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知识归纳

半角模型 --- 邻补四边形

知识点一

邻补四边形内含半角（邻边相等，对角互补的四边形）

1. 如图：四边形 ABCD 中，E、F 分别是 CD、AD 上的点， \angle ABC = \angle ADC = 90^\circ 且 \angle EBF = 45^\circ 。猜想并证明线段 EF、CE、AF 之间的数量关系。

Diagram for Problem 1: Quadrilateral ABCD with right angles at B and D. E is on CD, F is on AD. Segment BF is drawn, and angle EBF is marked as 45 degrees.
Diagram for Problem 1: Quadrilateral ABCD with right angles at B and D. E is on CD, F is on AD. Segment BF is drawn, and angle EBF is marked as 45 degrees.

备注： 用旋转法和截长补短法两种方法证明。

2. 如图：四边形 ABCD 中，E、F 分别是 CD、AD 上的点， \angle ABC + \angle ADC = 180^\circ 且 \angle EBF = \frac{1}{2} \angle ABC 。猜想并证明线段 EF、CE、AF 之间的数量关系。

Diagram for Problem 2: Quadrilateral ABCD where opposite angles B and D are supplementary. E is on CD, F is on AD. Segment BF is drawn, and angle EBF is marked as half of angle ABC.
Diagram for Problem 2: Quadrilateral ABCD where opposite angles B and D are supplementary. E is on CD, F is on AD. Segment BF is drawn, and angle EBF is marked as half of angle ABC.

备注： 用旋转法和截长补短法两种方法证明。

2. 如图：等腰直角 \triangle ABC 中， \angle ABC = 90^\circ ，E、F 都是 AC 上的点，且 \angle EBF = 45^\circ ，猜想并证明线段 EF、CE、AF 之间的数量关系。

Diagram for Problem 3: Right isosceles triangle ABC with right angle at B. E and F are points on the hypotenuse AC. Segment BF is drawn, and angle EBF is marked as 45 degrees.
Diagram for Problem 3: Right isosceles triangle ABC with right angle at B. E and F are points on the hypotenuse AC. Segment BF is drawn, and angle EBF is marked as 45 degrees.

备注： 用旋转法和截长补短法两种方法证明。