Plane Geometry
- Three consecutive angles of a cyclic quadrilateral are in the ratio of 1 : 4 : 5. The measure of fourth angle is :
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Here , Ratio of angles = 1 : 4 : 5
The sum of the opposite angles of a cyclic quadrilateral is 180°.
For a quadrilateral ABCD,
∠A = y°; ∠B = 4y°; ∠C = 5y°
∴ x + 5y = 180° ⇒ 6y = 180°⇒ y = 180° = 30° 6
Correct Option: B
Here , Ratio of angles = 1 : 4 : 5
The sum of the opposite angles of a cyclic quadrilateral is 180°.
For a quadrilateral ABCD,
∠A = y°; ∠B = 4y°; ∠C = 5y°
∴ x + 5y = 180° ⇒ 6y = 180°⇒ y = 180° = 30° 6
∴ ∠B + ∠D = 180°
⇒ 4 × 30° + ∠D = 180°
⇒ ∠D = 180° – 120° = 60°
- ABCD is a quadrilateral in which BD and AC are diagonals then
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According to question , we draw a figure of quadrilateral ABCD
We know that the sum of two sides of a triangle is greater than the third side.
∴ AB + BC > AC
BC + CD > BD
CD + AD > AC
DA + AB > BD
On adding, we getCorrect Option: B
According to question , we draw a figure of quadrilateral ABCD
We know that the sum of two sides of a triangle is greater than the third side.
∴ AB + BC > AC
BC + CD > BD
CD + AD > AC
DA + AB > BD
On adding, we get
2 (AB + BC + CD + DA) > 2 (AC + BD)
⇒AB+BC + CD + DA > (AC + BD)
- In a cyclic quadrilateral ABCD, ∠BCD = 120° and passes through the centre of the circle. Then ∠ABD = ?
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As per the given in question , we draw a figure of cyclic quadrilateral ABCD
As we know that the sum of opposite angles of a concyclic quadrilateral is 180°.
∴ ∠BCD + ∠BAD = 180°
⇒ 120° + ∠BAD = 180°
⇒ ∠BAD = 180° – 120° = 60°
The angle in a semi-circle is a right angle.Correct Option: A
As per the given in question , we draw a figure of cyclic quadrilateral ABCD
As we know that the sum of opposite angles of a concyclic quadrilateral is 180°.
∴ ∠BCD + ∠BAD = 180°
⇒ 120° + ∠BAD = 180°
⇒ ∠BAD = 180° – 120° = 60°
The angle in a semi-circle is a right angle.
∴ ∠BDA = 90°
∴ In ∆ABD,
∴ ∠ABD = 90° – 60° = 30°
- The measures of three angles of a quadrilateral are in the ratio 1 : 2 : 3. If the sum of these three measures is equal to the measrue of the fourth angle, find the smallest angle.
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Given , The measures of three angles of a quadrilateral are in the ratio 1 : 2 : 3
Let three angles of quadrilateral = y°, 2y° and 3y°
According to question ,
∴ Fourth angle = y + 2y + 3y = 6y°
∴ y + 2y + 3y + 6y = 360°
⇒ 12y = 360°Correct Option: A
Given , The measures of three angles of a quadrilateral are in the ratio 1 : 2 : 3
Let three angles of quadrilateral = y°, 2y° and 3y°
According to question ,
∴ Fourth angle = y + 2y + 3y = 6y°
∴ y + 2y + 3y + 6y = 360°
⇒ 12y = 360°⇒ y = 360 = 30° = Smallest angle 12
- If ABCD is a cyclic quadrilateral with ∠A = 50°, ∠B = 80°, then ∠C and ∠D are
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In a concyclic quadrilateral ABCD,
We know that , The sum of opposite angles of a concyclic quadrilateral is 180
∠A + ∠C = 180°
⇒ 50° + ∠C = 180°
⇒ ∠C = 180° – 50° = 130°
Again,
∠B + ∠D = 180°Correct Option: D
In a concyclic quadrilateral ABCD,
We know that , The sum of opposite angles of a concyclic quadrilateral is 180
∠A + ∠C = 180°
⇒ 50° + ∠C = 180°
⇒ ∠C = 180° – 50° = 130°
Again,
∠B + ∠D = 180°
⇒ 80° + ∠D = 180°
⇒ ∠D = 180° – 80° = 100°
