Question

In a quadrilateral ABCD, ∠A +∠C=140, ∠A:∠C=1:3 and ∠B:∠D=5:6, find angles A, B, C and D.

Answer 1

Given:

In a quadrilateral ABCD,

∠A +∠C = 140,

∠A : ∠C = 1 : 3

and

∠B : ∠D = 5 : 6

To find:

The angles A, B, C & D

Solution:

∵ Since ∠A : ∠C = 1 : 3, so let's assume,

"x" → represents the measure of the ∠A

"3x" → represents the measure of the ∠C

But, ∠A + ∠C = 140° (given)

∴ x + 3x = 140°

⇒ 4x = 140°

⇒ x = $\frac{140}{4}$

⇒ x = 35°

∴ ∠A = x = 35°

and

∴ ∠C = 3x = 3 × 35° = 105°

We know, in quadrilateral ABCD, we have

∠A + ∠B + ∠C + ∠D = 360° ..... [angle sum property of a quadrilateral]

⇒ ∠B + ∠D + 140° = 360°

⇒ ∠B + ∠D = 360° - 140°

⇒ ∠B + ∠D = 360° - 140°

⇒ ∠B + ∠D = 220°

∵ Since ∠B : ∠D = 5 : 6, so let's assume,

"5y" → represents the measure of the ∠B

"6y" → represents the measure of the ∠D

∴ 5y + 6y = 220

⇒ 11y = 220

⇒ y = $\frac{220}{11}$

⇒ y = 20°

∴ ∠B = 5y = 5 × 20 = 100°

and

∴ ∠D = 6y = 6 × 20 = 120°

Thus, the measure of angles are:

${\boxed{\bold{\angle A = 35\°}}}\\{\boxed{\bold{\angle B = 100\°}}}\\\\{\boxed{\bold{\angle C = 105\°}}}\\\\{\boxed{\bold{\angle D = 120\°}}}$

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Answer 2

Answer:

In a quadrilateral ABCD,

∠A +∠C = 140,

∠A : ∠C = 1 : 3

and

∠B : ∠D = 5 : 6

To find:

The angles A, B, C & D

Solution:

∵ Since ∠A : ∠C = 1 : 3, so let's assume,

"x" → represents the measure of the ∠A

"3x" → represents the measure of the ∠C

But, ∠A + ∠C = 140° (given)

∴ x + 3x = 140°

⇒ 4x = 140°

⇒ x = \frac{140}{4}

4

140

​

⇒ x = 35°

∴ ∠A = x = 35°

and

∴ ∠C = 3x = 3 × 35° = 105°

We know, in quadrilateral ABCD, we have

∠A + ∠B + ∠C + ∠D = 360° ..... [angle sum property of a quadrilateral]

⇒ ∠B + ∠D + 140° = 360°

⇒ ∠B + ∠D = 360° - 140°

⇒ ∠B + ∠D = 360° - 140°

⇒ ∠B + ∠D = 220°

∵ Since ∠B : ∠D = 5 : 6, so let's assume,

"5y" → represents the measure of the ∠B

"6y" → represents the measure of the ∠D

∴ 5y + 6y = 220

⇒ 11y = 220

⇒ y = \frac{220}{11}

11

220

​

⇒ y = 20°

∴ ∠B = 5y = 5 × 20 = 100°

and

∴ ∠D = 6y = 6 × 20 = 120°

Thus, the measure of angles are:

\begin{gathered}{\boxed{\bold{\angle A = 35\°}}}\\{\boxed{\bold{\angle B = 100\°}}}\\\\{\boxed{\bold{\angle C = 105\°}}}\\\\{\boxed{\bold{\angle D = 120\°}}}\end{gathered}

∠A=35\°

​

∠B=100\°

​

∠C=105\°

​

∠D=120\°

​

​

Thank you ❤❤