1
Name ________________________________________ Period _______
Quadrilaterals – Chapter 6 - GEOMETRY Section 6.1 Polygons
GOAL 1: Describing Polygons 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. _____________________
Ex. 1 Decide whether the figure is a polygon. If not, explain why.
1. 2. 3. 4. 5.
A vertex is _______________________________________________. The plural of vertex is ________.
You can name a polygon by listing its vertices consecutively.
PQRST is one way to name this polygon. What is another way? ________________
Polygons are also named by the number of sides they have.
A polygon is convex if ___________________________________________________
______________________________________________________________________
A polygon is concave if __________________________________________________
Ex. 2 Use the number of sides to tell what kind of polygon the shape is. Then state whether the polygon is
convex orconcave.
6. 7. 8.
A diagonal of a polygon is a _____________________________________________________________.
# of sides Type of polygon # of sides Type of polygon # of sides Type of polygon
3 Triangle 7 Heptagon 12 Dodecagon
4 Quadrilateral 8 Octagon n n-gon
5 Pentagon 9 Nonagon
6 Hexagon 10 Decagon
2
Ex. 3 Use the diagram at the right to answer the following.
9. Name the polygon by the number of sides it has.
10. Polygon MNOPQR is one name. State two other names.
11. Name all of the diagonals that have vertex M as an endpoint.
12. Name the consecutive angles to .N
A polygon is equilateral if _______________________________________________________________.
A polygon is equiangular if ______________________________________________________________.
A polygon is regular if ____________________________________________________________.
Ex. 4 State whether the polygon is best described as equilateral, equiangular, regular, or none of these.
13. 14. 15. 16.
GOAL 2: Interior Angles of Quadrilaterals
Ex. 5 Use the information in the diagram to solve for x.
17. 18. 19.
Section 6.2 Properties of Parallelograms
GOAL 1: Properties of Parallelograms A parallelogram is a ___________________________________________________________________.
Theorem 6.1 Interior Angles of a Quadrilateral
The sum of the measures of the interior angles
of a quadrilateral is 360°.
.3604321 mmmm
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Ex. 1 Mark the congruences for the theorems below.
Ex. 2 Lets prove Theorem 6.3 in a paragraph proof.
Given: ABCD is a parallogram.
Prove: DBCA and
Opposite sides of a parallelogram are congruent, so __________________ and __________________.
By the Reflexive Property of Congruence, __________________. CDBABD because of the
________ Congruence Postulate. Because _________________ parts of congruent triangles are congruent,
.CA Now draw diagonal AC. By use of the same reasoning, .DB
Ex. 3 Decide whether the figure is a parallelogram. If it is not, explain why not.
1. 2. 3.
Theorem 6.2
If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
QRSPRSPQ and
Theorem 6.3
If a quadrilateral is a parallelogram, then its
opposite angles are congruent. SQRP and
Theorem 6.4
If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
180 ,180
,180 ,180
PmSmSmRm
RmQmQmPm
Theorem 6.5
If a quadrilateral is a parallelogram, then its
diagonals bisect each other.
RMPMSMQM and
4
Ex. 4 Use the diagram of parallelogram MNOP at the right. Complete the statement and give a reason.
4. MN 5. MN P
6. ON 7. MPO
8. PQ 9. QM
10. MQN 11. NPO
Ex. 5 Find the measure in the parallelogram HIJK. Explain your reasoning.
12. HI 13. KH
14. GH 14. HJ
16. mKIH 17. mJIH
18. mKJI 19. mHKI
Ex. 6 Find the value of each variable in the parallelogram.
20. 21.
Section 6.3 Proving Quadrilaterals are Parallelograms
GOAL 1: Proving Quadrilaterals are Parallelograms Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive
angles, then the quadrilateral is a parallelogram.
Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel,
then the quadrilateral is a parallelogram.
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We can also use the definition of a parallelogram to prove that a quadrilateral is a parallelogram.
If both pairs of opposite sides are parallel,
then the quadrilateral is a parallelogram.
Ex. 1 Name 6 ways to prove that a quadrilateral is a parallelogram.
Ex. 2 Are you given enough information to determine whether the quadrilateral is a parallelogram?
1. 2. 3.
4. 5. 6.
Ex. 3 What additional information is needed in order to prove that quadrilateral ABCD is a parallelogram?
7. ABPDC 8. AB DC
9. DCA BAC 10. DE EB
11. mCDAmDAB 180
Ex. 4 What value of x and y will make the polygon a parallelogram?
12. 13. 14.
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GOAL 2: Using Coordinate Geometry When a figure is in the coordinate plane, you can use the Distance Formula to prove that sides are
congruent and you can use the slope formula to prove that sides are parallel.
Ex. 5 Prove that the points represent the vertices of a parallelogram. Use two different methods.
A( 2, -1), B( 1, 3), C( 6, 5), D( 7, 1)
Ex. 6 Draw the quadrilateral ABCD.
If the hat rack were expanded outward, would
ABCD still be a parallelogram? Explain.
Section 6.4 Rhombuses, Rectangles, and Squares
GOAL 1: Properties of Special Parallelograms In this lesson you will study three special types of parallelograms: rhombuses, rectangles, and squares.
A rhombus is a A rectangle is a A square is a
______________________ ______________________ ________________________
______________________ ______________________ ________________________
You can use the following corollaries to prove that a quadrilateral is a rhombus, rectangle, or square without
proving first that the quadrilateral is a parallelogram.
Rhombus Corollary
A quadrilateral is a rhombus if and only if it has four congruent sides.
Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four right angles.
Square Corollary
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
Ex. 1 Decide whether the statement is sometimes, always, or never true.
1. A square is a rectangle. 2. A parallelogram is a rhombus.
3. A rectangle is a square. 4. A rhombus is a rectangle.
5. A parallelogram is a rectangle. 6. A square is a parallelogram.
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GOAL 2: Using Diagonals of Special Parallelograms
The following theorems are about diagonals of rhombuses and rectangles.
Theorem 6.11
A parallelogram is a rhombus if and only if its diagonals are perpendicular. ABCD is a rhombus if and only if AC BD
Theorem 6.12
A parallelogram is a rhombus if and only if each diagonal bisects a pair
of opposite angles. ABCD is a rhombus if and only if
AC bisects DAB and BCD and
BD bisects ADC and CBA
Theorem 6.13 A parallelogram is a rectangle if and only if its diagonals are congruent. ABCD is a rectangle if and only if AC BD
Remember that is a square is both a rectangle and a rhombus.
Ex. 2 List everything you know about squares. (Hint: List everything about parallelograms, rectangles
and rhombuses.
Ex. 3 Match the properties of a quadrilateral with all of the types of quadrilateral which have that property.
7. The diagonals are congruent. A. Parallelogram
8. Both pairs of opposite sides are congruent. B. Rectangle
9. Both pairs of opposite sides are parallel. C. Rhombus
10. All angles are congruent. D. Square
11. All sides are congruent.
12. Diagonals bisect the angles.
Ex. 4 Decide whether the statement is sometimes, always, or never true.
13. A rhombus is equilateral.
14. The diagonals of a rectangle are perpendicular.
15. The opposite angles of a rhombus are supplementary.
16. The diagonals of a rectangle bisect each other.
17. The consecutive angles of a square are supplementary.
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Ex. 5 Find the value of x.
18. MNOP is a square 19. DEFG is a rhombus. 20. WZYZ is a rectangle.
Section 6.5 Trapezoids and Kites
GOAL 1: Using Properties of Trapezoids
A trapezoid is a ____________________________________________________
____________________________________________. The parallel sides are the
___________. A trapezoid has two pairs of ____________________. The
nonparallel sides are the ________ of the trapezoid. If the legs of a trapezoid are
congruent, then the trapezoid is an ____________________________________.
Ex. 1 Match the pairs of segments or angles with the term, which describes them in trapezoid PQRS.
1. S and P A. bases
2. QS and PR B. legs
3. QR and PS C. diagonals
4. Q and S D. base angles
5. PQ and RS E. opposite angles
Theorem 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent.
A B, C D
Theorem 6.15
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
ABCD is an isosceles trapazoid.
Theorem 6.16
A trapezoid is isosceles if and only if its diagonals are congruent.
ABCD is isosceles if and only if AC BD
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Ex. 2 Complete the statement with always, sometimes or never.
6. A trapezoid is _________________________ a parallelogram.
7. The bases of a trapezoid are ____________________parallel.
8. The base angles of an isosceles trapezoid are ____________________ congruent.
9. The legs of a trapezoid are _____________________ congruent.
Ex. 3 Find the angle measures of ABCD.
10. 11.
The midsegment of a trapezoid is the ________________________________________________________.
Theorem 6.17 Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base and its length is
one half the sum of the lengths of the bases.
MN PAD, MN PBC, MN 1
2(AD BC)
Ex. 4 Find the length of the midsegment .RT
12. 13. 14.
GOAL 2: Using Properties of Kites
A kite is a _________________________________________________________
___________________________________________.
Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular.
Theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite
angles are congruent.
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Ex. 5 Find the length of the sides to the nearest hundredth or the measure of the angles in kite KITE.
15. 16. 17.
Section 6.6 Special Quadrilaterals
GOAL 1: Summarizing Properties of Quadrilaterals
Ex. 1 Summarize the seven special types of quadrilaterals in a diagram.
Ex. 2 Put an X in the box if the shape always has the given property.
Property gram Rectangle Rhombus Square Kite Trapezoid Isosceles
Trapezoid Both pairs of opp.
sides are
Exactly 1 pair of
opp. sides are
Diagonals are
Diagonals are
Diagonals bisect
each other
Both pairs of opp.
Sides are
Exactly 1 pair of
opp. Sides are
All sides are
Both pairs of opp.
's are
Exactly 1 pair of
opp. 's are
All 's are
Area
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Ex. 3 Identify the special quadrilateral. Use the most specific name.
1. 2. 3.
Ex. 4 What quadrilateral meet the conditions shown? ABCD is not drawn to scale.
4. 5. 6.
Section 6.7 Areas of Triangles and Quadrilaterals
GOAL 1: Using Area Formulas
Area Postulates
Postulate 22 Area of a Square Postulate
The area of a square is the square of the length of its side, or A s2
Postulate 23 Area Congruence Postulate
If two polygons are congruent, then they have the same area.
Postulate 24 Area Addition Postulate
The area of a region is the sum of the areas if its nonoverlapping parts.
Area Theorems
Theorem 6.20 Area of a Rectangle
The area of a rectangle is the product of its base and height.
A bh
Theorem 6.21 Area of a Parallelogram
The area of a parallelogram is the product of a base and its corresponding height. A bh
Theorem 6.22 Area of a Triangle
The area of a triangle is one half the product of a base and its corresponding height.
A 1
2bh
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Ex. 1 Find the area of the polygon.
1. 2. 3.
GOAL 2: Areas of Trapezoids, Kites, and Rhombuses
Theorem 6.23 Area of a Trapazoid
The area of a trapezoid is on half the product of the height and the sum of the basses.
A 1
2h(b1 b2 )
Theorem 6.24 Area of a Kite
The area of a kite is one half the product of the lengths of its diagonals.
A 1
2d1d2
Theorem 6.25 Area of a Rhombus
The area of a rhombus is equal to one half the product of the lengths of the diagonals.
A 1
2d1d2
Ex. 2 Find the area of the polygon.
4. 5. 6.
7. 8. 9.