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Authoritative PPT Summary
"Special Parallelograms" Quadrilateral PPT Courseware 3
learning target
1. Can use the comprehensive method to prove the property theorem and determination theorem of rectangles and related conclusions.
2. Can use the properties of rectangles to perform simple proofs and calculations.
3． Understand the mathematical thinking methods such as induction, analogy, and transformation used in the proof process.
New lesson import
Quadrilateral---two opposite sides are parallel---parallelogram---one angle is a right angle---rectangle
We already know that a parallelogram is a special quadrilateral. Therefore, in addition to the properties of a quadrilateral, a parallelogram also has its special properties. There are also special cases for parallelograms, that is, special parallelograms. In this class, we will study one A special parallelogram - a rectangle
Knowledge explanation
Analysis: (1) The formation process of a rectangle is the change process of one corner of the parallelogram from quantitative change to qualitative change.
(2) A rectangle only has one more condition than a parallelogram: "One angle is a right angle." A rectangle cannot be defined by "a parallelogram with four right angles is a rectangle."
Definition: A parallelogram with one right angle is a rectangle
(3) A rectangle is a special parallelogram that has all the properties (common characteristics) of a parallelogram and also has its own special properties (personality).
Known: As shown in the figure, quadrilateral ABCD is a rectangle.
Prove: ∠A=∠B=∠C=∠D=90°.
Analysis: Based on the definition of rectangle, the problem can be proved by using the equal angles and complementary adjacent angles.
Proof: ∵ Quadrilateral ABCD is a rectangle,
∴∠A=90°, quadrilateral ABCD is a parallelogram.
∴∠C=∠A=90°, ∠B=180°-∠A=90°,
∠D=180°-∠A=90°.
∴ Quadrilateral ABCD is a rectangle.
【Theorem】The four corners of a rectangle are right angles.
fill in the blank
⑴A quadrilateral with four equal sides is a rhombus, and a quadrilateral with three right angles is _______
⑵ A ________ with one right angle is a rectangle.
⑶The parallelogram with diagonal _______ is a rectangle
⑷A quadrilateral whose diagonals bisect each other and are equal is _______
⑸A quadrilateral with one angle being a right angle and diagonal ___________ is a rectangle.
Practice in class
1. (2010·Bazhong High School Entrance Examination) As shown in the figure, given □ABCD, the following conditions: ①AC=BD, ②AB=AD, ③∠1=∠2, ④AB⊥BC, it can be explained that □ABCD is a rectangle. ________ (fill in the serial number).
Analysis: A parallelogram with equal diagonals is a rectangle; definition of rectangle.
Answer: ① ④
2. (2010·Yiyang High School Entrance Examination) As shown in the figure, in △ABC, AB=AC=8, AD is the height on the base, E is the midpoint of AC, then DE=_____.
Analysis: According to the midline of the hypotenuse of a right triangle is equal to half of the hypotenuse, DE is equal to half of AC, so DE=4.
Answer: 4
Summary of this lesson
Through the study of this class, we need to master:
1. Unique properties of rectangle:
(1) The four corners of the rectangle are right angles;
(2) The diagonals of the rectangle are equal;
(3) Corollary: The midline on the hypotenuse of a right triangle is equal to half of the hypotenuse.
2. Determination theorem of rectangle:
(1) Definition: A parallelogram with one right angle is a rectangle;
(2) A parallelogram with equal diagonals is a rectangle;
(3) A quadrilateral with three right angles is a rectangle.
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