Download Geometry Chapter 6 Blank Notes

6.1: ________________________________________________
Geometry
Date: __________
Theorem 6-1: Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is:
Ex 1). What is the sum of the angle measures of a 10-gon?
Ex 2). What is the sum of the angle measures of a 13-gon?
An ________________ polygon
An ________________ polygon
A ________________ polygon
is a polygon with all sides
is a polygon with all angles
is a polygon that is both
congruent.
congruent.
equilateral & equiangular.
Corollary to the Polygon Angle-Sum Theorem
The measure of _______ angle of a _____________ polygon is
Ex 3). Marcy creates a floor tile pattern using squares, regular hexagons, and regular
dodecagons (12-sided polygon). What is the measure of each angle in one regular dodecagon?
Ex 4). What is 𝑚∠𝐷 in quadrilateral ABCD?
Theorem 6-2: Polygon Exterior Angle-Sum Theorem
The sum of the measures of the ____________ angles of a polygon,
one at each vertex, is __________.
Ex 5). What is the measure of an exterior angle of a regular hexagon?
Ex 6). Find the value of each variable.
Homework: pg. 374 #1 – 3, 5, 7 – 13, 15 – 23, 26 – 28
Ex 7). Find the value of each variable.
6.2 Day 1: ________________________________________________
Geometry
Properties of Parallelograms
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Date: __________
Ex 1). KLMN is a parallelogram. Find MN and KN.
Ex 2). STVW is a parallelogram. Find the missing angle measures.
Ex 3). JKLM is a parallelogram. Find JN.
Ex 4). Find the values of x and y in the parallelogram.
Homework: pg. 381 #1 – 4, 7 – 10, 12 – 16, 21 – 23
6.2 Day 2: ___________________________________________
Geometry
Date: __________
Ex 1). Solve a system of linear equations to find the values of a and b in parallelogram HIJK. What
are HJ and IK?
Ex 2). Solve a system of linear equations to find the values of x and y in the parallelogram.
Ex 3). Solve a system of linear equations to find the values of 𝑥 and 𝑦 in the parallelogram.
Theorem 6-7:
If three (or more) parallel lines cut off congruent
segments on one transversal, then they cut off
congruent segments on every transversal.
Ex 4). In the figure below, ̅̅̅̅̅
𝑅𝑊 ∥ ̅̅̅̅
𝑆𝑉 and ̅̅̅̅
𝑆𝑉 ∥ ̅̅̅̅
𝑇𝑈. If RS = ST = 5 and WV = 7, what is WU?
Ex 5). Find each length.
a) EB: ___________
d) BD: ___________
b) AF: ___________
e) AK: ___________
c) CD: ___________
f) GJ: ___________
Homework: pg. 385 #1, 2 – 15, 21 – 28
6.3: _______________________________________________
Geometry
Date: __________
The following theorems are converses of the theorems from lesson 6-2. If a quadrilateral has any
of the following properties, then it must be a ________________________.
Theorem 6-8: “If both pairs of opposite sides of a quadrilateral are ____________, then the
quadrilateral is a parallelogram.”
Theorem 6-9: “If an angle of a quadrilateral is supplementary to both of its _______________ angles,
then the quadrilateral is a parallelogram.”
Theorem 6-10: “If both pairs of opposite angles of a quadrilateral are ___________, then the
quadrilateral is a parallelogram.”
Theorem 6-11: “If the diagonals of a quadrilateral __________ _____ ________, then the quadrilateral
is a parallelogram.”
Theorem 6-12: “If one pair of opposite sides is both congruent and ____________, then the
quadrilateral is a parallelogram.”
Ex 1). For what value of x must RSTU be a parallelogram?
Ex 2). For what values of x and y must EFGH be a parallelogram?
Ex 3). Can you prove that the quadrilateral is a parallelogram based on the given information?
Explain.
a)
b)
c)
Ex 4). Can you prove the quadrilateral is a parallelogram based on the given information?
Explain.
a) Given: AE = CE = 14, DB = 2DE
b) Given: 𝑚∠𝑌 = 49, 𝑚∠𝑍 = 131, 𝑚∠𝑊 = 49
Ex 5). A table tray can be adjusted up or down by raising or lowering the table on its hinged legs
as shown. Will the table always remain parallel to the surface it sits on? Explain.
Ex 6). For what value of x must the quadrilateral be a parallelogram?
a)
b)
c)
d)
Homework: pg. 392 #1 – 3, 7 – 14, 19 – 21, 27, 28
6.4: _________________________________________________
Geometry
Date: __________
Special Parallelograms:
A ______________ is a parallelogram with ______ ____________ sides.
A ______________ is a parallelogram with ______ _________ angles.
A ______________ is a parallelogram with ______ ____________ sides
and _______ __________ angles.
Ex 1). Is parallelogram FGHJ a rhombus, a rectangle, or a square? Explain.
Ex 2). List the quadrilaterals that have the given property. Choose among parallelogram,
rhombus, rectangle, and square.
a) Opposite angles are supplementary
b) Consecutive sides are congruent
c) Consecutive sides are perpendicular
d) Consecutive angles are congruent
Ex 3). Decide whether the parallelogram is a rhombus, rectangle, or square. Explain.
a)
b)
c)
More on Rhombuses:
Theorem 6-13:
If a parallelogram is a rhombus, then its diagonals
are _________________________.
Theorem 6-14:
If a parallelogram is a rhombus, then each diagonal
___________ a pair of opposite angles.
Ex 4). What are the measures of the numbered angles in the rhombus?
a)
b)
More on Rectangles:
If a parallelogram is a rectangle, then its
diagonals are ______________.
Ex 5). In rectangle MNOP, PN = 7x – 8 and MO = 4x + 10. What is the length of PN?
FYI: Everything that is true for rhombuses and rectangles is true for a square!
Homework: pg. 400 #1 – 4, 9 – 23, 24 – 32(e), 34, 35
6.5: _______________________________________________
Geometry
Date: __________
The following theorems are converses of lesson 6.4. Each theorem allows you to prove that a
parallelogram is either a rhombus or a rectangle.
Theorem 6-16: “If the diagonals of a parallelogram are ___________________________, then the
parallelogram is a _____________.”
Theorem 6-17: “If one diagonal of a parallelogram ____________ a pair of opposite angles, then
the parallelogram is a ___________.”
Theorem 6-18: “If the diagonals of a parallelogram are _____________, then the parallelogram is a
_______________.
*Notice that if a parallelogram is ________ a rectangle and a rhombus, then it is a ____________.
Ex 1). Can you conclude that the parallelogram is a rhombus, rectangle, or square? Explain.
a)
Ex 2). For what value of x is ABCD a square?
b)
Ex 3). For what value of x is DEFG a rhombus?
Ex 4). LN = 54. For what values of x is LMNO a rectangle?
Ex 5). For what value of x is RSTU a rhombus? What is 𝑚∠𝑆𝑅𝑇? What is 𝑚∠𝑈𝑅𝑆?
Homework: pg. 407 #1 – 5, 8 – 13, 16 – 18, 23 – 29
6.6: ________________________________________________
Geometry
Date: __________
TRAPEZOID
Theorem 6 -19: “If a quadrilateral is an isosceles trapezoid,
then each pair of base angles is __________________.”
Ex 1). RSTU is an isosceles trapezoid and 𝑚∠𝑆 = 75.
What are 𝑚∠𝑅, 𝑚∠𝑇, 𝑎𝑛𝑑 𝑚∠𝑈?
Ex 2). LMNO is an isosceles trapezoid and
𝑚∠𝐿 = 75. What are 𝑚∠𝑀, 𝑚∠𝑁, 𝑎𝑛𝑑 𝑚∠𝑂?
Theorem 6-20: “If a quadrilateral is an isosceles trapezoid,
then its diagonals are ________________.”
Ex 3). WXYZ is an isosceles trapezoid,
and WY = 12, what is XZ?
Ex 4). AC = x + 5 and DB = 2x – 2.
Find the value of x and each diagonal.
Theorem 6-21: “If a quadrilateral is a trapezoid then,
1. The midsegment is ________________ to the bases
2. The length of the midsegment is ______________ of the
lengths of the bases.
Ex 5). TU is the midsegment of
trapezoid WXYZ. What is x?
Ex 6). Find GH.
Ex 7). Find the lengths of the segments with variable expressions.
KITE
Theorem 6-22: “If a quadrilateral is a kite, then
its diagonals are ____________________.
Ex 8). ABCD is a kite. What are 𝑚∠1 𝑎𝑛𝑑 𝑚∠2?
The following are kites, find the missing angles:
Ex 9).
Ex 10).
Find the value of the variable in each kite.
Ex 11).
Ex 12).
Ex 13). Determine whether each statement is true or false. Explain.
a) All kites are quadrilaterals
b) A kite is a parallelogram
c) A kite can have congruent diagonals
d) Both diagonals of a kite bisect angles at the vertices.
Homework: 6.6 Practice Worksheet
6.7: ________________________________________________
Geometry
Date: __________
You will need …
Distance Formula:
Slope Formula:
Classifying Triangles:
Scalene
Isosceles
Ex 1). Is Triangle RST scalene, isosceles, or equilateral?
Ex 2). Is parallelogram ABCD a rhombus? Explain.
Equilateral
Ex 3). Parallelogram MNPQ has vertices M(0, 1), N(-1, 4), P(2, 5), and Q(3, 2). Is MNPQ a
rectangle? Is it a square?
Ex 4). Determine whether the parallelogram is a rhombus, rectangle, square, or none. Explain.
P(-1, 2), O(0, 0), S(4, 0), T(3, 2)
Homework: pg. 428 #1, 2, 5 – 7, 10, 12, 23 (4 graphs)