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Area Of Parallelograms Definition: A parallelogram is a quadrilateral with opposite sides parallel.

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Review of Characteristics of Parallelograms Opposite sides are congruent. Opposite angles are congruent. Opposite sides are parallel. Consecutive angles are supplementary. (SSI angles)

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Review of Characteristics of Parallelograms Opposite sides are congruent. Opposite angles are congruent. Opposite sides are parallel. Consecutive angles are supplementary. (SSI angels) Diagonals bisect each other.

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Areas of Parallelograms Find the area by counting squares. Full blocks = Half blocks = Total Area = 8 4 10 su.

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Areas of Parallelograms Find the area by counting squares. Full blocks = Half blocks = Total Area = 12 6 15 su.

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Areas of Parallelograms Find the area by counting squares. Full blocks = Half blocks = Total Area = 20 8 24 su.

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It is tough to compute all the partial blocks. There must be an easier way! There is!!! I like rectangles much better! They are real easy! Don’t you agree? ? ? ? ? ? ? ? ?

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Doesn’t a parallelogram look like a rectangle with it’s side kicked in ? Let’s cut off a corner and start to make a rectangle.

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Wow! We have part of a rectangle. Now watch what else we have.

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Let’s move the triangle to the other side. The two triangles are congruent.

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We have now created a rectangle with the same area as the parallelogram. What has happened?

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We have now created a rectangle with the same area as the parallelogram. Therefore, the formula for the area of a parallelogram is the same as that of a rectangle.

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Parallelogram b h

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b2b2 h2 Note that there is another base and another height. Sometimes you must use the other height.

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Triangle Now, we can use the parallelogram formula to derive the area of a triangle. No, we are not going to add up squares again.

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Triangle Let’s construct a line through the vertex parallel to the base.

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Triangle Let’s construct another line through the right vertex parallel to another side. We have just created a parallelogram.

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Triangle We can do this for any triangle. Note that both the triangles are congruent by… SAS, SSS, ASA, or AAS

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SAS Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent.

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SSS Opposite sides of a parallelogram are congruent. Reflexive property.

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ASA Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. AIA: Alternate Interior Angles are congruent.

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AAS Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. AIA: Alternate Interior Angles are congruent.

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Triangle All triangles are just Half a parallelogram. Therefore…

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Triangle Note where the height is located. h It is the height of both the parallelogram and triangle.

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Sample Problems 14.75 8.25 Find the area. A= 121.6875 8.2514.75 A= 121.7

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Finding Areas of Parallelogram 12 20 16 1216

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Find the area: fractions

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Convert to decimals.

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Find the area: mixed modes Convert to decimals.

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Find the area: mixed modes A = 59.754 Convert to decimals. A = 59.8

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Find Area of Triangle 4 3 5 4 3

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Triangle ABC has three altitudes or heights. They are … Each side is a corresponding base. Find the base associated with the corresponding height.

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If the area = 96 and …. Find the values of AD and FC. This is a backwards problem. You always start with the formula. h h

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If the area = 96 and …. Find the values of AD and FC. This is a backwards problem. You always start with the formula. h h

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Backward Problems If the area is 125 sf, and the base is 25, find the height. 25 125 h 2 2 Divide by 25 h

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If the area is 182 sf, and the base is 30, find the height. 30 h h182

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With these problems, you… 1 Plan solution with equations or written strategy as if you have all the information needed. 2. Write the equation, leaving empty parentheses to insert the needed values. 4. When all values are found, complete the original strategy or equation. 3. Go find the needed values in sidebar stages, substituting back into the original strategy or equation. Compound Complex Multiple Stage Problems

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Find Area of Triangle 13 12 5 5, 12, 13 triangle 12 5

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Find Area of Triangle 8 8 8 60 0 8 4

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Find Area of Triangle 8 60 0 15 30-60-90 triangle

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Find Area of Triangle 11 b 5 5 b 5

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Find Area of Parallelogram 8 16 h Need to find value of height.

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Find Area of Triangle 8 15 45-45-90 triangle 15 8

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Find the area of pentagon. Add line and label figure. 10 60 0 5

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Which triangle has the largest area? They all have the same area. Why?

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They each have the same base: AB They each have the same height.

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M is midpoint of h h is 13 cm. Find the area of triangles ABM and ACM. h is 13 for both triangles.

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M is midpoint of h h is 13 cm.

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Summary 1. The area of a parallelogram is… 2. The area of a triangle is half the area of parallelogram. 3. A triangle has three heights or altitudes. 4. A triangle has three bases (sides) to correspond with each height.

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