Question

19. In the following figure, ABCD is a rhombus and DCFE is a square.
If abc = 56°, find : (1) /_DAE (ii) /_FEA (iii) /_EAC (iv)/_AEC​

Answer 1

Given that, ABCD is a rhombus and CDEF is a square. Also ∠ABC=56o

To find out: The measure of ∠DAG, ∠FEG, ∠GAC and ∠AGC

In the rhombus ABCD,

AB=BC=CD=AD

Also, in square CDEF,

CD=DE=EF=FC

Hence, AB=BC=CD=AD=DE=EF=FC

DE=AD

We know that, opposite angles of a parallelogram are equal.

∴ ∠ABC=∠ADC

Also, ∠ADE=∠EDC+∠ADC

∴ ∠ADE=90o+56o=146o

(i)InΔADE,

DE=AD

We know that, angles opposite to equal sides of a triangle are equal.

∴ ∠DEA=∠DAE=x(Let)

Hence, by interior angle sum property, ∠ADE+x+x=180o

∴ 2x=1800−1460

⇒2x=34o

∴ x=17o

Hence, ∠DAG=17o

ii)∠FEG=∠DEF−∠DEG

⇒900−17o=73o

Hence, ∠FEG=73o

(iii) We know that, adjacent angles of a parallelogram are supplementary.

∴∠DAB+∠ABC=180o

∴ ∠DAB=180o−56o=124o

Also, the diagonal of a rhombus bisects the vertex angle.

∴ ∠DAC=21∠DAB

⇒ ∠DAC=21124o=62o

Now, ∠GAC=∠DAC−∠DAG

∴ ∠GAC=62o−170=45o

Hence, ∠GAC=45o

iv)In ΔAGC,

∠AGC+∠GCA+∠GAC=180o    [Interior angle sum property]

Also, ∠GCA=21∠DCB        [Diagonal of a rhombus bisects the vertex angle]

And ∠DCB+∠ABC=180o           [Adjacent angles of a parallelogram are supplementary]

∴ ∠GCA=211240=620

Hence, ∠AGC+62o+45o=180o

∴ ∠AGC=180o−1070=73o

Hence, ∠AGC=73o.

Step-by-step explanation:

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