| Category | Format | Size |
|---|---|---|
| People's Education Press Eighth Grade Mathematics Volume 2 | pptx | 6 MB |
"Special Parallelograms" Quadrilateral PPT Courseware 2
Description
"Special Parallelograms" Quadrilateral PPT Courseware 2
1. Rectangle:
Definition: A parallelogram with one right angle is called a rectangle.
Properties: Angles: Four angles are right angles.
Diagonals: equal.
Symmetry: It is both a centrally symmetrical figure and an axially symmetrical figure.
The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Judgment: Parallelogram---one angle is right and diagonals are equal---rectangle
Quadrilateral - three angles are right angles - rectangle
Example 1. An interior angle bisector of a rectangle divides one side of the rectangle into two parts of 3cm and 5cm, then the area of the rectangle is 24 or 40cm2.
Example 2. Known: As shown in the figure, in the rectangle ABCD, E is a point on BC, and AE=AD. Let DF⊥AE be in F.
Prove: CE=FE.
Proof: even DE.
∵AE=AD, ∴∠1=∠ADE.
∵DF⊥AE, ∴∠1+∠2=90o.
∵Rectangle ABCD,
∴∠ADC=90o. ∠ADE+∠3=90o. ∴ ∠2=∠3.
And ∵∠DFE=∠C=90o,
∴CE=FE (the distance from the point on the angle bisector to both sides of the angle is equal)
2. Diamond shape:
Definition: A set of parallelograms with equal adjacent sides is called a rhombus.
Properties: Sides: Four sides are equal.
Diagonals: Perpendicular to each other, each diagonal bisects a set of opposite angles.
Symmetry: It is both a centrally symmetrical figure and an axially symmetrical figure.
Judgment: Parallelogram---a set of adjacent sides are equal and diagonals are perpendicular to each other---rhombus
Quadrilateral---four sides are equal---rhombus
Area: S=ah=1/2mn
Example 5. Known: As shown in the figure, in △ABC, ∠C=90o,
AD is the bisector of ∠BAC.
CE⊥AB is at E and AD is at O,
DF⊥AB in F.
Prove: Quadrilateral DCOF is a rhombus.
Prove: ∵AD is the bisector of ∠BAC, DF⊥AB, ∠C=90o.
∴DF=DC. And ∵AD=AD.
∴Rt△ADF≌Rt△ADC (HL). ∴∠1=∠2.
∵CE⊥AB, DF⊥AB. ∴DF//CE, ∠1=∠3.
∴∠2=∠3. ∴OC=DC.
∴DF OC. ∴The quadrilateral DCOF is a parallelogram (a set of quadrilaterals with parallel and equal opposite sides is a parallelogram).
Also ∵DF=CD.
∴DCOF is a rhombus. (A set of parallelograms with equal adjacent sides is a rhombus)
3. Square:
Properties: Angles: four right angles.
Sides: All four sides are equal.
Diagonals: are equal, bisect each other perpendicularly, and bisect a set of opposite angles.
Axiality: It is both a centrally symmetrical figure and an axially symmetrical figure.
Judgment: rectangle---a set of adjacent sides are equal---square
Rhombus---one angle is a right angle---square
Example 7. Known: As shown in the figure, ABCD is a square, CE and BF intersect at O. and CE=BF.
Verify: CE⊥BF.
Prove: ∵ABCD is a square,
∴CD=BC. ∠D=∠BCD=90o.
And ∵CE=BF. ∴Rt△CDE≌Rt△BCF (HL)
∴∠1=∠2.
∵∠1+∠3=90o,
∴∠2+∠3=90o,
∴∠COB=90o. That is CE⊥BF.
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For more information about the "Special Parallelograms and Quadrilaterals" PPT courseware, please click on the Special Parallelograms ppt Quadrilaterals ppt tab.
"Special Parallelograms" PPT courseware 6:
"Special Parallelogram" PPT Courseware 6 Creation Scenario 1 The square is a special rhombus Scenario 2 Questions: 1. When the CD in the picture is translated, what kind of shape does this shape always look like? 2. When CD moves to the CD position, and AD=AB, is the quadrilateral ABCD still a rectangle?
"Special Parallelograms" PPT courseware 5:
"Special Parallelograms" PPT Courseware 5 Scenario Creation We have studied parallelograms and rectangles before, and we know what shape it will become if one of the angles of a parallelogram is a right angle? (A rectangle is obtained by changing the angle) If the parallelogram is transformed from the angle of the side special..
"Special Parallelogram" PPT Courseware 4:
"Special Parallelograms" PPT Courseware 4 Review Review Definition: A parallelogram with one right angle is called a rectangle. Give it a try. It is known that △ABC is Rt△, ABC=Rt, and BD is the midline on the hypotenuse AC. 1. If BD=3cm, then AC=________cm 2. If C=30, A..
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