unit 7 quadrilaterals 2014-15 geometry. unit outline 6-1 the polygon angle-sum theorems 6-2...
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Unit 7Quadrilaterals
2014-15 Geometry
Unit Outline
6-1 The Polygon Angle-Sum Theorems
6-2 Properties of Parallelograms
6-3 Proving that a Quadrilateral is a Parallelogram
6-4 Properties of Rhombuses, Rectangles and Squares
6-5 Conditions for Rhombuses, Rectangles and Squares
6-6 Trapezoids and Kites
6-7 Polygons in the Coordinate Plane
Unit Standards
MAFS.912.G-CO.3.11 (DOK 3)
Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
MAFS.912.G-GPE.2.4 (DOK 2)
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
6-1 The Polygon Angle-Sum Theorems
Unit 7 - Quadrilaterals
Vocabulary
Interior Angles of a Polygon
The angles on the inside of a polygon are called interior angles.
Exterior Angles of a Polygon
The exterior angles of a polygon are those formed by extending sides. There is one exterior angle at each vertex.
Vocabulary
Equilateral Polygon
Polygon with all sides congruent
Equiangular Polygon
Polygon with all angles congruent
Regular Polygon
A Polygon that is both equilateral and equiangular.
Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is (n 2)180.
You can write this as a formula. This formula works for regular and irregular polygons.
Sum of angle measures = (n 2)180
Polygon Angle-Sum Theorem
What is the sum of the measures of the angles in a hexagon?
Solution: There are six sides, so n = 6.
Sum of angle measures = (n 2)180
= (6 − 2)180 Substitute 6 for n
= 4(180) Subtract.
= 720 Multiply.
The sum of the measures of the angles in a hexagon is 720
Note: You can use the formula to find the measure of one interior angle of a regular polygon if you know the number of sides.
Polygon Angle-Sum Theorem
What is the measure of each angle in a regular pentagon?
Solution: A pentagon has 5 sides, so n = 5.
Sum of angle measures = (n 2)180
= (5 − 2)180 Substitute 5 for n
= 3(180) Subtract.
= 540 Multiply.
Divide by the number of angles:
Measure of each angle = 540 5
= 108 Divide.
Each angle of a regular pentagon measures 108.
Your Turn!
Find the sum of the interior angles of each polygon.
Quadrilateral Solution: Sum of interior angles = 360
Decagon Solution: Sum of interior angles =1440
Find the measure of an interior angle of each regular polygon.
Decagon Solution: Interior Angle = 144
32-gon Solution: Interior Angle = 168.75
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon is 360.
Example:
A pentagon has five exterior angles. The sum of the measures of the exterior angles is always 360, so each exterior angle of a regular pentagon measures 72.
Your Turn!
Find the measure of an exterior angle for each regular polygon.
Octagon Solution: Exterior Angle Measure = 45
Hexagon Solution: Exterior Angle Measure = 60
6-2 Properties of Parallelograms
Unit 7 - Quadrilaterals
Vocabulary
Parallelograms
A quadrilateral with both pairs of opposite sides parallel. The opposite sides are congruent. (Theorem 6-3)
The consecutive angles are supplementary. (Th. 6-4)
The opposite angles are congruent. (Th. 6-5)
The diagonals bisect each other. (Th. 6-6)
Solving Parallelograms
Find the value of x.
Solution: Because the consecutive angles are supplementary,
x + 60 = 180
x = 120
Solving Parallelograms
Find the value of x.
Solution: Because opposite sides are congruent,
x + 7 = 15
x = 8
Solving Parallelograms
Find the value of x and y.
Solution: Because the diagonals bisect each other, y = 3x and 4x = y + 3.
4x = y + 3
4x = 3x 3 Substitute for y.
x = 3 Subtraction Property of =
y = 3x Given
y = 3(3) Substitute for x.
y = 9 Simplify.
Your Turn!
Find the value of x in each parallelogram.
Solution: x = 8
Solution: x = 3, y = 6
Solution: x = 21
Solution: x = 50
6-3 Proving that a Quadrilateral is a Parallelogram
Unit 7 - Quadrilaterals
Vocabulary
Definition of a Parallelogram
If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram.
Theorem 6-8
If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
Theorem 6-10
If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.
Vocabulary
Theorem 6-11
If the diagonals bisect each other, then the quadrilateral is a parallelogram.
Theorem 6-12
If one pair of sides is both congruent and parallel, then the quadrilateral is a parallelogram.
Solving Parallelograms
For what value of x and y must figure ABCD be a parallelogram?
Solution: In a parallelogram, the two pairs of opposite angles are congruent. So, in ABCD, you know that x 2y and 5y + 54 4x. You can use these two expressions to solve for x and y.
Step 1: Solve for y. 5y + 54 4x
5y + 54 4(2y) Substitute 2y for x.
5y + 54 8y Simplify.
54 3y Subtract 5y from each side.
18 y Divide each side by 3.
Step 2: Solve for x. x 2y Opposite angles of a parallelogram are congruent.
x 2(18) Substitute 18 for y.
x 36 Simplify.
For ABCD to be a parallelogram, x must be 36 and y must be 18.
Your Turn!
For what value of x must the quadrilateral be a parallelogram?
Solution: x = 18 Solution: x = 3
6-4 Properties of Rhombuses, Rectangles and Squares
Unit 7 - Quadrilaterals
Vocabulary
Rhombus
A parallelogram with four congruent sides.
Special Features: The diagonals are perpendicular.
The diagonals bisect a pair of opposite angles.
Rectangle
A parallelogram with four congruent angles. These angles are all right angles.
Special Features: The diagonals are congruent.
Vocabulary
Square
A parallelogram with four congruent sides and four congruent angles. A square is both a rectangle and a rhombus. A square is the only type of rectangle that can also be a rhombus.
Special Features: The diagonals are perpendicular.
The diagonals bisect a pair of opposite angles (forming two 45 angles at each vertex).
The diagonals are congruent.
Finding Angle Measures
Determine the measure of the numbered angles in rhombus DEFG.
Solution: 1 is part of a bisected angle. mDFG = 48, so m1 = 48.
Consecutive angles of a parallelogram are supplementary.
mEFG = 48 + 48 = 96, so mDGF = 180 96 = 84.
The diagonals bisect the vertex angle, so m2 = 84 2 = 42.
Finding Diagonal Length
In rectangle RSBF, SF = 2x + 15 and RB = 5x – 12. What is the length of the diagonal?
Solution: The length of the diagonals of a rectangle are congruent, so SF = RB.
Step 1. Solve for x. SF = 2x + 15 and RB = 5x – 12
SF = RB
2x + 15 = 5x – 12 Substitute values of SF and RB.
15 = 3x – 12
27 = 3x
x = 9 Simplify.
Step 2. Solve for the length of a diagonal.
SF = 2(9) + 15 Substitute the value of x.
SF = 18 + 15
SF = 33 Simplify.
Your Turn!
Determine the measure of the numbered angle.
Solution: 1 = 78 and 2 = 90
TUVW is a rectangle. Find the value of x and the length of each diagonal.
TV = 10x – 4 and UW = 3x + 24
Solution: x = 4; TV = 36; UW = 36
6-5 Conditions for Rhombuses, Rectangles and Squares
Unit 7 - Quadrilaterals
Vocabulary
A parallelogram is a rhombus if either:
The diagonals of the parallelogram are perpendicular. (Theorem 6-16)
A diagonal of the parallelogram bisects a pair of opposite angles. (Th. 6-17)
A parallelogram is a rectangle if the diagonals of the parallelogram are congruent.
Using Properties of Special Parallelograms
For what value of x is DEFG a rhombus?
Solution: In a rhombus, diagonals bisect opposite angles.
So, m DGDF = m DEDF.
(4x + 10) = (5x + 6) Set angle measures equal to each other.
10 = x + 6 Subtract 4x from each side.
4 = x Subtract 6 from each side.
Your Turn!
SQ = 14. For what value of x is PQRS a rectangle? Solve for PT. Solve for PR.
Solution: x = 6
For what value of x is RSTU a rhombus? What is m SRT? What is m URS?
Solution: x = 48
6-6 Trapezoids and Kites
Unit 7 - Quadrilaterals
Vocabulary
Trapezoid
A quadrilateral with exactly one pair of parallel sides. The two parallel sides are called bases. The two nonparallel sides are called legs.
Midsegment
Parallel to the bases, the length of the midsegment is half the sum of the lengths of the bases.
Vocabulary
Isosceles Trapezoid
A trapezoid in which the legs are congruent.
An isosceles trapezoid has some special properties: Each pair of base angles is congruent.
The diagonals are congruent
Finding Angle Measures in Trapezoids
CDEF is an isosceles trapezoid and m C = 65. What are m D, m E, and m F?
Solution:
m C + m D = 180 Same-side interior angles are supplementary
65 + m D = 180
m D = 115 Simplify
m C = m F = 65 Base Angles are Congruent
m D = m E = 115
C F
D E
Vocabulary
Kite
A quadrilateral in which two pairs of consecutive sides are congruent and no opposite sides are congruent.
In a kite, the diagonals are perpendicular. (Theorem 6-22)
Notice that the sides of a kite are the hypotenuses of four right triangles whose legs are formed by the diagonals.
Proving Congruent Triangles in a Kite
Statement Reasoning
1) Given
2) mFKG = mGKH = mHKJ = mJKF = 90 Th. 6-22
3) Reflexive Property
4) HL Theorem
5) CPCTC
6) Reflexive Property
7) SAS Postulate
8) Given
9) Reflexive Property
10) SSS Postulate
H
Your Turn!
In kite FGHJ in the problem, m JFK = 38 and m KGH = 63. Find the following angle and side measures.
m FKJ Solution: 90
m GHK Solution: 27
If FG = 4.25, what is JF? Solution: 4.25
If HG is 5, what is JH? Solution: 5
H
6-7 Polygons in the Coordinate Plane
Unit 7 - Quadrilaterals
Key Concepts
Distance Formula
Midpoint Formula
Slope Formula
Classifying a Triangle
Is scalene, isosceles or right?
Solution: Find the lengths of the sides using the Distance Formula.
The sides are all different lengths. So, ABC is scalene.
Classifying a Parallelogram
Is quadrilateral GHIJ a parallelogram?
Solution: Find the slopes of the opposite sides.
So, and . Therefore, GHIJ is a parallelogram.
Your Turn!
has vertices at Determine whether is scalene, isosceles or equilateral. Explain.
Solution: The triangle is isosceles because the measure of two sides of the triangle are the same.
Trapezoid ABCD has vertices at . Which formula would help you find out if this trapezoid is isosceles? Is this an isosceles trapezoid? Explain.
Solution: The slope formula can be used to determine the slopes of each base. The distance formula can be used to determine the length of the legs.
Yes, trapezoid ABCD is an isosceles trapezoid. (Determined from calculating the slopes of AB and CD as well as the lengths of the legs AD an BC.)
