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7. 如图,在菱形ABCD中,点E是对角线AC上一点.求证:△ABE≌△ADE.
答案
7. 证明:在菱形$ABCD$中,$AB=AD$,$∠ DAE=∠ BAE$,
在$△ ABE$和$△ ADE$中,$\because\begin{cases} AB=AD,\\ ∠ DAE=∠ BAE,\\ AE=AE,\\ \end{cases}$
$\therefore△ ABE≌△ ADE(\mathrm{SAS})$.
在$△ ABE$和$△ ADE$中,$\because\begin{cases} AB=AD,\\ ∠ DAE=∠ BAE,\\ AE=AE,\\ \end{cases}$
$\therefore△ ABE≌△ ADE(\mathrm{SAS})$.
8. 如图,在菱形ABCD中,过点D作DE⊥AB于点E,作DF⊥BC于点F,连结EF.
(1)求证:△ADE≌△CDF.
(2)求证:∠BEF=∠BFE.
(1)求证:△ADE≌△CDF.
(2)求证:∠BEF=∠BFE.
答案
8. 证明(1):在菱形$ABCD$中,$DC=AD$,$∠ A=∠ C$,
$\because DE⊥ AB$,$DF⊥ BC$,
$\therefore∠ DEA=∠ DFC$,
$\therefore△ ADE≌△ CDF(\mathrm{AAS})$.
(2)$\because△ ADE≌△ CDF$,$\therefore AE=CF$,
$\because$在菱形$ABCD$中,$AB=BC$,
$\therefore AB - AE=BC - CF$,$\therefore BE=BF$,
$\therefore∠ BEF=∠ BFE$.
$\because DE⊥ AB$,$DF⊥ BC$,
$\therefore∠ DEA=∠ DFC$,
$\therefore△ ADE≌△ CDF(\mathrm{AAS})$.
(2)$\because△ ADE≌△ CDF$,$\therefore AE=CF$,
$\because$在菱形$ABCD$中,$AB=BC$,
$\therefore AB - AE=BC - CF$,$\therefore BE=BF$,
$\therefore∠ BEF=∠ BFE$.
9. 若菱形的周长是8 cm,相邻两内角的度数之比是1:2,则菱形的面积是
$2\sqrt{3}$
.答案
9. $2\sqrt{3}$
10. 如图,在菱形ABCD中,AB=5,对角线AC=6,过点A作AE⊥BC,垂足为E,则AE的长为
$\dfrac{24}{5}$
.答案
10. $\dfrac{24}{5}$
11. 如图,在菱形ABCD中,AB=2,∠A=120°,点P,Q,K分别为线段BC,CD,BD上的任意一点,则PK+QK的最小值为(
A.1
B.2
C.$\sqrt{3}$
D.$\sqrt{3}+1$
C
)A.1
B.2
C.$\sqrt{3}$
D.$\sqrt{3}+1$
答案
11. C
