In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find
(i) ∠DAG
(ii) ∠FEG
(iii) ∠GAC
(iv) ∠AGC.

Question

In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find (i) ∠DAG (ii) ∠FEG (iii) ∠GAC (iv) ∠AGC.

KnowledgeBoat · Accepted Answer

(i) We know that, Each angle of a square = 90°. As ABCD is a rhombus so, AB = BC = DC = AD .......(i) Also, CD = ED = FC = EF (As CDEF is a square) … (ii) From (i) and (ii), we have AB = BC = DC = AD = EF = FC = ED … (iii) ∠ABC = 56° [Given] ⇒ ∠ADC = ∠ABC = 56° [Opposite angle in rhombus are equal] From figure, ⇒ ∠EDA = ∠EDC + ∠ADC = 90° + 56° = 146° In ∆ADE, ⇒ DE = AD [From (iii)] ⇒ ∠DEA = ∠DAE [Equal sides have equal opposite angles] From figure, ⇒ ∠DEA = ∠DAE = ∠DAG ⇒ ∠DAE + ∠DEA + ∠EDA = 180° ⇒ ∠DAG + ∠DAG + ∠EDA = 180° ⇒ 2∠DAG + 146° = 180° ⇒ 2∠DAG = 180° - 146° = 34° ⇒ ∠DAG = ⇒ ∠DAG = 17°. Hence, ∠DAG = 17°. (ii) Also, ⇒ ∠DEG = 17° ⇒ ∠FEG = ∠E – ∠DEG = 90° – 17° = 73° Hence, ∠FEG = 73°. (iii) In rhombus ABCD, ⇒ ∠A + ∠B = 180° (As AD || BC, the sum of co-interior angles = 180°.) ⇒ ∠A = 180° - ∠B = 180° - 56° = 124° ⇒ ∠DAC = [∵ Diagonal AC bisects ∠A] ⇒ ∠DAC = 62° ⇒ ∠GAC = ∠DAC – ∠DAG = 62° – 17° = 45°. Hence, ∠GAC = 45°. (iv) In ∆EDG, ⇒ ∠D + ∠DEG + ∠DGE = 180° [Angles sum property of a triangle] ⇒ 90° + 17° + ∠DGE = 180° ⇒ ∠DGE = 180° – 107° = 73° Thus, ∠AGC = ∠DGE [Vertically opposite angles are equal] ⇒ ∠AGC = 73°. Hence, ∠AGC = 73°.

Mathematics

In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find
(i) ∠DAG
(ii) ∠FEG
(iii) ∠GAC
(iv) ∠AGC.

Rectilinear Figures

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Mathematics

In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find(i) ∠DAG(ii) ∠FEG(iii) ∠GAC(iv) ∠AGC.

Rectilinear Figures

Related Questions

Using ruler and compasses only, construct a parallelogram ABCD with AB = 5 cm, AD = 2.5 cm and ∠BAD = 45°. If the bisector of ∠BAD meets DC at E, prove that ∠AEB is a right angle.

Find the size of each lettered angle in the following figure:

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If one angle of a rhombus is 60° and the length of a side is 8 cm, find the lengths of its diagonals.

In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find
(i) ∠DAG
(ii) ∠FEG
(iii) ∠GAC
(iv) ∠AGC.