Download presentation
1
Chapter 6 Notes
2
6.1 – Polygons
3
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.
4
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! P R E T L B Hexagon PRTBLE Or Hexagon TBLEPR Or other names
5
Convex A polygon such that no line containing a side of a polygon contains a point in the interior of the polygon.
6
Equilateral Sides are the same.
Equiangular Angles are the same. Regular Both
7
Diagonal – Segment that joins two nonconsecutive vertices
Is it a polygon? If so, name it and say if it is convex or concave. Diagonal – Segment that joins two nonconsecutive vertices
8
Interior angles of a Quadrilateral
The sum of the measures of the interior angles of a quadrilateral is 360o. Solve for x xo xo 2xo 2xo x+20o x+20o 100o 100o
9
(3x + 2)o (2x – 10 )o (x + 5)o (2x – 7)o
10
6.2 – Properties of Parallelograms
11
Definition of a parallelogram Both pairs of opposite sides are parallel.
C M
12
A B D C M Find all information
13
x + 24 z 65 y 4x – 12
14
zo 7y wo 9q + 4 3y + 28 xo 30o 30o write
15
Proving Theorem 6.2 A B D C
16
A B D C E F
17
B X C 1 F E Y 2 A Z D
18
6.3 – Proving Quadrilaterals are Parallelograms
19
Opposite sides are parallel, then it’s a ||-gram by def.
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 T Q S R 1 4 2 3 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
20
Let’s discuss how to prove two of these theorems
THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram. T Q S R 1 4 2 3 M THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram
21
Yes parallelogram, not a parallelogram, and why?
THRM 6.10, If one pair of opposite sides are both CONGRUENT and PARALLEL, then the quadrilateral is a ||-gram T Q S R 1 4 2 3 M Yes parallelogram, not a parallelogram, and why?
22
Prove that the following coordinates make a parallelogram by the given theorems\definitions using slope formula and distance formula. 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 (-1, 3) (3, 2) (-4, -1) (0, -2) A B D C
23
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)
24
6.4 – Rhombuses, Rectangles, and Squares
25
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?
26
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
27
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.
28
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. A B D C
29
Thrm 6.13 A ||-gram is a rectangle iff its diagonals are congruent.
B D C Prove this theorem. Stuwork
30
Which shape could it be? What can be true about it?
31
Given the figure is a rectangle,
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
32
Given the figure is a square,
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
33
A ||-gram is a rectangle iff its diagonals are congruent
A ||-gram is a rectangle iff its diagonals are congruent. We’ll now prove this using coordinate proof ( , ) ( , ) A B D C ( , ) ( , ) Stuwork
34
Well do some coordinate proof stuff with rhombus, rectangles, and squares added on.
35
6.5 – Trapezoids and Kites
36
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.
37
Theorem 6.14 Base angles of an isosceles trapezoid are congruent.
ONLY TRUE FOR ISOS TRAPEZOID, NOT REGULAR TRAPEZOID! X Y A B Z Congruent, opp sides ||-gram congruent. Transitive to work all sides congruent. Corresponding, base angles thrm, transitive. Same side interior angles, measure, supp, subtraction.
38
Theorem 6.15 If a trapezoid has one pair of congruent base angles, then it’s an isosceles trapezoid. A B C D Theorem 6.16 A trapezoid is isosceles iff its diagonals are congruent. A B C D
39
Just like in a triangle, the midsegment will go through the midpoint of the legs.
B C D b1 b2 midsegment E F
40
A B C D 10 18 x E F A B C D 12 y 15 E F
41
A B C D z 14 11 E F A B C D 14 2x+2 5y+10 7y-10
42
x 13 y 19 z A B C D R AD = 15 AR = 6 BC = _____ BR = __
Write in A B C D R AD = 15 AR = 6 BC = _____ BR = __ RC = ____ RD = __
44
55 22 y x 44 45 z
45
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
46
A B D C x 12 5 A B D C 60o 140o zo yo
47
A B D C 25 24 x A B D C zo 100o 70o yo
48
Go Over HW Verify Quadrilateral while HW checked Properties of shapes of Parallelogram, Rhombus, Rectangle, Square Discuss what’s on quiz
49
6.6 – Special Quadrilaterals
50
Quad-Tree-Laterals
51
For which shape(s) is/are the following always true?
Make a table with the following shapes as the heading: Parallelogram Rectangle Rhombus Square Trapezoid Isosceles Trapezoid Kite
52
6.7 – Areas of Triangles and Quadrilaterals
53
Postulate: The area of a square is the square of the length of its side A = s2
Area congruence Postulate If two figures are congruent, they have the same area.
54
Area Addition Postulate The area of a region is the sum of the areas of the non overlapping part. Just means you can cut up the parts and add them together.
55
In a parallelogram, a side can be considered the base
In a parallelogram, a side can be considered the base. The line perpendicular to the base and going to the other side is the altitude. The length of the altitude is called the height. (Altitude segment, height number) a h b b Area of a rectangle equals the product of the base and height. (A = bh)
56
Area of Parallelogram A = bh h b
57
b1 h b2
58
d2 d1 Kite works the same way.
59
22 22 5 10 10 4 10 15 4 8 5 10 Write bottom two
60
The sides of a rectangle are x and x – 14. The area is 120
The sides of a rectangle are x and x – 14. The area is Find the value of x. x – 14 x
61
Area is 40 ft2. Find h Area is 80 ft2. Find x 10 x h x 16
62
12 17 8 8 12 6
63
4 5 11 4 14 6 14
64
Find Area Write bottom two
65
Find Area Write bottom two
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.