Quadrilateral: Angle Sum Property - Equation, Solved Example and FAQs - Infinity Learn by Sri Chaitanya (2024)

Sorts of Quadrilaterals

There are four different types of quadrilaterals: parallelograms, rectangles, squares, and rhombuses.

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite angles of a parallelogram are equal, and the sum of the squares of the two shorter sides is equal to the square of the longer side.

A rectangle is a quadrilateral with four right angles. The opposite sides are equal in length, and the sum of the squares of the two shorter sides is equal to the square of the longer side.

A square is a rectangle with four equal sides. The angles of a square are all right angles.

A rhombus is a parallelogram with four equal sides. The angles of a rhombus are not all right angles.

Prove that the Sum of the Angles of a Quadrilateral is 360 Degree

The sum of the angles of a quadrilateral is 360 degrees. This can be proven using basic geometry.

First, draw a diagram of a quadrilateral.

Next, draw in the angles.

Then, using basic geometry, we can add up the angles.

The sum of the angles is 360 degrees.

What is a Quadrilateral

A quadrilateral is a four-sided polygon. It has four angles and four sides.

Parallelogram:

A parallelogram is a four-sided figure with two pairs of parallel sides.

Square:

A square is a simple geometric shape that has four equal sides and four right angles. Squares can be found in a variety of places, including in buildings, parks, and even in nature.

One of the most well-known squares is probably Times Square in New York City. This square is home to some of the most famous tourist attractions in the world, including the New York Times building, the TKTS booth, and the Toys “R” Us store.

Another famous square is the Piazza San Marco in Venice, Italy. This square is surrounded by beautiful architecture, including the Basilica di San Marco and the Doge’s Palace. It is also a popular spot for tourists to visit.

Squares can also be found in nature. One example is the Grand Canyon, which is a large, rectangular canyon that is over 1,000 feet deep.

Rhombus:

A rhombus is a parallelogram with four equal sides. Each of the four angles is equal to 90 degrees.

Trapezium:

A trapezium is a quadrilateral with two pairs of parallel sides.

Angle Sum Property of a Quadrilateral

As shown by the angle sum property of a Quadrilateral, the sum of the huge number of four inside angles is 360 degrees.

Confirmation: In the quadrilateral named PQRS,

∠PQR, ∠QRS, ∠RSP, and ∠SPQ are the internal angles.

QR is a diagonal

QR disengages the quadrilateral into two triangles, ∆PQR and ∆PSR

We have found that the sum of internal angles of a quadrilateral is 360°, that is, ∠PQR + ∠QRS + ∠RSP + ∠SPQ = 360°.

We should exhibit that the sum of the overall huge number of four angles of a Quadrilateral is 360 degrees.

We understand that the sum of angles in a triangle is 180°.

As of now consider triangle PSR,

∠S + ∠SPR + ∠SRP = 180° (Sum of angles in a triangle)

As of now consider triangle PQR,

∠Q + ∠QPR + ∠QRP = 180° (Sum of angles in a triangle)

On adding both the conditions procured above we have,

(∠S + ∠SPR + ∠SRP) + (∠Q + ∠QPR + ∠QRP) = 180° + 180°

∠S + (∠SPR + ∠QPR) + (∠QRP + ∠SRP) + ∠Q = 360°

We see that (∠SPR + ∠QPR) = ∠SPQ and (∠QRP + ∠SRP) = ∠QRS.

Displacing them we have,

∠S + ∠SPQ + ∠QRS + ∠Q = 360°

That is,

∠S + ∠P + ∠R + ∠Q = 360°.

Or of course, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.

What is a Quadrilateral Equation

Area of the quadrilateral is the absolute space involved by the figure. The zone recipe for the various quadrilaterals is given underneath:

Solved Example

Example: The angles of a quadrilateral are as follows

(3a + 2)°

(a – 3)

(2a + 1)°

2(2a + 5) °

Calculate the value of ‘a’ and how much each angle measures.

Solution:

Applying the angle sum property of quadrilateral, we obtain

(3a + 2)°+ (a – 3)° + (2a + 1)° + 2(2a + 5)°= 360°

⇒ 3a + 2 + a – 3 + 2a + 1 + 4a + 10 = 360°

⇒ 10a + 10 = 360

⇒ 10a = 360 – 10

⇒ 10a = 350

⇒ a = 350/10

⇒ a = 35

Thus, (3a + 2)

= 3 × 35 + 2

= 105 + 2 = 107°

(a – 3) = 35 – 3 = 32°

(2a + 1) = 2 × 35 + 1 = 70 + 1 = 71°

2(2a + 5)

= 2(2 × 35 + 5)

= 2(70 + 5)

= 2 × 75 = 150°

Hence, the four angles of the quadrilateral measures 32°, 150°,107°, 71° respectively.

For more visit Area of Trapezium – Definition, Formulas, Properties and Solved Examples