Parallelogram Trapezoid Rectangle Rhombus Quadrilaterals Kite

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Geometry Unit 11 Quadrilaterals Notes Packet
The Quadrilateral Family Tree
Quadrilaterals
- Four angles
- All four-sided shapes
- Sum of the angles is 360°
Trapezoid
Kite
Parallelogram
- Only 2 parallel sides
- Opposite sides parallel
- Opposite sides congruent
- Opposite angles congruent
- Diagonals bisect each other
- Consecutive angles supp.
- 2 pairs adjacent sides congruent
- Diagonals perpendicular
- Angles between non-congruent
sides are congruent
Rhombus
Rectangle
- A parallelogram
- Four right angles
- Diagonals are congruent
- A parallelogram
- Equilateral sides
- Diagonals bisect angles
- Diagonals perpendicular
Isosceles
Trapezoid
- Non-parallel sides (legs) are
congruent
- Diagonals are congruent
- Base angles are congruent
- Legs are congruent
- All properties of a rectangle
- All properties of a rhombus
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Square
1
Coordinate Geometry Reference Sheet:
Important Points:
1. IT IS NOT ENOUGH to just draw a graph!!
You MUST use slope, midpoint, or distance formulas to receive full (any!) credit for the problem!
2. Always write the formulas before plugging in numbers.
3. Be extremely neat and organized when showing your work.
4. Write a concluding statement (sentence) at the end of the proof.
Graphing Instructions:
1.
2.
3.
4.
Always use graph paper.
Always label your axes, scale, equations (if any), and the coordinates of the points plotted.
Always use a straightedge.
Always use pencil.
Formulas:
Name:
Formula:
Slope
m
y2  y1
x2  x1
What it finds:
How its used in proofs:
The slope of a line
1. To prove two lines parallel
(Show 2 equal slopes)
2. To prove two lines perpendicular
(Show 2 slopes that are negative
reciprocals)
Midpoint
d
 x2  x1    y2  y 1 
2
x x y y 
midpt   1 2 , 1 2 
2 
 2
2
The length of a line
segment
The midpoint of a line
segment
2. To prove lines are not congruent
(Show 2 unequal distances)
1. To prove two line segments
bisect each other
(Show they have the same
midpoint, 2 equal midpoints)
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Distance
1. To prove two lines congruent
(Show 2 equal distances)
2
Coordinate Geometry Explanations
HOW to prove a…
1.  is isosceles
a.) prove only 2  sides (distance 3x)
2.  is a right 
a.) Find the lengths of all three sides (distance 3x) and show a2+b2=c2 … OR
b.) Find the slopes of the 2 sides that look  and show they are negative
reciprocals forming a right angle* (slopes 2x – easiest of these!)
3. quad is a
a.) opposite sides  (distance 4x)
b.) opposite sides (slopes 4x)
c.) one pair of opposite sides  and
(distance 2x & slopes 2x ON THE SAME SIDES)
d.) diagonals bisect each other* (midpoint 2x – for diags – easiest of these!)
e.) opposite angles congruent (not used in a coordinate proof)
4. quad is a rectangle
a.) it has 4 right angles* (slopes 4x – easiest of these!)
b.) if it is a
(do ONE of the above for
) AND has 1 right angle (slopes 2x)
*also easy – do 4 slopes, show opposite sides are || and 2 are negative recips
c.) if it is a
(do ONE of the above for
) AND diagonals  (distances 2x)
5. quad is a rhombus
a.) it has 4  sides* (distances 4x – easiest of these!)
b.) if it is a (do ONE of the above for
) AND diagonals are  (slopes 2x)
c.) if it is a (do ONE of the above for
) AND 2 adjacent sides  (dist 2x)
6. quad is a square
a.) if it has 4  sides and 1 right angle (distance 4x & slopes 2x)
b.) if it has 4 right angles and 2 adjacent sides  (slopes 4x, distance 2x)
c.) if it has diagonals that bisect each other (makes it a ) and
are  (makes it a rectangle) and
are  (makes it a rhombus) (midpoint 2x, distance 2x, slopes 2x,)
(diags bisect each other, are congruent, and are  )
d.) if it has four right angles and  diagonals (slopes 6x)
e.) if it has four congruent sides and congruent diagonals*
(distances 6x – easiest)
7. quad is a trapezoid
a.) if only one pair of opposite sides are
8. quad is isos. trap.
a.) if only one pair of opposite sides are AND the 2 non-parallel sides are 
(slopes 4x (sides) & distance 2x (non-|| sides))
b.) if only one pair of opposite sides are AND the 2 diagonals are 
(slopes 4x (sides) & distance 2x (diagonals))
(slopes 4x, 2 will be =, 2 will be  )
***Include phrases like:
“ = slopes → || lines ”
“ negative reciprocal slopes →  lines ”
“  slopes → non-|| lines ”
“ diags have same midpt → bisect each other ”
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** For all proofs above, give a sentence to “explain” your work (from the list above!)
3
Geometry Unit 11 Quadrilaterals
Interior and Exterior Angles of Polygons
Warm-up:
CW 11.1
HW 11.1 on pages: 8 & 9
The angle measures of a quadrilateral are x  5 , 2 x  10 , 2 x  4 , and 3x  1 . Solve for x.
Explore: Complete the chart – draw a conclusion
# of
Number
Sum of Interior
Polygon
of Sides
Triangles
Angles
1 Interior
Angle
1 Exterior
Angle
Sum of
Exterior Angles
Triangle
Quadrilateral
Pentagon
Hexagon
Conclusions:
# s = ______________
___________ ____________ ____________
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Octagon
4
Definitions:
1.
Convex:
polygon where every interior angle is _________
Diagram:
2.
Concave:
polygon that has at least
3.
Regular:
polygon with equal _____________ and equal _____________
interior angle _________ Diagram:
Theorems: (where n is the number of sides)
FORMULA:
WHAT IT FINDS / DIAGRAM:
Sum of the interior angles of a polygon
v  w x  y  z
The measure of a specific interior angle of a regular
polygon
(finds one of these)
Sum of the exterior angles of ANY polygon
Exterior Angles add to __________
1 Exterior Angle:_______________
v  w  x  y  z  360
u  v  w  x  y  z  360
... because they are a linear pair!
An interior  + its exterior  = ________
Example:
Find the sum of the measures of the interior angles of a convex octagon.
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x  y  180
5
Practice:
Find the sum of the measures of the interior angles of the indicated convex polygon.
__________1. 13-gon
__________2. 18-gon
__________3. 25-gon
__________4. 34-gon
The sum of the measures of the interior angles of a convex polygon is given.
Classify the polygon by the number of sides.
_________5. 3060
__________6. 1260°
__________7. 3240°
__________8. 7560°
Find the measure of ONE exterior angle of each regular polygon.
__________9. Decagon
__________10. 20-gon
__________11. 72-gon
__________12. 15-gon
__________13. 168
_________14. 174
__________
_________
__________15. 135
_________16. 140
__________
_________
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If an interior angle of a polygon is given, find the exterior angle and find the number of sides:
6
17.
What is the value of the variables in the diagram shown?
z: ______
x: ______
v: ______
w: ______
18.
Error Analysis/Reasoning:
Your friend says she has another way to find the sum of the interior angle measures of a polygon. She
picks a point inside the polygon, draws a segment to each vertex, and counts the number of triangles.
She multiplies the total by 180, and then subtracts 360 from the product. Does her method work?
Explain.
Regents Multiple Choice Practice:
__________19.What is the measure of an exterior angle of a regular octagon?
(1) 1080°
(2) 180°
(3) 135°
(4) 45°
__________20.What is the measure of an interior angle of a regular hexagon?
(1) 540°
(2) 720°
(3) 120°
(4) 6°
__________21.For any regular polygon, what is the sum of one of its interior angles
and one of its exterior angles?
(2) 180°
(3) 90°
(4) 540°
__________22.What is the measure of an exterior angle of a regular nonagon?
(1) 180°
(2) 40°
(3) 1260°
(4) 140
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(1) 360°
7
Geometry Unit 11 Quadrilaterals
**TWO PAGES!!
Interior and Exterior Angles of Polygons
HW 11.1
Find the sum of the measures of the interior angles of the indicated convex polygon.
__________1. 14-gon
__________2. 23-gon
The sum of the measures of the interior angles of a convex polygon is given.
Classify the polygon by the number of sides.
_________3. 3060
__________4. 1260°
Find the measure of ONE exterior angle of each regular polygon. Round to nearest tenth if needed.
__________5. Nonagon
__________6. 22-gon
If an interior angle of a polygon is given, find the exterior angle and find the number of sides:
__________7. 162
_________8. 171
__________
_________
What is the value of x in the diagram shown?
__________a.)
__________b.)
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9.
8
Interior and Exterior Angles Practice:
__________10. Find the measure of each interior angle of a regular decagon.
(1) 144
(2) 135
(3) 120
(4) Cannot be determined
__________11. How many degrees are there in each interior angle of a regular hexagon?
(1) 108
(2) 120
(3) 144
(4) Cannot be determined
__________12. If a polygon has six sides, how many degree are there in each of its exterior angles?
(1) 60
(2) 30
(3) 120
(4) Cannot be determined
__________13. If each interior angle of a regular polygon measures 162°, how many sides
does the polygon have?
(1) 20
(2) 18
(3) 16
(4) Cannot be determined
__________14. How many sides does a polygon have if each of its interior angles measures 174°?
(1) 20
(2) 40
(3) 60
(4) Cannot be determined
__________15. How many degrees are there in the sum of the exterior angles of a dodecagon ?
(1) 4320
(2) 2160
(3) 1800
(4) 360
__________16. Find the number of degrees in each exterior angle of a regular pentagon?
(1) 36
(2) 72
(3) 108
(4) 360
__________17. If each exterior angle of a regular polygon contains 40°, how many sides does the
polygon have?
(2) 10
(3) 11
(4) 12
__________18. If the sum of the interior angles of a regular polygon is 900°, how many sides
does the polygon have?
(1) 7
(2) 9
(3) 10
(4) 11
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(1) 9
9
Geometry Unit 11 Quadrilaterals
Properties of Parallelograms
Warm-up:
CW 11.2
HW 11.2 on pgs: 13 & 14
Complete the following statements about two lines cut by a transversal:
1) Alternate Interior Angles are congruent
_______________________________
2) Alternate Exterior Angles are congruent
_______________________________
3) Corresponding Angles are congruent
_______________________________
4) Co-Interior Angles are supplementary
_______________________________
Notes:
*Go over Quadrilateral Family Tree
The 5 Properties of a Parallelogram:
1) Opposite sides ________
Diagram of the 5 Properties:
2) Opposite sides ________
3) Opposite angles _______
4) Diagonals bisect __________ _________
5) Consecutive Angles are _____________________
Practice Examples: (**Draw a picture)
__________1. Given parallelogram RSTW with RS  2 x  7 , ST  3 y  5 , TW  25 , and WR  16 ,
__________ solve for the values of x and y.
__________2. Given parallelogram RSTW such that diagonals SW and RT intersect at Z, if WZ  4 x  3 ,
__________
ZS  13 , RZ  17 , and ZT  7 y  3 , solve for the values of x and y.
__________3. Given parallelogram RSTW, with mWRS  24x , mRST  14 y  4 , mSTW  15x  27 ,
and mTWR  11y  20 , solve for the values of x and y.
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__________
10
__________4. Given parallelogram RSTW, with mWRS  75 , mRST  7 x , and mSTW  11y  9 ,
__________
solve for the values of x and y.
__________5. Given parallelogram ABCD, if mDAB  4 x  4 and mBCD  74  x , find mCDA .
__________6. Given parallelogram ABCD, if mDCB  2x  9 and mABC  5x  3 , find mDAB .
__________7. Given parallelogram ABCD where diagonals AC and BD intersect at E, if AC  4 y  6
and EC  3 y  1 , find AC .
__________8. Given parallelogram ABCD where diagonals AC and BD intersect at E, if DE  y  4
and DB  5 y  10 , find DB .
To Complete a Coordiante Proof of a Parallelogram:
1.
↔ Diagonals bisect each other – Show the midpoints of diagonals are same
2.
↔ 2 pairs opp sides || - Show 4 slopes and opposite sides parallel
3.
↔ 2 pairs opp sides  - Show 4 distances and opposite sides = in length
4.
↔ 1 pair opp sides  and || - Show 2 distances and 2 slopes of SAME OPP. SIDES
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- MUST SHOW ONE OF THESE:
11
Coordinate Geometry Proofs:
9. The vertices of quadrilateral ABCD are given:
A(1,2), B(2,5), C(5,7) D(4,4)
Graph it and use slopes to show that it is a parallelogram.
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10. Quadrilateral LMNO has vertices L  2, 4  , M (5, 2) , N  2, 1 , O  5,1 .
Use distances to prove that LMNO is a parallelogram.
12
Geometry Unit 11 Quadrilaterals
**TWO PAGES!!
Interior and Exterior Angles of Polygons
HW 11.2
__________1. Given parallelogram JKLM where diagonals MK and JL
__________ intersect at N, name four pairs of congruent segments.
__________
__________
__________2. Given parallelogram JKLM with mMJK  65 , find mJKL , mKLM , and mLMJ .
__________
__________
__________3. Given parallelogram JKLM where diagonals MK and JL
intersect at N, if NJ  7 , find JL .
__________4. Given parallelogram JKLM where diagonals MK and JL
intersect at N, if MK  10 , find NK .
__________5. Given parallelogram JKLM, if mMJL  37 and mLJK  27 , find mJKL .
__________7. Given parallelogram ABCD, if DA  2 y  5 and CB  14  y , find DA.
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__________6. Given parallelogram JKLM, if mJMK  71 and mKML  42 , find mJKL and mMKL .
__________
13
__________8. Given parallelogram ABCD , if mDAC  4 x  7 , mCAB  5x  8 and,
mDCB  7 x  13 , find mDAC .
9. Quadrilateral LMNO has vertices L  2, 4  , M (5, 2) , N  2, 1 , O  5,1 .
Use midpoints to prove that LMNO is a parallelogram.
(**Look up what you should be doing midpoints of)
Identify the error(s) in planning the solution or solving the problem.
Then write the correct solution.
A student is trying to find the sum of the angle measures of a regular 27-gon. They write
(n  2)180
n
(27  2)180
=
27
25  180
=
27
2
=166
3
Sum 
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10.
14
Geometry Unit 11 Quadrilaterals
Properties of Parallelograms – Proof Emphasis
CW 11.3
HW 11.3 on pgs: 17 & 18
Warm-up:
__________1. Given parallelogram ABCD where diagonals AC and BD intersect at E,
what congruence postulate shows that ABE  CDE ?
__________2. Given parallelogram EFGH, if mE  62 , what is mH ?
Parallelogram Proofs:
Tell why each quadrilateral ABCD is a parallelogram.
_________________ _________________ _________________ _________________ _______________
To Prove a Parallelogram:
- May need to prove two triangles congruent
- May need to use facts about parallel lines
- MUST SHOW EITHER:
1.
↔ Diagonals bisect each other
2.
3.
↔ 2 pairs opp sides 
4.
5.
↔ 2 pairs opp angles 
6.*
→ consec  ' s supp
(additional proof reason, not used to prove a
Proof Example:
↔ 2 pairs opp sides ||
1 pair opp sides  and || →
)
Given: Quadrilateral ABCD
1  2
BD bisects AC
Prove: ABCD is a parallelogram
D
C
2
4
A
1
E
B
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3
15
Proof Practice:
1.
Given: E is the midpoint of BD
BD bisects AC
Prove: ABCD is a parallelogram
2.
Given: BC || AD , A  C
Prove: ABCD is a parallelogram
From the August 2009 Regents… [6 pts]
Given: Quadrilateral ABCD, diagonal AFEC ,
AE  FC , BF  AC , DE  AC , 1  2
Prove: ABCD is a parallelogram.
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3.
16
Geometry Unit 11 Quadrilaterals
TWO PAGES!!
Properties of Parallelograms – Proof Emphasis
HW 11.3
__________1. In parallelogram EFGH, mG is 25 degrees less than mH . Find mH .
__________2. Which statement is not always true about a parallelogram?
(1) The diagonals are congruent.
(2) The opposite sides are congruent.
(3) The opposite angles are congruent.
(4) The opposite sides are parallel.
__________3. In parallelogram QRST, diagonals QS and RT intersect at point E.
Which statement is always true?
(1)
(3)
Prove:
(2)
(4)
RQS  SQT
TQE  RQE
PQRS
PE  SQ , RF  SQ
SE  QF
OVER
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4. Given:
QS  RT
RES  TEQ
17
5.
Given: ABCD is a quadrilateral
AB  CD
1  2
Prove: ABCD is a parallelogram
Statement
1. ABCD is a quadrilateral
AB  CD
1  2
2. DC AB
3. ABCD is a parallelogram
Reason
1.
2.
3.
6. The vertices of quadrilateral ABCD are given.
Draw ABCD in the coordinate plane and show that it is a parallelogram.
A(-2, 3), B(-5, 7), C(3,6), and D(6,2)
7. Find the error:
What is the mT in pentagon PQRST?
The student writes:
350  mT  540
mT  190
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130  60  160  mT  (5  2)180
18
Geometry Unit 11 Quadrilaterals
Properties of Parallelograms – Proof Emphasis
CW 11.4
HW 11.4 on pgs: 21 & 22
Warm-up:
__________a.)
Which of the following does not prove a parallelogram?
(1)
Quadrilateral with two pairs of opposite sides congruent
(2)
Quadrilateral with diagonals congruent
(3)
Quadrilateral with one pair opposite sides congruent and parallel
(4)
Quadrilateral with diagonals that bisect each other
__________b.)
Which of the following is false about a parallelogram?
(1)
It has opposite angles congruent
(2)
It has consecutive angles that are complementary
(3)
It has diagonals that bisect each other
(4)
It has opposite sides parallel
1. Quadrilateral ABCD has vertices A  2, 2  , B(1, 4) , C  2,8 , and D  1, 6  .
Use midpoints to prove that ABCD is a parallelogram.
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2. Quadrilateral ABCD has vertices A  2, 1 , B(1,3) , C  6,5 , and D  7,1 .
Prove that ABCD is a parallelogram.
19
Algebraic Proof: For what value of the variables must ABCD be a parallelogram?
3
4.
Summary:
5.
To show a parallelogram by distances, you must show the _____ distances of the _________.
To show a parallelogram by midpoints, you must show the _____ midpoints of the __________.
To show a parallelogram by slopes, you must show the _____ slopes of the __________.
Challenging Proofs:
6.
Given: Parallelogram ABCD
FG bisects DB
Prove: DB bisects FG
Given: FDEC ,
ABCD
BE  FC , AF  FC
Prove: ABEF is a parallelogram
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7.
20
Geometry Unit 11 Quadrilaterals
TWO PAGES!!
Properties of Parallelograms – Proof Emphasis
HW 11.4
For what value of the variables must ABCD be a parallelogram?
1.
2.
3. Given:
PQRS is a quadrilateral
1  2
3  4
4. Given:
Prove:
PQRS is a parallelogram
LM is a median in GKL
LM  MJ
GJKL is a parallelogram
OVER
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Prove:
21
5. Three of the vertices of parallelogram ABCD are A(-4, 1), B(-1, 5), C(6, 5), and D(x, y).
Find the coordinates of point D. Show your method.
6. The vertices of quadrilateral ABCD are given. Draw ABCD in the coordinate
plane and show that it is a parallelogram.
A(0,1), B(4,4), C(12,4), and D(8,1)
7. Find the error in the student’s conclusion: In the two pictures below, can you prove the quadrilateral is
a parallelogram based on the given information?
The student writes:
Yes, One pair of opposite sides is
congruent and one pair is parallel
b.
The student writes:
No, One pair of opposite sides is
congruent but that is not enough
information.
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a.
22
Geometry Unit 11 Quadrilaterals
Properties of Rhombi, Rectangles, Squares
Warm-up:
Properties:
CW 11.5
HW 11.5 on pgs: 25 & 26
*Review the Quadrilateral Family Tree
List the 5 properties of a parallelogram
List the 2 extra properties for a Rectangle
1.________________________________
1.__________________________
2.________________________________
2.__________________________
3.________________________________
List the 3 extra properties for a Rhombus
4.________________________________
1.__________________________
5.________________________________
2.__________________________
Square
3.__________________________
Practice:
__________1. Given rhombus ABCD, if AD  3w  7 and AB  2(w  8) , find AD and DC.
__________ Justify your answer.
__________2. Given rhombus ABCD such that diagonals AC and DB intersect at E, what is mAEB ?
__________3. Given rhombus ABCD with AB  3k  1 and the perimeter of rhombus ABCD
is 13k  1 , find AB.
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__________4. In the diagram below of rhombus ABCD, diagonals AC and
DB intersect at E. If mDAB  7 x  14 and m2  5x  5 , find m1 .
23
__________5. Given rectangle RSTW, what is mRWT ?
__________6.Given rectangle PQRS such that diagonals PR and QS intersect at T,
if PR  7a  2 and ST  4a  3 , find PT .
__________7. Using the diagram to the right of rectangle RSTW, if m1  61 ,
find mRZW .
Cool Square Shortcut: **remember the special right triangle from Trig Unit 10.**
__________8.If the length of the side of a square is 6, find the length of a diagonal of the square in
simplest radical form. Fill in the measurements on your diagram.
__________9. If the length of the side of a square is 8, find the length of a diagonal of the square in
simplest radical form. Fill in the measurements on your diagram.
What is the shortcut for finding the length of a diagonal of a square?
__________10.The diagonals of square DEFG intersect at H. Given that EH  5 , find the
perimeter of the square in simplest radical form.
If UV  z  2 , what kind of parallelogram is TUVW and why?
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__________11.In parallelogram TUVW, TU  3z 14 and WV  2 z  6 .
Find the value of z.
24
Geometry Unit 11 Quadrilaterals
TWO PAGES!!
Properties of Rhombi, Rectangles, Squares
HW 11.5
1. True or False:
__________a.) All parallelograms are quadrilaterals.
__________b.) All quadrilaterals are parallelograms.
__________c.) All rhombuses are parallelograms. (i.e., they come from parallelograms on the family tree)
__________d.) All rectangles are squares.
__________e.) All squares are rectangles.
__________f.) All squares are parallelograms.
__________2. In the diagram below of rhombus ABCD, diagonals AC and
__________
DB intersect at E. If m1  50 , find m2 and m3 .
__________3. Given rhombus ABCD such that diagonals AC and DB intersect at E,
if AC  2 y  8 and EC  2 y  1 , find EC.
4. Using the diagram to the right of rectangle RSTW,
__________a.) Find m1  m2 .
__________b.) If m1  3x  12 and m2  2 x  7 , find m1 .
6. Fill in all of the angle measures in the diagram of the square:
3/9/2015 8:22 PM
__________5. Given rectangle PQRS, if PR  5 y  2 , SQ  11y  10 , find SQ .
25
7. Based upon the markings on each figure below, determine the most precise name for each quadrilateral.
a)_____________
b)________________
c)__________________
d) __________________
8. In the table below, check the quadrilateral(s) that have the following properties
Property
Parallelogram
Rhombus
Rectangle
a. All sides 
b. Opposite sides //
c. All ’s are rt. ’s
d. Diag.s bisect ea other
e. Diags are 
f. Opposite sides 
g. Consec ’s are sup.
h. Diags are 
Square
Review:
a)_________9. a) What is the sum of the interior angle measures of a regular octagon?
b)_________ b) What is one exterior angle of a regular octagon?
___________10. What is the measure of one interior angle of a regular 12-gon?
___________11. What is the value of x in the regular polygon at the right?
___________12. If the measure of an exterior angle of a regular polygon is 24,
how many sides does the polygon have?
13. Find the errors:.
A student was asked to find the
measures of the numbered angle
in rhombus ABCD.
They wrote:
m1 = 90, Diags of rhom. 
BDC  ACD, so m2 = 40
b) A student was asked to find the
value of x in parallelogram
PQRS that makes it a rectangle.
They wrote:
5x – 2 = 4x + 1
9x = 3
x = 1/3
3/9/2015 8:22 PM
a)
26
Geometry Unit 11 Quadrilaterals
Properties of Rhombi, Rectangles, Squares – Proof Emphasis
Warm-up:
CW 11.6
HW 11.6 on pgs: 29 & 30
The diagonals of a rhombus measure 16 cm and 30 cm. Find the perimeter of the rhombus.
1. Quadrilateral ABCD has vertices A  4, 2  , B(7,3) , C  8, 6  , and D  5,5 .
Prove that ABCD is a rhombus.
3/9/2015 8:22 PM
2. The vertices of quadrilateral PQRS are P  0, 2  , Q(4,8) , R  7, 6  , and S  3, 0  .
Use slopes to prove that PQRS is a rectangle.
27
3. Quadrilateral MATH has vertices M  1, 4  , A(4, 7) , T  7, 2  , and H  2, 1 .
Prove that MATH is a square.
4.
Given: ABCD is a rhombus
AE  CE
Prove: ADE  CDE
In a parallelogram ABCD, AB  2 x  3 , BC  4 x  5 , and CD  5x  9 .
Show that ABCD is a rhombus.
3/9/2015 8:22 PM
5.
28
Geometry Unit 11 Quadrilaterals
TWO PAGES!!
Properties of Rhombi, Rectangles, Squares – Proof Emphasis
1.
HW 11.6
Given: ABCD is a rectangle
M is the midpoint of AB
Prove: DM  CM
OVER
3/9/2015 8:22 PM
2. Using the appropriate formulas, prove quadrilateral PQRS is a rectangle.
P( -1, -2 ), Q( 5 , 2 ), R( 7, -1 ), S( 1, -5 )
29
__________3. The measures of five of the interior angles of a hexagon are 150°, 100°, 80°, 165°, and 150°.
What is the measure of the sixth interior angle?
(1)
(2)
(3)
(4)
75°
105°
180°
80°
__________4. The measures of two consecutive angles of a parallelogram are in the ratio of 5:4. What is the
measure of an obtuse angle of the parallelogram?
(1)
(2)
(3)
(4)
20°
80°
100°
160°
__________5. In parallelogram ABCD, diagonals AC and BD intersect at E. If BE 
ED  x  10 , what is the value of x?
(1)
(2)
(3)
(4)
2
x and
3
30
-6
6
-30
__________6. In the diagram below of parallelogram ABCD with diagonals AC and BD , m1  45 and
mDCB  120 . What is the measure of 2 ?
(1)
(2)
(3)
(4)
30°
15°
60°
45°
(1)
(2)
(3)
(4)
12
37
40
50
3/9/2015 8:22 PM
__________7. In the accompanying diagram of rectangle ABCD, mBAC  3x  4
and mACD  x  28 . What is mCAD ?
30
Geometry Unit 11 Quadrilaterals
Properties of Rhombi, Rectangles, Squares – Proof Emphasis
Warm-up:
CW 11.7
HW 11.7 on pgs: 33 & 34
Given: ABCD is a rhombus
Prove: BFA  DFC
1.
Quadrilateral ABCD has vertices A  3, 6  , B(6, 0) , C  9, 9  , and D  0, 3 .
Prove that ABCD is a parallelogram but NOT a rhombus.
2.
Quadrilateral ABCD has vertices A  0, 2b  , B  0, 0  , C  4a, 0  , and D  4a, 2b  .
Prove that ABCD is a rectangle using slopes.
3/9/2015 8:22 PM
Practice:
31
3.
4.
Quadrilateral ABCD has vertices A  2, 1 , B(2,3) , C  4,1 , and D  0, 3 .
Prove that ABCD is a rectangle.
Given: DFEC , AGE , BGF
Rectangle ABCD
DF  CE
Prove: ADE  BCF
Given: Square ABCD
Prove: Diagonals are congruent
(By means of proving triangles congruent)
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5.
32
Geometry Unit 11 Quadrilaterals
Properties of Rhombi, Rectangles, Squares – Proof Emphasis
1.
HW 11.7
Given: Rectangle ABCD
Isosceles ALD with vertex L
Prove: L is the midpoint of BC
Given: Rhombus ABCD
CB bisects DF
Prove: BF  AD
Statement
1. Rhombus ABCD
CB bisects DF
2. DC AB
3. DCE  FBE
4. DEC  FEB
5. DE  FE
6. DCE  FBE
OVER
Reason
1.
2.
3.
4.
5.
6.
7. BF  CD
7.
8. CD  AD
8.
9. BF  AD
9.
3/9/2015 8:22 PM
2.
33
Given rectangle ABCD where diagonals AC and BD intersect at E, complete the following:
__________a.)
If AC  4 x  60 and BD  30  x , find BD .
__________b.)
If AC  4 x  60 and AE  x  5 , find EC .
__________c.)
If mBAC  4 x  5 and mCAD  5x 14 , find mCAD .
__________d.)
If AE  2 x  3 and BE  12  x , find BD .
4.
Given square ABCD where diagonals AC and BD intersect at E, complete the following:
__________a.)
If mBAC  9 x , find x.
__________b.)
If AB  x2  15 and BC  2 x , find the perimeter of the square.
5.
In a parallelogram TUVW with diagonals that intersect at X, TX  2 y  11 , VX  y  9 , WU  y  18 .
__________a.)
b.)
Find TV .
Can TUVQ be a rectangle? Why?
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3.
34
Geometry Unit 11 Quadrilaterals
Properties of Trapezoids and Kites
CW 11.8
HW 11.8 on pgs:37 & 38
Warm-up:
In the diagram to the right, XY || AB . Solve for angles x and y.
If AX and BY intersect at P, what kind of triangle is XPY ?
Properties:
*Review Family Tree
Midsegment (Median) of Trapezoid: ________________________________________________________
_______________________________________________________________________________________
__________1. The measures of the bases of a trapezoid measure 64 and 82. Find the length of the
midsegment of the trapezoid.
a._________2. Find EF in the given trapezoid below.
b._________
a)
b)
HG________
EF________
CD________
CD________
EF________
HG________
__________4. In an isosceles trapezoid, the smaller base measures 2 x  4 , the larger base measures 4 x  8 ,
and
__________ the midsegment measures 24. Find the measures of the bases.
3/9/2015 8:22 PM
3. Find the lengths of the segments with variable expressions..
35
__________5. In trapezoid RSUT, RS TU , x is the midpoint of RT , and v is the midpoint of SU .
If RS = 30 and XV = 44, find TU.
__________6. In isosceles trapezoid FJHG with JH || FG and where JH is the smaller base, if mJ  110 ,
__________ find mF , mG , and mH .
__________
Find the value(s) of the variable(s) in each kite.
7._________
8. _________
_________
__________9.Given kite STUV with ST  SV , VU  TU , diagonals SU and VT intersect at R, VR  RT ,
__________ VR  9 , UR  7 ,and SR  8 , find VU and SV in simplest radical form.
b.) Determine if MATH is an isosceles trapezoid.
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10. Quadrilateral MATH has vertices M 1,1 , A(2,5) , T  5, 7  , and H  7,5 .
a.) Prove that MATH is a trapezoid.
36
Geometry Unit 11 Quadrilaterals
Properties of Trapezoids and Kites
TWO PAGES!!!
HW 11.8
Find EF in each trapeoid
__________1.
_________2.
__________3.Given kite JKLM with JM  JK and ML  KL , if mM  88 and mL  120 , find mK
__________ and mJ .
__________4.Given kite JKLM with JM  JK and ML  KL , if mJ  60 and mL  50 , find mK .
__________5. Given kite STUV with ST  SV , VU  TU , diagonals SU and VT intersect at R,
SR  TR  VR ,
__________ UR  12 ,and SR  5 , find VU and SV in simplest radical form.
6. Find the Errors:
A) The student is given: segment MS is the midsegment
of trapezoid WXYZ. What is the value of x?
B) The student is given: Quad TUVW is a kite.
What the mTUV and mTWY?
They write:
They write:
mTUV + mUTW + mUVW = 180
x + 40 + 60 = 180
x = 80
mTUV = mTWV = 80
3/9/2015 8:22 PM
MS = WX + ZY
4x – 4 = 18 + 3x + 4
x – 4 = 22
x = 26
37
7. Use the given vertices to graph parallelogram JKLM. Classify the figure and explain your reasoning.
Then, find the perimeter of parallelogram JKLM.
3/9/2015 8:22 PM
J( -4, 2 ), K( 0 , 3 ), L( 1 , -1 ), M( -3 , -2 )
38
Geometry Unit 11 Quadrilaterals
Properties of Trapezoids and Kites – Proof Emphasis
Warm-up:
CW 11.9
HW:11.9 on pgs: 41 & 42
Name each type of quadrilateral for which the statement is true (parallelogram, rhombus,
rectangle, square).
a.)
Both pairs of opposite angles are congruent. _____________________________________________
b.)
The quadrilateral is equilateral. ________________________________________________________
c.)
Given rhombus MNOP, the diagonals intersect at Q and mQNO  48 , find mNPO .
Proof Reasons & Theorems:
Trap  quad w/ only 1 pr // sides
Isos Trap  Trap w/ non // sides 
Isos Trap  Trap w/ base ’s 
Isos Trap  Trap w/ diags 
Median of trap = average of bases it’s || to
Kite → diags 
Kite → only 1 pr opp ’s 
Kite  2 pr adjacent sides 
Median of trap meets midpts of non-|| legs
Practice:
Given: TRAP is a trapezoid
TA  RP
Prove: RPA  TAP
3/9/2015 8:22 PM
1.
39
2.
Given: ABCD is a kite
Prove: BE bisects CBA
Quadrilateral ABCD has vertices A  0, 4  , B(0, 8) , C  3, 4  , and D  3,1 .
Prove that ABCD is a trapezoid but NOT an isosceles trapezoid.
3/9/2015 8:22 PM
3.
40
10R Unit 11 Quadrilaterals
TWO PAGES!!!
Properties of Trapezoids and Kites – Proof Emphasis
1. Complete the following proof:
DE || AV
DAV  EVA
Prove: DAVE is an isosceles trapezoid
2. DAVE is a
trapezoid
3.
4. DAVE is an
isosceles
trapezoid
2.
Reason
1.
2.
3. CPCTC
4.
Show that quadrilateral A  0, 2  , B  9,1 , C  4,6  , D 1,5
is an isosceles trapezoid. SHOW ALL WORK.
OVER
3/9/2015 8:22 PM
Given:
Statement
1. DE || AV
DAV  EVA
HW 11.9
41
3.
Given: Isosceles trapezoid ABCD with AB || CD
Prove: 1  2
Points P, Q, R, S are the vertices of a quadrilateral. Give the most specific name for PQRS.
Justify your answer: P 1,0  , Q 1, 2  , R  6,5 , S 3,0 
3/9/2015 8:22 PM
4.
42
Geometry Unit 11 Quadrilaterals
Special Quadrilaterals
Warm-up:
#
2
3
4
5
6
7
8
9
10
Property
Rectangle
Rhombus
Square
Kite
Trapezoid
Isosceles
Trapezoid
Both pairs
opposite sides
congruent
Both pairs
opposite angles
congruent
Exactly one pair
of opposite sides
congruent
Exactly one pair
of opposite sides
parallel
Exactly one pair
of opposite angles
congruent
Consecutive
angles are
supplementary
Diagonals bisect
each other
Diagonals are
congruent
Diagonals are
perpendicular
Diagonals bisect
the angles
Practice:
1.
Put an X in the box if the quadrilateral always has the given property.
Give the most specific name for each quadrilateral. Explain your reasoning.
2.
3.
4.
3/9/2015 8:22 PM
1
CW 11.10
HW: 11.10 on pgs:
43
Tell whether enough information is given in the diagram to classify the quadrilateral by the indicated name.
5.
Rectangle
6.
Isosceles Trapezoid
7.
Rhombus
8.
Kite
Which two segments or angles must be congruent so that you can prove that FGHJ is the indicated
quadrilateral? There may be more than one correct answer.
9.
Kite
10.
Isosceles Trapezoid
Directions: In #11-16, use the information in and below each diagram and the properties of various
quadrilaterals to find x, y, and z, as required. Label answers with appropriate units.
C
120°
A
70°
U
12.
x
60°
B
R
Quadrilateral ABCD
D
13.
A
Rectangle ABCD
y
T
S
Parallelogram RSTU
C
x
y
z
70°
O
14.
z
50°
x
N
130°
x
B
z
y
L
M
Isosceles Trapezoid LMNO
3/9/2015 8:22 PM
D
11.
44
15.
16.
T
S
H
y
120°
30°
x
G
x
z
Q
E
R
Rhombus QRST
y
20°
F
Trapezoid EFGH
__________17. For what value of x is the quadrilateral a parallelogram?
__________18. Find the length of the midsegment or find the value of x.
__________
__________19. JKLM is a kite. Find mK .
Which statement is true?
(1)
(2)
(3)
(4)
__________21.
Which quadrilateral does not necessarily have congruent diagonals?
(1)
(3)
__________22.
All parallelograms are quadrilaterals.
All parallelograms are rectangles.
All quadrilaterals are trapezoids.
All trapezoids are parallelograms.
isosceles trapezoid
rhombus
(2)
(4)
square
rectangle
True or False: All squares are similar to each other.
(1)
True
(2)
False
3/9/2015 8:22 PM
__________20.
45
__________23.
To prove that a parallelogram is a rectangle, it is
sufficient to show that the
(1)
(3)
__________24.*
diagonals are congruent
(2)
opposite sides are congruent (4)
A parallelogram must be a rectangle if the opposite angles are
(1) congruent
(3) supplementary
__________25.
congruent and bisect the angles to which they are drawn
congruent and do not bisect the angles to which they are drawn
not congruent and bisect the angles to which they are drawn
not congruent and do not bisect the angles to which they are drawn
A quadrilateral must be a square if
(1)
(2)
(3)
(4)
27.
(2) equal in measure
(4) complementary
A parallelogram must be a square if the diagonals are
(1)
(2)
(3)
(4)
__________26.
diagonals are perpendicular
adjacent sides are congruent
diagonals are congruent
sides and angles are congruent
opposite sides and opposite angles are congruent
diagonals bisect each other and are perpendicular to each other
In quadrilateral ABCD, mA  x  10 , mB  2x  10 , mC  2 x  70 , and mD  3x  50 .
What kind of quadrilateral is ABCD and why?
Type of Quadrilateral: ____________________
Why? ____________________________________________________________________________
In parallelogram ABCD, AB  3x  2 , DC  10 x  12 , and AD  5x  2 .
What kind of parallelogram is ABCD? Justify your answer.
Type of Parallelogram: ____________________
Justification: _____________________________________________________________________
3/9/2015 8:22 PM
28.
46
29.
__________30.
In rectangle ABCD, diagonals AC and BD intersect at P. If CP  40 and BD  2 x 12 .
What is the value of x?
__________31.
WXYZ is a parallelogram, YA is an altitude to WX , and YA  AX . Find mZ .
W
X
Y
__________32.
The lengths of the diagonals of a rhombus are 10 and 24.
Find the perimeter of the rhombus.
__________33.
The lengths of the diagonals of a rhombus are 14 and 48.
Find the perimeter of the rhombus.
__________34.
__________
In an isosceles trapezoid, the smaller base measures 2 x  4 , the larger base measures
4 x  8 , and the midsegment measures 24. Find the measures of the bases.
3/9/2015 8:22 PM
Z
A
47
__________35.
__________
The sum of the measures of the interior angles of a convex polygon is given.
Classify the polygon.
a.)
1080°
b.)
__________37.
2520°
__________36.
Solve for x.
Solve for x.
__________38.
__________
In an isosceles trapezoid, the smaller base measures 3x  1 , the larger base measures
6 x  8 , and the midsegment measures 36. Find the measures of the bases.
__________39.
In isosceles trapezoid ABCD, diagonal AC measures 2 x  4 and diagonal BD
measures 4 x  8 . Find the measure of AC .
Study Polygons:
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Regular Polygon:
40.
Pentagon
41.
Heptagon
42.
Decagon
43.
Nonagon
44.
Octagon
8 sides
9 sides
10 sides
11 sides
12 sides
Sum of its interior angles:
Octagon
Nonagon
Decagon
11-gon
Dodecagon
Measure of an interior angle:
3/9/2015 8:22 PM
3 sides
4 sides
5 sides
6 sides
7 sides
48
Geometry Unit 11 Quadrilaterals
Quadrilaterals Review Packet – Proofs Practice
1.
Quadrilateral QUAD has vertices Q  1,1 , U (3, 4) , A 1,5 , and D  3, 2  .
Prove that QUAD is a parallelogram.
2. Given: PQRS is a parallelogram
PS  QT
Prove: QRT is isosceles
Quadrilateral ABCD has vertices A  5, 0  , B(2,9) , C  4, 7  , and D  1, 2  .
Prove that ABCD is a rectangle.
3/9/2015 8:22 PM
3.
49
4.
Given: Rectangle ABCD
E is the midpoint of DC
Prove: 1  2
6.
Quadrilateral ABCD has vertices A  3, 2  , B(2, 6) , C  2, 7  , and D 1,3 .
Prove that ABCD is a rhombus.
Given: Rhombus ABCD
E is the midpoint of DF
Prove:
AD  BF
3/9/2015 8:22 PM
5.
50
7.
Quadrilateral PQRS has vertices P  0, 0  , Q(4,3) , R  7, 1 , and S  3, 4  .
Show that PQRS is a square.
8.
Quadrilateral DEFG has vertices D  4, 0  , E (0,1) , F  4, 1 , and G  4, 3 .
Prove that ABCD is a trapezoid but NOT an isosceles trapezoid
9.
Given: Trapezoid ABCD, BC || AD ,
BE and CF are altitudes, AE  DF
3/9/2015 8:22 PM
Prove: Trapezoid ABCD is isosceles
51
Geometry Reasons for Proofs:
Transitive
Reflexive
Symmetric
Substitution
Add Post of Seg
Add Post of ’s
Sub Post of Seg
Sub Post of ’s
//lines  alt int ’s 
//lines  alt ext ’s 
//lines  corresp ’s 
//lines  co-int ’s supp
//lines   to same line
Seg bisector  2  seg
Midpt  2  seg
 bisector  2  ’s
 bisector  2  seg & 2  rt ’s
 lines  2  rt ’s
10R: 2 steps:
 lines  rt ’s AND All rt ’s 
Median  2  seg
Altitude  2  rt ’s
10R: 2 steps
Altitude  2 rt ’s AND All rt ’s 
Vertical ’s 
Halves of  ’s are 
SSS  SSS
SAS  SAS
ASA  ASA
AAS  AAS
HL(R)  HL(R)
CPCTC
Isosc    w/ 2  sides & 2  ’s
Sides opp  ’s 
’s opp  sides 
3/9/2015 8:22 PM
Comp ’s  + to 90
Comp of  ’s 
Rt ’s   ’s
Rt  has 1 rt 
All rt ’s are 
Supp ’s  + to 180
Linear pairs  supp ’s
Supp of  ’s 
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