chapter 6 polygons and quadrilaterals practice and problem solving exercises

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Chapter 6: Polygons and Quadrilaterals

8th - 11th grade, mathematics.

37 questions

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• 1. Multiple Choice Edit 5 minutes 1 pt Which of the following statements is not true?  A trapezoid is a rectangle.  A square is also a rectangle.  A rectangle is a quadrilateral.  A square is also a rhombus. 
• 3. Multiple Choice Edit 2 minutes 1 pt A parallelogram is a rhombus. always sometimes never
• 4. Multiple Choice Edit 2 minutes 1 pt A trapezoid is a parallelogram. always sometimes never
• 11. Multiple Choice Edit 2 minutes 1 pt What is the SUM of the angle measures in a nonagon (9 sides)? 1440° 900° 1080° 1260°
• 12. Multiple Choice Edit 2 minutes 1 pt What is the measure of  ONE interior angle  in a regular hexagon (6 sides)? 60° 80° 108° 120°
• 13. Multiple Choice Edit 2 minutes 1 pt Which formula is used to find the sum of the interior angles of a polygon? 180 360 (n-2)180 n(n-3)/2
• 14. Multiple Choice Edit 2 minutes 1 pt What is the measure of a single exterior angle for a regular pentagon? 360 108 72 36

Find the measurement of a single anterior angle of a 25-gon

• 16. Multiple Choice Edit 5 minutes 1 pt What is the sum of the exterior angles of a 20-gon? 180 360 3240 162
• 22. Multiple Choice Edit 1 minute 1 pt Which is NOT a property of a parallelogram? Diagonals bisect each other Diagonals are congruent Both pairs of opposite sides are parallel Both pairs of opposite angles are congruent
• 23. Multiple Choice Edit 1 minute 1 pt Which shape is NOT a type of parallelogram? trapezoid rectangle rhombus square
• 29. Multiple Choice Edit 1 minute 1 pt Base angles of an isosceles trapezoid are _____. neither supplementary congruent

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Unit 5: Quadrilaterals

About this unit, quadrilateral types.

• Intro to quadrilateral (Opens a modal)
• Right angles in shapes (informal definition) (Opens a modal)
• Identifying quadrilaterals (Opens a modal)
• Quadrilateral properties (Opens a modal)
• Quadrilateral types (Opens a modal)
• Classifying quadrilaterals (Opens a modal)
• Kites as a geometric shape (Opens a modal)
• Quadrilaterals review (Opens a modal)
• Identify quadrilaterals 7 questions Practice
• Analyze quadrilaterals 4 questions Practice
• Quadrilateral types 4 questions Practice

Quadrilateral proofs & angles

• Proof: Opposite sides of a parallelogram (Opens a modal)
• Proof: Diagonals of a parallelogram (Opens a modal)
• Proof: Opposite angles of a parallelogram (Opens a modal)
• Proof: Rhombus diagonals are perpendicular bisectors (Opens a modal)
• Whether a special quadrilateral can exist (Opens a modal)
• Rhombus diagonals (Opens a modal)
• Quadrilateral angles 7 questions Practice

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Exercise 7 Page   421

The measure of each interior angle of a regular n -gon is (n-2)180n.

Check the answer

Interior Angle: 108 Exterior Angle: 72

Practice exercises

Asses your skill.

Let's find the measure of an interior and an exterior angle of the given regular polygon one at a time.

Measure of Interior Angle

Measure of exterior angle.

To find the measure of an exterior angle, notice that the interior and exterior angles of any polygon form a linear pair .

This means that their measures add to 180. Let's call the exterior angle of our regular polygon x. To find x, we can write an equation using the measure of the interior angle and the fact that these will add to 180. 108+x=180 ⇔ x=72 The measure of the exterior angle is 72.

Quadrilaterals Questions

Quadrilateral questions and answers are provided here for students to understand the topic better. The quadrilateral is an important topic for students because these concepts are studied in more depth in higher education. Here, we have provided questions involving quadrilaterals and related formulas that students can simply solve. The problems framed here follow the CBSE and NCERT syllabus. Practising these questions can assist students in solving difficult questions and achieving higher exam scores. Learn more about quadrilaterals here.

The quadrilateral is one of the most common geometrical shapes we observe in everyday life. As a result, students must be taught about quadrilaterals. The questions in this section have been prepared, so that students can do well not only in academic exams but also in competitive exams.

Go through the below quadrilaterals questions and understand the concept quickly.

Quadrilateral Questions with Solutions

1. A quadrilateral with four equal sides and four right angles is a ______.

Answer: Square.

A square is one of the quadrilaterals with four equal sides and four right angles.

2. The sum of interior angles of a quadrilateral is ____.

Answer: 360°​

The quadrilateral is a four-sided polygon, and hence the sum of the interior angles of a quadrilateral is 360°​. A quadrilateral may be square, rectangle, parallelogram, rhombus, trapezium or kite-shaped.

3. The three angles of a quadrilateral are 60°, 90°, 110°. Determine the fourth angle.

We know that the sum of interior angles of a quadrilateral is 360°.

Given three angles are 60°, 90° and 110°.

Let the unknown angle be “x”.

By using the property of quadrilateral,

60° + 90° + 110° + x = 360°

260° + x = 360°

x = 360° – 260°

Hence, the fourth angle of a quadrilateral is 100°.

4. The two angles of a quadrilateral are 76° and 68°. If the other two angles are in the ratio of 5: 7, then find the measure of each of them.

Given two angles are 76° and 68°.

Let the other two angles be 5x and 7x.

As we know, the sum of interior angles of a quadrilateral is 360°.

Therefore, 76°+68°+5x + 7x = 360°

144° + 12x = 360°

12x = 360° – 144°

x = 216°/12

Hence, the other two angles are:

5x = 5(18)° = 90°

7x = 7(18°) = 126°.

5. The dimension of the rectangular field is 30 m and 50 m. Find its area.

Given: Length = 50 m

Breadth = 30 m

As the field is in the shape of a rectangle,

Area = Length × Breadth square units

Area = 50 × 30 m 2

Area = 1500 m 2

Hence, the area of the rectangular field is 1500 m 2 .

6. ABCD is a quadrilateral, whose angles are ∠A = 5(a+2)°, ∠B = 2(2a+7)°, ∠C = 64°, ∠D = ∠C-8°. Determine the value of ∠A.

Given that, ∠A = 5(a+2)°, ∠B = 2(2a+7)°, ∠C = 64°, ∠D = ∠C-8°

Hence, ∠D = 64° – 8°

As we know,

∠A+∠B+∠C+∠D = 360°

Now, substitute the values, we get

5(a+2)° + 2(2a+7)° + 64°+56° = 360°

5a°+10°+4a°+14° +64° +56° = 360°

9a° + 144° = 360°

9a° = 360° – 144°

a° = 216°/9

Hence, the value of ∠A is:

∠A = 5(a+2)° = 5(24°+2°) = 5 (26°) = 130°

Therefore, ∠A = 130°.

7. The angles of a quadrilateral are in the ratio of 1: 2: 3: 4. Find the measure of each angle.

Given angle ratio is 1: 2: 3: 4.

Let the four angles be 1x, 2x, 3x, 4x.

Hence, 1x+2x+3x+4x = 360°

Hence, the measure of four angles are:

⇒ 2x = 2(36°) = 72°

⇒ 3x = 3(36°) = 108°

⇒ 4x = 4(36°) = 144°

Therefore, the angles of a quadrilateral are 36°, 72°, 108° and 144°.

8. The lengths of adjacent sides of a parallelogram are 3 cm and 4 cm. Find its perimeter.

Given that the length of the adjacent side of a parallelogram = 3 cm and 4 cm.

That is, base = 3 cm and side = 4 cm.

The formula to calculate the perimeter of a parallelogram is P = 2 (Base + Side) units

P = 2 (3+4)

P = 2 (7) = 14 cm

Hence, the perimeter of a parallelogram is 14 cm.

9. The diagonals of a rhombus are 12 cm and 7.5 cm. Find the area of a rhombus.

Given: Length of diagonal 1 = 12 cm

Length of diagonal 2 = 7.5 cm

We know that,

Area of a rhombus = (1/2) × Diagonal 1× Diagonal 2 square units

A = (½)×12×7.5

A = 45 cm 2

Hence, the area of a rhombus is 45 cm 2 .

10. A quadrilateral has three acute angles, each measuring 75°. Find the measure of the fourth angle.

Let A, B, C and D be the four angles of a quadrilateral.

If A=B=C=75°, we have to find the angle D

As we know that A+B+C+D = 360°

Therefore, 75°+75°+75°+D = 360°

225°+D = 360°

D = 360° – 225°

Hence, the measure of the fourth angle is 135°.

Practice Questions

• If one angle of a parallelogram is 30° less than twice the smallest angle, find the measure of each angle.
• The adjacent sides of a parallelogram are in the ratio of 4: 5 and its perimeter is 72 m. Find the side lengths of a parallelogram.
• In a quadrilateral ABCD, ∠D = 150°, and ∠A = ∠B = ∠C. Find the measure of ∠A, ∠B and ∠C.

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Quadrilateral Practice Problems

Introduction of Quadrilateral

Introduction to quadrilateral.

When we see a tabletop or a handkerchief, the basic quadrilateral shapes—a rectangle or a square—come to mind. There are many numerous varieties of quadrilaterals, and the shapes of the quadrilaterals themselves can recognize the types of quadrilaterals . The word is made up of two Latin words, Quadri, which is a variation of four, and latus, which signifies sides, as suggested by the name. A quadrilateral is a two-dimensional polygon with four sides. Quadrilaterals include the following 2D shapes: square, rectangle, rhombus, trapezium, parallelogram, and kite.

Quadrilaterals 

Definition of Quadrilaterals

A quadrilateral is a form of a polygon that consists of four sides, four vertices, four angles, and two diagonals.

Properties of Quadrilaterals

The properties of a quadrilateral set it apart from a normal polygon in various ways. We have listed some common properties of quadrilaterals below:

They must have four sides.

They must have four vertices.

They must have two diagonals.

360° is the total sum of interior angles.

Quadrilateral shapes explain a lot about their properties.

Types of Quadrilaterals

There are various types of quadrilaterals , depending on different characteristics, and unique shapes. Just by looking at the quadrilateral shapes, one can tell the differences and qualities between them. Let's briefly discuss some of the types of quadrilaterals in this section.

Square: A quadrilateral with four equal sides and angles is called a square. The fact is that its sides and angles are equal, making it a regular quadrilateral. A square has four 90° angles. It can alternatively be viewed as a rectangle with equal lengths on its two neighbouring edges.

Sides: $A B=C D$ and $B C=A D$

Diagonals: $\mathrm{AO}=\mathrm{OC}$ and $\mathrm{DO}=\mathrm{OB}$

Angles: $\angle A=\angle C$ and $\angle B=\angle D=90^{\circ}$

If a square's side is "a," then

The square's area is equal to $a\times a =a^{2}$

The Square's perimeter is equal to 4a.

Rectangle: A rectangle contains four corners and four sides where opposite sides are of the same length and parallel to each other. The angles of a rectangle are equal in measure and are right-angled i.e. they measure $90^{\circ}$.

Some of the properties of the rectangle are given below:

Two pairs of parallel sides.

All four angles are right angles, that is, they measure 90 degrees.

Opposite sides are of equal lengths.

Two equal diagonals

In a rectangle, the two diagonals bisect each other in equal halves In the rectangle $P Q R S, P Q\|R S, P Q=R S, P S\| Q R$, and $P S=Q R$. All the angles are $90^{\circ}$ angles.

Kite: A kite has various names such as a dart or an arrowhead because of the shape. A kite has two pairs of equal-length sides and these sides are adjacent to each other. 

Some of the properties of the kite are given below:

Contains four edges and four vertices.

Contains one line of symmetry.

Contains two pairs of congruent and consecutive sides.

Diagonals are perpendicular to each other.

In the kite $P Q R S, P Q=Q R$, and $P S=S R$.

Parallelogram: As the name implies, a parallelogram is a simple quadrilateral with opposite sides as parallel. It, therefore, has two sets of parallel sides. A parallelogram also has its opposite angles as equal. In addition, the diagonals of a parallelogram cut each other. The sum of any two adjacent angles equals 180°.

Parallelogram

Angles: $\angle A=\angle C$ and $\angle B=\angle D$

If a parallelogram's length is I, breadth is $b$, and height is $h$, then

Parallelogram's perimeter equals $2 \times(\mathrm{l}+\mathrm{b})$.

Parallelogram's area equals $\mathrm{l} \times \mathrm{h}$

Rhombus: A quadrilateral called rhombus has four equal sides that are non-parallel to one another. The angles are not exactly 90 degrees. Right angles transform a rhombus into a square. In recognition of its similarities to the diamond card of playing cards, the rhombus is sometimes known as the "diamond."

A rhombus's perimeter is equal to 4a if its side is a.

If two of the rhombus' diagonals are $d_{1}$ and $d_{2}$, then the area of a rhombus is equal to $\dfrac{1}{2} \times d_{1} \times d_{2}$

Trapezium: One pair of parallel sides make up a quadrilateral known as a trapezium. The two other sides are referred to as "legs" or "lateral sides," whereas the parallel sides are known as "bases."

As demonstrated in the diagram above, if a trapezium's height is "h," then

Trapezium perimeter = total lengths of all sides, which are AB + BC + CD + DA.

The area of the trapezium equals $\dfrac{1}{2} \times (AB + CD) \times h$

Quadrilateral Solved Problems

Q1. Find the base of a parallelogram, if its area is 144 square units and the height is 6 units.

Ans: Given, a parallelogram with;

Area = 144 square units

Height = 6 units

Therefore the area of the parallelogram = Base × Height

144 = Base $\times$ 6

Base = $\dfrac{114}{6}$ = 24 units

Q2. Calculate the quadrilateral's perimeter, which has sides of 2, 4, 9, and 15 cm.

Ans: We have the sides of a quadrilateral: 2 cm, 4 cm, 9 cm and 15 cm.

Therefore, the perimeter of a quadrilateral is given by the sum of all its sides, i.e.

P =  2 cm + 4 cm + 9 cm + 15 cm 

Q 3. Find the missing angle x.

Quadrilateral

Ans: We are aware that in a quadrilateral, the sum of all the angles is equal to $360^{\circ}$.

Therefore, we can express it as follows:

$x+77^{\circ}+101^{\circ}+67=360^{\circ}$

$x+245^{\circ}=360^{\circ}$

$x=360^{\circ}-245^{\circ}$

Hence, $x=115^{\circ}$.

Q1. The quadrilateral has three sides that measure 9 cm, 13 cm, and 17 cm in length, and its perimeter is 50 cm. Find the quadrilateral's missing side.

Ans: 11 cm .

Q 2. Find $\angle B C D$, if the figure at left, $\mathrm{ABCD}$ is a cyclic quadrilateral in which $\mathrm{AB} \| \mathrm{DC}$. If $\angle \mathrm{BAD}= 100^{\circ}$

ABCD is a cyclic quadrilateral

Ans : $80^{\circ}$

Let us summarise what we have learned throughout this article. A quadrilateral is a polygon with four sides and four angles, such as a square, parallelogram, rectangle, trapezium,  kite or rhombus. There are various types of quadrilaterals, depending on different characteristics, and shapes. The properties of quadrilaterals , namely square, parallelogram, rhombus, and trapezium are discussed, along with their solved examples, in brief in the above article. Try out the given practice questions to get a better understanding of the topic, quadrilateral.

FAQs on Quadrilateral Practice Problems

1. Are all the quadrilaterals parallelograms?

No, all the quadrilaterals can never be parallelograms. For a quadrilateral to be a parallelogram, it must follow some conditions, such as the two sets of opposite sides must be parallel. Thus, we can say that every parallelogram is a quadrilateral but every quadrilateral need not be a parallelogram.

2. How do you calculate a quadrilateral's area?

The space occupied by a quadrilateral in two dimensions is referred to as the area of a quadrilateral. According to their characteristics, different kinds of quadrilaterals have different formulas for calculating area. For instance, the area of a square is equal to the square of its length, while a rectangle's area is equal to the product of length and width.

3. What are convex quadrilaterals and concave quadrilaterals, respectively?

A quadrilateral is considered concave if at least one of its internal angles is greater than 180 degrees. A concave quadrilateral diagonal is located outside the enclosed figure.

A quadrilateral with four sides and internal angles that are each less than 180 degrees is said to be a convex quadrilateral. All of the diagonals fit within these quadrilaterals. Convex quadrilaterals can be divided into several subcategories according to the lengths of their sides and angles.

Geometry: Common Core (15th Edition)

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