ABCD is a parallelogram and angle cab=35° .calculate the angles of the parallelogram ABCD
Answers
Answer:
Step-by-step explanation:Given, ABCD is a parallelogram having ∠BAO = 35°, ∠DAO = 40° and ∠COD = 105°
Now, ∠COD = ∠AOB = 105° [vertically opposite angles]
In ΔAOB, by angle sum property of triangle,
⇒ ∠AOB + ∠OAB + ∠ABO = 180°
⇒ 105° + 35° + ∠ABO = 180°
⇒ ∠ABO = 40°
Again, adjacent angles of a parallelogram are supplementary.
⇒ ∠DAB + ∠ABC = 180°
⇒ ∠DAO + ∠OAB + ∠ABO + ∠CBO = 180°
⇒ 40° + 35° + 40° + ∠CBO = 180°
⇒ ∠CBO = ∠CBD = 180° - 115° = 65°
⇒ ∠CBD = 65°
In ΔABC, by angle sum property of triangle,
⇒ ∠CAB + ∠ABC + ∠ACB = 180°
⇒ 35° + ∠ABO + ∠CBO + ∠ACB = 180°
⇒ 35° + 40° + 65° + ∠ACB = 180°
⇒ ∠ACB = 180° - 140° = 40°
⇒ ∠ACB = 40°
Now, opposite angles of a parallelogram are equal
⇒ ∠A =∠C
⇒ ∠C = 75°
On applying angle sum property of triangle in BCD, we get
⇒ ∠C + ∠CBD + ∠CDB = 180°
⇒ 75° + 65° + ∠CDB = 180°
⇒ ∠CDB = 40°
or ∠ODC = 40°
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Step-by-step explanation:
Data: ABCD is a parallelogram AD ∥ BC and DC ∥ AB.
The diagonals AC and BD intersect at O
∠DAC = 40
∘
∠CAB = 35
∘
and ∠DOC = 110
∘
To find : (1)∠ABO (2) ∠ADC (3) ∠ACB (4)∠CBD
Proof:∠DAC + ∠CAB = ∠A
40
∘
+ 35
∘
= ∠A
∠A = 75
∘
∠C = ∠A=75
∘
(Opposite angles of parallelogram are equal)
∠D = ∠A = 180
∘
(Supplementary angles)
∠D = 180
∘
-75
∘
= 105
∘
∠B = ∠D = 105
∘
(Opposite angles of parallelogram are equal)
∠DOC = ∠AOB = 110
∘
Vertically opposite angles)
In AOB, ∠A + ∠O + ∠B = 180
∘
(Sum of all angles of a triangle is 180
∘
)
35
∘
+ 110
∘
+ ∠B = 180
∘
∠B = 180
∘
-145
∘
(1) ∠ABO = 35
∘
(2) ∠ADC = 105
∘
(Proved)
(3) ∠ACB = ∠CBD = 40
∘
(Alternate angles, AD || BC)
∠CBD = 105
∘
-35
∘
(4) ∠CBD = 70
∘
(1) ∠ABO = 35
∘
(2) ∠ADC = 105
∘
(3) ∠ACB = 40
∘
(4)∠CBD = 70
∘