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1
Ch 6: Quadrilaterals
6‐1 Classifying Quadrilaterals
6‐2 Properties of Parallelograms
6‐3 Proving that a Quadrilateral is a Parallelogram
6‐4 Special Parallelograms
6‐5 Trapezoids and Kites
6‐6 Placing Figures in the Coordinate Plane
6‐7 Proofs Using Coordinate Geometry (optional)
6‐1 Classifying Quadrilaterals
Focused Learning Target: I will be able to
Define and classify special types of quadrilaterals
CA Standards: Geo 12.0: Students find and use measures of sides and or interior angles of polygons to classify figures and solve problems
Vocabulary:
Parallelogram
Rhombus
Rectangle
Square
Kite
Trapezoid
Isosceles trapezoid
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Example 1: Classifying a Quadrilateral I’ll do one:
Judging by appearance, classify ABCD in as many ways as possible.
We’ll do one together:
Judging by appearance, classify WXYZ in as many ways as possible.
You Try:
Judging by appearance, classify this polygon in as many ways as possible.
Example 2: Classifying by Coordinate Methods I’ll do one:
Determine the most precise name for the quadrilateral LMNP Step 1: Find the slope of each side:
Slope of LM =
Slope of MN =
Slope of NP =
Slope of LP = Step 2: Use the distance formula to see if any pairs of sides are congruent:
LM =
MN =
NP =
LP =
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We’ll do one together:
Determine the most precise name for the quadrilateral with vertices G(‐4, 4), E(‐2,9), O(8,9), and M(10,4) Step 1: Graph the quadrilateral with given vertices: Step 2: Find the slope of each side:
Slope of EG =
Slope of EO =
Slope of MO =
Slope of MG = Step 3: Use distance formula to see if any pairs of sides are congruent:
Distance of EG =
Distance of EO =
Distance of MO =
Distance of MG =
You Try:
Determine the most precise name for the quadrilateral ABCD Step 1: Find the slope of each side:
Slope of AB =
Slope of BC =
Slope of CD =
Slope of AD = Step 2: Use distance formula to see if any pairs of sides are congruent:
Distance of AB =
Distance of BC =
Distance of CD =
Distance of AD =
Example 3: Using the Properties of Special Quadrilaterals I’ll do one:
Find the values of the variables for the rhombus. Then find the lengths of the sides.
4
We’ll do one together:
Find the values of the variables for the kite. Then find the lengths of the sides.
You Try:
Find the values of the variables for the parallelogram. Then find the lengths of the sides.
5
6‐2 Properties of Parallelograms
Focused Learning Target: I will be able to
Use the relationships among sides and among angles of parallelograms
Use relationships involving diagonals of parallelograms or transversals
Vocabulary:
Consecutive angles
CA Standards: Geo 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, and the properties of quadrilaterals Geo 13.0 Students prove basic relationships between angles in polygons by using properties of supplementary angles.
Vocabulary & Key Concepts:
Example 1: Using Consecutive Angles
I’ll do one:
We’ll do one together:
6
You Try:
Example 2: Using Algebra I’ll do one:
We’ll do one together:
You Try:
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Example 3: Using Algebra I’ll do one:
If 2 , 4, 2,PT x TR y QT x TS y , find x and y.
We’ll do one together:
You Try:
Find the values of x and y in the parallelogram below:
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Example 4:
I’ll do one:
We’ll do one together:
Find XZ and TZ
You try :
Find XV and YV.
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6‐3 Proving that a Quadrilateral is a Parallelogram
Focused Learning Target: I will be able to
Determine whether a quadrilateral is a parallelogram.
CA Standards: Geo 7.0: Students write prove and use theorems involving the properties of quadrilaterals Geo 12.0: Students find and use measures of sides and of interior angles of polygons to classify figures and solve problems.
Parallelogram: A Quadrilateral with 2 pairs of parallel sides.
Properties of Parallelograms: (All) Opposite sides are congruent. (All) Opposite angles are congruent. (All) Consecutive angles are supplementary. Diagonals bisect each other. I’ll do one:
Find the value of the variables for which ABCD must be a parallelogram.
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We’ll do one together:
Find the values of the variables for which ABCD must be a parallelogram.
You do one:
Find the value of the variables for which LPMN must be a parallelogram.
6‐4 Special Parallelograms Focused Learning Target: I will be able to
Use properties of diagonals of rhombuses and rectangles
Determine whether a parallelogram is a rhombus or a rectangle
CA Standard(s): Geo 7.0 Students prove and use theorems involving the properties of quadrilaterals Geo 12.0 Students find and use measure of sides and of interior angles of polygons to classify figures and solve problems. Geo 13.0 Students prove relationships between angles in polygons by using properties of complementary and supplementary angles
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Notice: In the figure, diagonal AC forms &ABC ADC which are both isosceles triangles. This means that angles 1 and 3 are congruent (base angles) and angles 2 and 4 are also congruent. Example 1: Finding Angle Measures I’ll do one:
Find the measure of angles 1, 2, 3 & 4. Justify each step.
Statement Justification/Reason
We’ll do one together:
Find the measure of angles 1, 2, 3 & 4. Justify each step.
Statement Justification/Reason
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You try:
Find the measure of angles 1, 2, 3 & 4. Justify each step.
Statement Justification/Reason
Example 2: Finding Diagonal Length I’ll do one:
Find the values of x: Given: LMNP is a rectangle. Find the value of x and the length of each diagonal if LN = x and MP = 2x‐4.
We’ll do one together:
Find the values of x: Given: LMNP is a rectangle. Find the value of x and the length of each diagonal if LN = 3x+1 and MP = 8x‐4.
13
You Try one:
Find the values of x: Given: LMNP is a rectangle. Find the value of x and the length of each diagonal if LN = 7x‐2 and MP = 4x+3.
How can you tell when a quadrilateral is a rectangle or a rhombus?
Example 3: Identifying Special Parallelograms I’ll do one:
Determine if EFGH is a rectangle or a rhombus. Justify your answer.
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We’ll do one together:
Determine if the quadrilateral below is a rectangle or a rhombus. Justify your answer.
You Try:
Determine if the quadrilateral below is a rectangle or a rhombus. Justify your answer.
Question:
Is it possible for a quadrilateral to be both a rhombus and rectangle at the same time? Explain:
6‐5 Trapezoids and Kites
Focused Learning Target:
To verify and use properties of trapezoids and kites
Vocabulary:
Base angles of a trapezoid
CA Standard(s): Geo 7.0 Students prove and use theorems involving the properties of quadrilaterals Geo 12.0 Students find and use measures of sides and of interior angles of triangles and polygons to solve problemsGeo 13.0 Students prove relationships between angles in polygons by using properties of complementary and supplementary angles
15
Notice that since the bases of a parallelogram are parallel, the legs act as transversals creating same side interior angles that are supplementary. This fact, along with Theorem 6‐15, allows us to find missing angles of any trapezoid. Let’s see how these properties work in the next few examples.
Example 1: Finding Angle Measures in Trapezoids I’ll do one:
Find the value of the variable in the isosceles trapezoid.
We’ll try one together:
16
You try one:
Find the measures of each angle.
Example 2: Using Diagonals of an Isosceles Trapezoid I’ll do one:
Find the value of x in this isosceles trapezoid.
1 xSU
32 xTR You try one:
Find the value of x.
53 xSU
304 xTV Example 3: Finding Angle Measures in Kites I’ll do one:
Find ,2,1 mm and 3m in the kite.
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We’ll try one together:
Find ,2,1 mm and 3m in the kite.
You try one:
Find ,2,1 mm and 3m in the kite.
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6‐6 Placing Figures in the Coordinate Plane
Focused Learning Target:
To name coordinates of special figures by using their properties
CA Standard(s): Geo 17.0 Students prove theorems by using coordinate geometry
When finding the coordinates for a figure in a coordinate plane, use the origin (0, 0) as either the center or a vertex of the polygon and placing one side on the x‐axis or y‐axis. Also, when finding the midpoints, you should use multiples of 2 to avoid fractions.
Example: Naming Coordinates I’ll do one:
In the diagram, rectangle KLMN is centered at the origin with sides parallel to the axes. Find the missing coordinates without using any new variables.
We’ll try one together:
Use the properties of parallelogram OPQR to find the missing coordinates. Do not use any new variables.
You try one:
Use the properties of rectangle to find the coordinates for W and Z. Do not use any new variables.
You try another one:
Use the properties of the rhombus to find the coordinates for W and Z. Do not use any new variables.