Journal Chapter 6. Kirsten Erichsen 9-5 Polygons and Quadrilaterals INDEX: 6-1: Properties and Attributes of Polygons 6-2: Properties of Parallelograms 6-3: Conditions for Parallelograms 6-4: Properties of Special Parallelograms 6-5: Conditions for Special Parallelograms 6-6: Properties of Kites and Trapezoids
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Journal Chapter 6. Kirsten Erichsen 9-5 Polygons and Quadrilaterals INDEX: 6-1: Properties and Attributes of Polygons 6-2: Properties of Parallelograms.
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Journal Chapter 6.
Kirsten Erichsen 9-5
Polygons and Quadrilaterals
INDEX: 6-1: Properties and Attributes of Polygons 6-2: Properties of Parallelograms 6-3: Conditions for Parallelograms 6-4: Properties of Special Parallelograms 6-5: Conditions for Special Parallelograms 6-6: Properties of Kites and Trapezoids
POLYGONS.
WHAT IS A POLYGON? Definition: a closed figure formed by 3 or more segments,
whose endpoints have to touch another 2 endpoints.
Definition: it is a quadrilateral with 4 congruent sides.
A rhombus
THEOREM 6-4-3
If a quadrilateral is a rhombus, then it is a parallelogram.
A
B
C
D
H
I
J
K
YESYES
NONo because we don’t know if the other 2 sides are congruent to the other pair.
N
O
P
Q
THEOREM 6-4-4
If a parallelogram is a rhombus, then its diagonals are perpendicular.
A
B
C
D
HI
J
KYES
NONo because we don’t know if the bisectors actually bisect.
YES
THEOREM 6-4-5
If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.
A
B
C
D
HI
J
KYES
NO
No because we only know that one pair of angles are being bisected.
YES
SQUARES AND THEOREMS.
WHAT IS A SQUARE?
It is another special quadrilateral.
Definition: it is a quadrilateral with 4 right angles and 4 congruent sides.
A square has the properties of a parallelogram, a rectangle and a rhombus.
EXAMPLE 1.
Do you consider this a square?
No because we only know that it has 4 right angles and no 4 congruent sides.
EXAMPLE 2.
Do you consider this a square?
Yes because we have the 4 right angles and the 4 congruent sides.
EXAMPLE 3.
Do you consider this a square?
No because we only know that there are 4 congruent sides, but we don’t know anything about the angles.
RECTANGLES AND THEOREMS.
WHAT IS A RECTANGLE?
It is another special quadrilateral.
Definition: it is a quadrilateral or parallelogram with 4 right angles.
The diagonals in a rectangle are congruent.
THEOREM 6-4-2
If a parallelogram is a rectangle, then its diagonals are congruent.
Tell me if these are rectangles.
NO
3
3
4
4
YES NO
THEOREM 6-5-1
If one angle is a parallelogram is a right angle, then the parallelogram is a rectangle.
Tell me if these are rectangles.
YES
3
3
4
4
YES NO
RECTANGLES, RHOMBUS AND SQUARES
All of these shapes are quadrilaterals and parallelograms.
A rectangle has the properties of a parallelogram and a few more.
A rhombus has the properties of a rectangle, parallelogram and its individual properties.
A square has the properties of a rectangle, parallelogram, rhombus and a pair of its own properties.
TRAPEZOIDS AND THEOREMS.
WHAT IS A TRAPEZOID?
Definition: it is a quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid
PARTS OF A TRAPEZOID.
Base: each one of the parallel sides of the trapezoid.
Legs: the pair of non-parallel sides.
Base Angles: they are 2 consecutive angles whose the common side is a base.
Base
Base
Legs
Base Angles
Base Angles
ISOSCELES TRAPEZOID.
Definition: it is when the legs of the trapezoid are congruent.
THEOREM 6-6-3
If a quadrilateral is an isosceles, then each pair of base angles are congruent.
EXAMPLES. Tell me if each one of the isosceles trapezoid are
really isosceles.
YES
NO YES
THEOREM 6-6-5
A trapezoid is isosceles only if its diagonals are congruent.
EXAMPLES. Tell me if each one of the isosceles trapezoid are
really isosceles.
YES
NO
YES7
7
11
11
5
4
TRAPEZOID MISEGMENT THEOREM
The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the bases.
LM = 1/2 (HI + KJ)
H I
JK
L M
EXAMPLE 1.
Find the missing measurement.
32 cm.
38 cm.
A B
CD
E F
The distance between EF and DC is 6 cm, so since EF is in the middle of the trapezoid you subtract 6 from 32.
AB = 26 cm.
26 cm.
EXAMPLE 2.
Find the missing measurement.
16 cm.
24 cm.
G H
JK
L I
The distance between GH and JK is 8 cm, so since LI is in the middle of the trapezoid you divide 8 by 2, which is 4. Then you add 4 to 16, or subtract 4 from 24.
LI = 20 cm.
20 cm.
EXAMPLE 3.
Find the missing measurement.
6 cm.
13 cm.
M N
PQ
R O
The distance between MN and RO is 7 cm, so since QP is at the bottom of the trapezoid you add 7 to 13 cm or RO.
PQ = 20 cm.
20 cm.
KITES AND THEOREMS.
WHAT IS A KITE?
Definition: it is a quadrilateral with exactly two pairs of congruent consecutive sides.
It has 2 pairs of congruent adjacent sides, diagonals are perpendicular, and it has non-congruent adjacent angles
THEOREM 6-6-1
If a quadrilateral is a kite, then its diagonals are perpendicular. A
B
C
D
EXAMPLES.
Tell me if the following kites are really showing that they are kites.
D
E
F
G
6.5 6.5
2.5
3.5
8 8
YESNO YES
THEOREM 6-6-2
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
NOTE: only one of the perpendicular lines is being bisected.
A
B
C
D
EXAMPLES.
Tell me if the following kites are really showing that they are kites.