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Learn about polygons, parallelograms, and rhombuses, including their definitions, properties, and theorems. Explore how to identify and prove these shapes using coordinates and angle measures.
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Poly Many A polygon is a plane figure that meets the following conditions: 1) It is formed by three or more segments called sides such that no two sides with a common endpoint are collinear. 2) Each side intersects exactly two other sides, one at each endpoint.
Names of polygons # of sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 8 Octagon 10 decagon 12 dodecagon n n-gon When naming polygons, you list the vertices in order! R P T E L B Hexagon PRTBLE Or Hexagon TBLEPR Or other names
Convex A polygon such that no line containing a side of a polygon contains a point in the interior of the polygon.
Equilateral Sides are the same. Equiangular Angles are the same. Regular Both
Is it a polygon? If so, name it and say if it is convex or concave. Diagonal – Segment that joins two nonconsecutive vertices
Interior angles of a Quadrilateral The sum of the measures of the interior angles of a quadrilateral is 360o. Solve for x xo xo x+20o x+20o 2xo 2xo 100o 100o
(3x + 2)o (2x – 10 )o (x + 5)o (2x – 7)o
Definition of a parallelogram Both pairs of opposite sides are parallel. A D M C B
Find all information A D M C B
x + 24 z 65 y 4x – 12
zo 7y wo 9q + 4 3y + 28 xo 30o 30o write
Proving Theorem 6.2 A D B C
C B F A D E
X C B 1 E F Y 2 A D Z
Opposite sides are parallel, then it’s a ||-gram by def. S 3 T 2 1 R 4 Q THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram. THRM 6.8, If an angle of a quad is supplementary to both its consecutive angles, then the quadrilateral is a ||- gram THRM 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ||-gram THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram
Let’s discuss how to prove two of these theorems S 3 T 2 1 R 4 Q THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram. M THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram
THRM 6.10, If one pair of opposite sides are both CONGRUENT and PARALLEL, then the quadrilateral is a ||-gram S 3 T 2 M 1 R 4 Q Yes parallelogram, not a parallelogram, and why?
Prove that the following coordinates make a parallelogram by the given theorems\definitions using slope formula and distance formula. A B (-1, 3) (3, 2) D C (-4, -1) (0, -2) Opposite sides are parallel, then it’s a ||-gram by def. One pair of opposite sides are both CONGRUENT and PARALLEL. The diagonals of a quad bisect each other
Do these points make a parallelogram? (0,2) (-3,1) (-2, 3) (1,4) Do these points make a parallelogram? (-1,-3) (-2, 1) (2, 2) (1, -3)
Rectangle – Quad with 4 rt angles. Rhombus – Quad with 4 congruent sides Square – 4 rt angles AND 4 congruent sides, It’s a rectangle AND a rhombus!! Why are these parallelograms?
World O’ Parallelograms Answer with always, sometimes, or never A Rhombus is a Square Always Sometimes Never A Square is a Parallelogram Always Sometimes Never
Corollaries Rhombus Corollary A quadrilateral is a rhombus iff it has four congruent sides. Rectangle Corollary A quadrilateral is a rectangle iff it has four right angles. Square Corollary A quadrilateral is a square iff it is a rhombus and a rectangle.
A D M B C A B D C Thrm 6.11 A ||-gram is a rhombus iff its diagonals are perpendicular. Thrm 6.12 A ||-gram is a rhombus iff each diagonal bisects a pair of opposite angles.
Thrm 6.13 A ||-gram is a rectangle iff its diagonals are congruent. A D B C Prove this theorem. Stuwork
Which shape could it be? • What can be true about it?
Solve for x and y. Given the figure is a rectangle, Solve for x and y. 2y + 35 70o xo 2y + 16 (2x + 5)o 4y – 10 5y – 10 Stuwork
Solve for x and y. Given the figure is a square, Solve for x and y. 55o A B xo 10 yo 5 y D C AC = 4x – 10 BD = 2x + 2 Stuwork
A ||-gram is a rectangle iff its diagonals are congruent. We’ll now prove this using coordinate proof A D B C ( , ) ( , ) ( , ) ( , ) Stuwork
Well do some coordinate proof stuff with rhombus, rectangles, and squares added on.
A quadrilateral with EXACTLY one pair of parallel sides is called a TRAPEZOID. The parallel sides are called BASES. The other sides are LEGS Trapezoids have two pairs of base angles. ISOSCELES TRAPEZOID – LEGS are CONGRUENT! KITE – A quadrilateral with two pairs of consecutive sides, BUT opposite sides aren’t congruent.
Theorem 6.14 Base angles of an isosceles trapezoid are congruent. B A X Y Z ONLY TRUE FOR ISOS TRAPEZOID, NOT REGULAR TRAPEZOID! Congruent, opp sides ||-gram congruent. Transitive to work all sides congruent. Corresponding, base angles thrm, transitive. Same side interior angles, measure, supp, subtraction.
Theorem 6.15 If a trapezoid has one pair of congruent base angles, then it’s an isosceles trapezoid. A B D C A B D C Theorem 6.16 A trapezoid is isosceles iff its diagonals are congruent.
A B b1 E midsegment F D C b2 Just like in a triangle, the midsegment will go through the midpoint of the legs.
A A B B 10 x E F D D C C 18 12 15 E F y
A B z 11 E F D C A B 14 D C 14 2x+2 7y-10 5y+10
13 Write in x 19 y z A B R D C AD = 15 AR = 6 BC = _____ BR = __ RC = ____ RD = __
55 y 22 44 x 45 z
C Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular B D A Theorem 6.18 If a quadrilateral is a kite, then EXACTLY one pair of opposite angles are congruent. C B D A
C B D A C B D A x 12 5 60o 140o zo yo
C B D A C B D A 25 24 x zo 100o 70o yo
Go Over HW • Verify Quadrilateral while HW checked • Properties of shapes of Parallelogram, Rhombus, Rectangle, Square • Discuss what’s on quiz
