《2025年快乐过暑假江苏凤凰科学技术出版社八年级提优版》

7. 如图,$∠A= ∠B= 90^{\circ }$,P是AB的中点,且DP平分$∠ADC$,连接PC.
(1)求证:CP平分$∠BCD$.
(2)线段PD与PC有怎样的位置关系? 请说明理由.

答案：

(1)如图，过点 $P$ 作 $PQ \perp CD$ 于点 $Q$.
$\because P$ 是 $AB$ 的中点，$\therefore PA = PB$.
$\because DP$ 平分 $ \angle ADC$，$\angle A = 90^{\circ}$，$PQ \perp CD$，
$\therefore PA = PQ$，$\therefore PA = PQ = PB$.
$\because \angle B = 90^{\circ}$，$PQ \perp CD$，

$\therefore CP$ 平分 $ \angle BCD$.
(2) $PD \perp PC$. 理由如下：
$\because \angle A + \angle B = 180^{\circ}$，$\therefore AD // BC$，
$\therefore \angle ADC + \angle BCD = 180^{\circ}$.
$\because DP$ 平分 $ \angle ADC$，$CP$ 平分 $ \angle BCD$，
$\therefore \angle PDC = \frac{1}{2} \angle ADC$，$\angle PCD = \frac{1}{2} \angle BCD$，
$\therefore \angle PDC + \angle PCD = 90^{\circ}$，
$\therefore \angle DPC = 90^{\circ}$，$\therefore PD \perp PC$.

1. Some people say that life is like a journey, and we never know who we will meet during it.
2. After three hours of hiking, we finally got to the camp with tired bodies.
3. The main street in this town is about 2.3 kilometres long.
4. Travelling can help us learn about different cultures.
5. Without my parents' support, I can't be the winner.

答案： 1. journey 2. finally 3. main 4. learn about 5. support

二 阅读短文,从短文后 A 至 F 六个选项中选出五个可以填入空白处的最佳选项
A lot of people like taking a trip with no plans. They think it is very exciting and has more fun. But in fact, it is not good. 1E
2B If you visit a place but you don't book a hotel, you may have no place to stay at night.
Second, you have lots of things to take with you for a vacation. 3D For example, your ticket! You want to visit one place, but when you arrive at the airport(机场),you forget to bring your plane ticket! You may be late for the plane.
Third, making a vacation plan can help you save a lot of money. 4C If you surf the Internet before you go on a vacation, you may find some cheap plane tickets and hotel rooms.
Next time when you go on a vacation, try to write down some plans. 5F
A. We needn't make plans when we go on vacation.
B. First, a vacation plan can make sure that everything goes well.
C. No one wants to spend more money than they want to.
D. You need to plan what things you have to buy and what things you need to put in bags.
E. Making vacation plans is very important.
F. They can help you have more fun in your vacation.

答案： 1~5 EBDCF