Geometry 1 Unit 6
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Transcript of Geometry 1 Unit 6
Geometry 1 Unit 6
Quadrilaterals
Geometry 1 Unit 6
6.1 Polygons
Polygons Polygon
A closed figure in a plane Formed by connecting line segments endpoint to
endpoint Each segment intersects exactly 2 others Classified by the number of sides they have Named by listing vertices in consecutive order
Sides Line segments in a polygon
Vertex Each endpoint in a polygon
Polygons
polygons not polygons
Polygons
Pentagon ABCDE or pentagon CDEAB
A
D C
BE
PolygonsSides Name3 Triangle4 Quadrilateral5 Pentagon6 Hexagon7 Heptagon8 Octagon9 Nonagon10 Decagon11 Undecagon12 Dodecagonn n-gon (a 19 sided polygon is a 19-gon)
Polygons
Diagonals Line segments that connect non-
consecutive vertices.
Polygons
Convex polygons Polygons with no diagonals on the
outside of the polygon
Polygons
Concave polygons A polygon is concave if at least one
diagonal is outside the polygon These are also called nonconvex.
Polygons
Example 1 Identify the polygon and state whether it
is convex or concave.
Polygons
Equilateral Polygon all sides the same length
Equiangular Polygon all angles equal measure
Regular Polygon equilateral and equiangular
Polygons
Equilateral Equiangular Regular
Polygons
Example 2 Decide whether the polygon is regular.
Polygons
Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior
angles of a quadrilateral is 360°.
m1 + m2 + m3 + m4 = 360°
1
4 3
2
Polygons
Example 3 Find mF, mG, and mH.
x
x55°E
H
F
G
Polygons
Example 4 Use the information in the diagram to
solve for x
100°
2x + 30 3x – 5
120°
Geometry 1 Unit 6
6.2 Properties of Parallelograms
Properties of Parallelograms
ParallelogramQuadrilateral with two pairs of parallel
sides.
Properties of Parallelograms
Opposite Sides of a Parallelogram Theorem If a quadrilateral is a parallelogram, then
its opposite sides are congruent.
P
Q
S
RPQ RS and SP QR
Properties of Parallelograms
Opposite Angles in a Parallelogram Theorem If a quadrilateral is a parallelogram, then
its opposite angles are congruent.
P
Q
S
RP R and Q S
Properties of Parallelograms
Consecutive Angles in a Parallelogram Theorem If a quadrilateral is a parallelogram, then
its consecutive angles are supplementary
Add to equal 180°
P
Q
S
R
mP + mQ = 180°
mQ + mR = 180°
mR + mS = 180°
mS + mP = 180°
Properties of Parallelograms
Diagonals in a Parallelogram Theorem If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
P
Q
S
R
M
QM SM and PM RM
Properties of Parallelograms
Example 1 GHJK is a parallelogram. Find each
unknown length JH LH
G
K
H
J
L
68
Properties of Parallelograms
Example 2 In ABCD, mC = 105°. Find the
measure of each angle. mA mD
Properties of Parallelograms
Example 3 WXYZ is a parallelogram. Find the value
of x.
W
YZ
X
3x + 18°
4x – 9°
Properties of Parallelograms Example 4
Given: ABCD is a
parallelogram. Prove:
2 4
Statement Reasons
ABCD is a parallelogram
AD || BC
2 1
AB || CD
Alternate interior angles theorem
2 4
A
D
B
C
1
4
2
3
Properties of Parallelograms Example 5
Given: ACDF is a
parallelogram. ABDE is a
parallelogram. Prove:
∆BCD ∆EFA
Statement Reason
ACDF is a parallelogram.
ABDE is a parallelogram.
Opposite sides of a parallelogram are congruent
AC = DF
AB = DE
AC = AB + BC
DF = DE + EF
AC = DE + DF
AB + BC = AB + EF
BC = EF
Def of Congruent
∆BCD ∆EFA
A B C
F DE
Properties of Parallelograms
Example 6 A four-sided concrete slab has
consecutive angle measures of 85°, 94°, 85°, and 96°. Is the slab a parallelogram? Explain.
Geometry 1 Unit 6
6.3 Proving Quadrilaterals are Parallelograms
Proving Quadrilaterals are Parallelograms
Investigating Properties of Parallelograms Cut 4 straws to form two congruent pairs. Partly unbend two paperclips, link their smaller
ends, and insert the larger ends into two cut straws. Join the rest of the straws to form a quadrilateral with opposite sides congruent.
Change the angles of your quadrilateral. Is your quadrilateral a parallelogram?
Proving Quadrilaterals are Parallelograms
Converse of the Opposite Sides of a Parallelogram Theorem If a opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
D
A
C
BABCD is a parallelogram
Proving Quadrilaterals are Parallelograms
Converse of the Opposite Angles in a Parallelogram Theorem If both pairs of opposite angles of a
quadrilateral are congruent, then the quadrilateral is a parallelogram.
D
A
C
B ABCD is a parallelogram.
Proving Quadrilaterals are Parallelograms
Converse of the Consecutive Angles in a Parallelogram Theorem If an angle of a quadrilateral is supplementary to both
of its consecutive angles, then the quadrilateral is a parallelogram.
ABCD is a parallelogram
D
A
C
B
(180 – x)°
x°
x°
Proving Quadrilaterals are Parallelograms
Converse of the Diagonals in a Parallelogram Theorem If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
D
A
C
B
M
ABCD is a parallelogram
Proving Quadrilaterals are Parallelograms
Example 1 Given:
∆PQT ∆RST
Prove: PQRS is a
parallelogram.
Statements Reasons
∆PQT ∆RST
CPCTC
PT = RT
ST = QT
Def. of bisect
PQRS is a parallelogram
P Q
RS
T
Proving Quadrilaterals are Parallelograms
Example 2 A gate is braced as shown. How do you
know that opposite sides of the gate are congruent?
Proving Quadrilaterals are Parallelograms
Congruent and Parallel Sides Theorem If one pair of opposite sides of a
quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
A
B
D
C
ABCD is a parallelogram.
Proving Quadrilaterals are Parallelograms
To determine if a quadrilateral is a parallelogram, you need to know one of the following: Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent An angle is supplementary with both of its
consecutive angles Diagonals bisect each other One pair of sides is both parallel and congruent
Proving Quadrilaterals are Parallelograms
Example 3 Show that A(-1,2), B(3,2), C(1,-2), and
D(-3,-2) are the vertices of a parallelogram.
Geometry 1 Unit 6
6.4 Rhombuses, Rectangles, and Squares
Rhombuses, Rectangles, and Squares
RectangleParallelogram with four congruent angles
RhombusParallelogram with four congruent sides
SquareParallelogram with four congruent angles
and four congruent sides
Rhombuses, Rectangles, and Squares
Example 1Decide if each statement is always,
sometimes or never true.A rhombus is a rectangle
A parallelogram is a rectangle
A rectangle is a square
A square is a rhombus
Rhombuses, Rectangles, and Squares
Example 2Given FROG is a rectangle, what else do
you know about FROG?
F R
G O
Rhombuses, Rectangles, and Squares
Example 3EFGH is a rectangle. K is the midpoint of
FH. EG = 8z – 16, What is the measure of segment EK?What is the measure of segment GK?
Rhombuses, Rectangles, and Squares
Rhombus CorollaryA quadrilateral is a rhombus if and only if
it has four congruent sides.
Rhombuses, Rectangles, and Squares
Rectangle CorollaryA quadrilateral is a rectangle if and only if
it has four right angles.
Rhombuses, Rectangles, and Squares
Square CorollaryA quadrilateral is a square if and only if it
is a rhombus and a rectangle.
Rhombuses, Rectangles, and Squares
Perpendicular Diagonals of a Rhombus TheoremA parallelogram is a rhombus if and only if
its diagonals are perpendicular.
B
A
C
D
ABCD is a rhombus if and only if
AC BD.
Rhombuses, Rectangles, and Squares
Diagonals Bisecting Opposite Angles Theorem. A parallelogram is a rhombus if and only
if each diagonal bisects a pair of opposite angles.
A
B
D
C ABCD us a rhombus if and only if
AC bisects DAB and BCD
and
BD bisects ADC and CBA
Rhombuses, Rectangles, and Squares
Diagonals in a Rectangle Theorem A parallelogram is a rectangle if and only
if its diagonals are congruent.
A B
D C
ABCD is a rectangle if and only if
AC BD.
Rhombuses, Rectangles, and Squares
Example 4 You cut out a parallelogram shaped quilt
piece and measure the diagonals to be congruent. What is the shape?
An angle formed by the diagonals of the quilt piece measures 90°. Is the shape a square?
Geometry 1 Unit 6
6.5 Trapezoids and Kites
Trapezoids and Kites
Trapezoid A quadrilateral with exactly one pair of parallel sides.
Bases The parallel sides of a trapezoid.
Pairs of Base Angles Angles in a trapezoid that share a base.
Legs The nonparallel sides of a trapezoid.
Isosceles Trapezoid Trapezoid with congruent legs.
Trapezoids and Kites
Base Angles of an Isosceles Trapezoid Theorem If a trapezoid is isosceles, then each pair
of base angles is congruent.
A B, C D
D
A B
C
Trapezoids and Kites
Congruent Base Angles in a Trapezoid Theorem. If a trapezoid has a pair of congruent
base angles, then it is an isosceles trapezoid.
D
A B
C
ABCD is an isosceles trapezoid.
Trapezoids and Kites
Diagonals in an Isosceles Trapezoid Theorem A trapezoid is isosceles if and only if its
diagonals are congruent.
D
A B
C
ABCD is isosceles if and only if
AC BD.
Trapezoids and Kites
Midsegment of a trapezoid The segment that connects the
midpoints of a trapezoids legs.
midsegment
Trapezoids and Kites
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.
B C
D
NM
A
MN || AD, MN || BC,
MN = ½(AD + BC)
Trapezoids and Kites
Kite A quadrilateral with two distinct pairs of
consecutive congruent sides. Opposite sides are not congruent.
Diagonals of a Kite Theorem If a quadrilateral is a kite, then its
diagonals are perpendicular.
Opposite Angles in a Kite Theorem If a quadrilateral is a kite, then exactly
one pair of opposite angles are congruent.
Trapezoids and Kites
Example 4 GHJK is a kite. Find HP.
5
√29
G
H
PJ
K
Trapezoids and Kites
Example 5 RSTU is a kite. Find mR, mS, and
mT.
x + 30°
125°
x°R
S
T
U
Geometry 1 Unit 6
6.6 Special Quadrilaterals
Special Quadrilaterals
Special Quadrilaterals
Property Parallelogram Rectangle Rhombus Square Trapezoid Kite
1.Both pairs of opposite sides are congruent
2. Diagonals are congruent
3. Diagonals are perpendicular
Special QuadrilateralsProperty Parallelogra
mRectangle Rhombus Square Trapezoid Kite
4. Diagonals bisect each other
5. Consecutive angles are supplementary
6. Both pairs of opposite angles are congruent
Geometry 1 Unit 6
6.7 Areas of Triangles and Quadrilaterals
Area
Area is the number of square units in a figure.
Area-Rectangles
Count the number of squares to find the area.
A shortcut is to find the length and multiply it by the width.
Area-Parallelograms
Count the number of squares to find the area.
A shortcut is to find the length and multiply it by the width.
Rectangles and Parallelograms
Area = length times width Area = base times height A = bh
b b
h
Area-Triangles
Area-Triangles
A = ½ base times height A = ½ bh
hh
h
b
b
b
Area-Trapezoids
b2
h
b1
hb1
h
b2
A = triangle 1 + triangle 2
A = ½ h(b1) + ½ h(b2)
A = ½ h(b1 + b2)
Area-Trapezoids
A = ½ height times (base1 + base 2) A = ½ h(b1 + b2)
h h h
b2 b2 b2
b1 b1b1
Area- Kites
A = ½ (diagonal 1) times (diagonal 2) A = ½ (d1)(d2)
d1 and d2
Area-Rhombus
A = ½ (diagonal 1) times (diagonal 2) A = ½ (d1)(d2)
d1 and d2
Example 1
Find the area of ΔRST.
3
4 TS
R
Example 2
What is the base of a triangle that has an area of 48 and a height of 3?
Example 3
A rectangle has an area of 100 square meters and a height of 25 meters. Are all the rectangles with these dimensions congruent?
Example 4
Find the area of parallelogram RSTU.
U
R
6
6
3
S
T
Example 5
What is the height of a parallelogram that has an area of 96 square feet and a base length of 8 feet?
Example 6
Find the area of trapezoid EFGH.E(-2, 3), F(2, 4), G(2, -2), H(-2, -1)
Example 7
Find the area of kite ABCD.A(0, 5), B(3, 6), C(6, 5), D(3, 2)
Example 8
Use the information given in the diagram to find the area of kite ABCD.
A
D
C
B
8
8
8
16
Example 9
The tray below is designed to save space on cafeteria tables. How much table area does the tray use?
16 in
8 in
3 in
10 in
Example 10
Find the area of rhombus EFGH if EG = 10 and FH = 15.