Quadrilaterals room encountered all over in life, every square rectangle, any kind of shape with four sides is a quadrilateral. Us know, 3 non-collinear points make a triangle. Similarly, 4 non-collinear points take up a shape that is called a quadrilateral. It has 4 sides, 4 angles, and four vertices.

You are watching: Sum of angles of a quadrilateral Both the figures above are instances of quadrilaterals. ABCD is a quadrilateral. AB, BC, CD, and also DA are four sides of the quadrilateral. A, B, C, and also D are 4 vertices, and also ∠A, ∠B, ∠C, and ∠D space the angles of this quadrilateral.

### Some necessary Terminology

Let’s look at at some terms and conventions regarded quadrilaterals.Opposite Sides: Two political parties of the quadrilateral are dubbed opposite sides if they have actually no usual vertex.

For example: In the figure given above look at the quad ABCD. Here, abdominal and CD room opposite sides. Similarly, ad and BC are opposite sides.Opposite Angles: two angles the a quadrilateral space opposite if castle don’t have any common arm.For example: In the figure ABCD again, edge A and angle C don’t have any common arm. Thus, they have the right to be taken into consideration as the opposite angles. Similarly, angle B and D are also opposite angles.Adjacent Sides: Two sides room called surrounding if sides have actually a typical vertex.For example: AB and advertisement have typical vertex “A”. So, they room called adjacent sides. Similarly, AB, BC; BC, CD and AD, DC are surrounding sides. Adjacent Angles: Two angles, if they have a typical arm space called nearby angles.For example: ∠A, ∠B are surrounding angles.Question: list the pair of opposite sides and adjacent angles indigenous the quadrilateral given below. Answer:Pair of the opposite sides are the political parties which don’t have any common vertices.So, in this case (AB, CD) and (AC, BD) space two bag of opposite sides.Similarly, walking by the meaning given above. Pair of nearby sides are,(AC, AB); (AB, BD); (BD, DC); (CD, AC)

Quadrilaterals deserve to be share into five types:Parallelogram: it is quadrilateral which has actually its opposite sides parallel and congruent to every other. The opposite angles are also equal.Rectangle: It is a quadrilateral that has its opposite sides equal and also all the angles are at the appropriate angle(90°).Square: the is a quadrilateral that has all its political parties of equal length and also all the angles are at the best angle(90°).Rhombus: it is a parallelogram that has every one of its political parties of same length.Trapezium: It has one pair that parallel sides. That sides might or might not be of equal length. ### Angle sum Property

This building states the the amount of all angle of a square is 360°. Let’s prove this.Theorem: The sum of all the four angles of a quadrilateral is 360°.

Proof:Let ABCD be a quadrilateral. Join AC.Now notice,∠1 + ∠2 = ∠A∠3 + ∠4 = ∠CTherefore, indigenous triangle ABC∠4 + ∠2 + ∠B = 180oSimilarly, indigenous triangle ADC∠3 + ∠1 + ∠D = 180oAdding these 2 equations,∠4 + ∠2 + ∠B + ∠3 + ∠1 + ∠D = 360o⇒ (∠1 + ∠2) + (∠3+ ∠4) + ∠B + ∠D = 360o⇒ ∠A + ∠C + ∠B + ∠D = 360oThus, this proves that amount of all interior angles the a quadrilateral is 360°.

### Sample Problems

Question 1: The angle of a quadrilateral are 60°, 90°, 90°. Find the 4th remaining angle.Solution:We recognize from the angle sum home that the sum of the angles of a quadrilateral room 360o.Let the 4th angle it is in denoted by “x”.

So,60° + 90°+ 90° + x = 360°⇒ 180° + 60° + x = 360°⇒ 240° + x = 360°⇒ x = 120°Question 2: The angle of a square are given to it is in (3x)°, (3x + 30)°, (6x + 60)°, 90°. Discover the value of all the angle of quadrilaterals.Solution:We know, amount of all the angle of quadrilateral are 360°.3x + (3x + 30) + (6x + 60) + 90 = 360⇒ (3x + 3x + 6x) + (30 + 60 + 90) = 360⇒ (12x) + (180) = 360⇒ 12x = 360 – 180⇒ 12x = 180⇒ x = 15°Thus, the angles are 45°, 75°, 150° and also 90°Question 3: If the angle of a quadrilateral room in the ratio 1: 2: 3: 4, find the value of the largest angle of the quadrilateral?Solution:Since the amount of all 4 angles of a quadrilateral is 360°, we can equate the values (by multiplying with a constant) of this ratios come 360°Suppose the continuous that is acquiring multiplied is ‘x’We have the right to write, x+ 2x+ 3x+ 4x = 36010x = 360x = 36°Therefore, the largest angle will be 4x = 4×36 = 144°Question 4: In the given listed below trapezium, ∠A = 100°, ∠C = 80°, find the rest of the angles. Solution:We already know, In a Trapezium, two opposite sides space parallel to each other, here, abdominal muscle is parallel to CDThe interior angles developed by 2 parallel lines have actually a amount of 180°(Property the parallel lines)Therefore, we deserve to write, ∠A + ∠D = 180°100° + ∠D = 180°

∠D = 80°Similarly, ∠B+ ∠C = 180°∠B + 80° = 180°∠B = 100°Question 5: In the number below, the inner angles the the quadrilateral are provided as,∠ABC = 50°, ∠BAD = 20°, ∠BCD = 10°Find the worth of the exterior angle ∠ADC? Solution:In a quadrilateral, the amount of every the interior angles is 360°,∠ABC + ∠BAD + ∠BCD + ∠ADC = 360°50° + 20° + 10° + ∠ADC = 360°∠ADC = 280°The angle that come out is the interior angle, the sum of internal angle and the exterior angle will be 360,Exterior angle ∠ADC = 360 – 280 = 80°Question 6: In the offered parallelogram ABCD, the worth of an internal angle is 60°. Uncover the worths of all other angles. Solution:The value of ∠D is offered to it is in 60°. We need to uncover other angles.We understand that sum of nearby angles in a parallel is 180°. Therefore let the value of ∠A it is in x.x + 60° =180°x = 120°∠A = 120°We also know that opposite angles in a parallelogram room equal.So,∠A = ∠C and ∠D = ∠BSo, ∠A = 120°, ∠B = 60°, ∠C = 120° and ∠D = 60°Question 7: In the offered quadrilateral, ∠A = 2x°, ∠B = x°, ∠C = 90° and also ∠D = 3x°.

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find the value of the biggest angle. Solution:

We know that by the angle amount property, sum of all the interior angles of square is 360oSo, ∠A + ∠B + ∠C + ∠D = 360°Given that ∠C = 90°Let’s plugin the rest of the worths given,2x + x + 90 + 3x = 360⇒ 6x = 360 – 90⇒ 6x = 270⇒ x = 45°So, the biggest angle is ∠D = 3x = 3(45) = 135°Attention reader! Don’t stop discovering now. Join the First-Step-to-DSA course for class 9 to 12 college student , specifically design to present data structures and algorithms to the class 9 to 12 students