 1 pair of base angles 2nd pair of base angles V T Base Anatomy Of a Trapezoid • The legs are the non-parallel sides R S Leg Leg V T Isosceles Trapezoid A trapezoid with congruent legs. J M What do you think would the definition happen if Iisfolded based this on the figure diagram? in half? K L Theorem The base angles of an isosceles trapezoid are congruent. F Supplementary E G Supplementary What is something I can conclude about 2 of the angles (other than congruency) based on the markings of the diagram? H The Median of a Trapezoid A segment that joins the midpoints of the legs. B C X Y Note: this applies to any trapezoid A D Theorem The median of a trapezoid is parallel to the bases and its length is the average of the bases. Note: this applies to any trapezoid B X C Y How do we find an average of the bases ? A A D White Board Practice • Complete 19 1. AD = 25, BC = 13, XY = ______ B X A C Y D White Board Practice • Complete 22 DC = ____ 16 2. AX = 11, YD = 8, AB = _____, B X A C Y D White Board Practice • Complete 19 3. AD = 29, XY = 24, BC =______ B X A C Y D White Board Practice • Complete 3.5 4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____ B X A C Y D Test Review • Know what properties each type of quadrilateral has – i.e. – all sides are congruent: square and rhombus – #1 – 10 on pg 187 • Solving algebraic problems with a parallelogram – i.e. – finding the length of a side, angle, or diagonal • Proving a quad is a parallelogram – Do you have enough information to say that the quad is a parallelogram – Knowing the 5 ways to prove • Solving problems using TH. 5-9 – I.e. #10 – 15 on pg. 180 • Solving problems using Th. 5-11 and 5-15 – I.e. #1 – 4 on pg. 180 – I.ed # 14 , 17 on page 187 • Trapezoid Theorems – I.e. # 1 – 9 on pg. 192 – #11 pg 193