Exercise 15 - 11.15 | Chapter 3 : Quadrilaterals | CBSE Class 9 Maths
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Interactive Audio Lesson

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Understanding the angle properties of quadrilaterals

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0:00
Teacher
Teacher

Today, we'll start with the angle sum property of quadrilaterals. Can anyone tell me what the sum of the interior angles of any quadrilateral is?

Student 1
Student 1

Isn't it 360 degrees?

Teacher
Teacher

Exactly! Now, if we have three angles in a quadrilateral measuring 90 degrees, 85 degrees and 95 degrees, how do we find the fourth angle?

Student 2
Student 2

We add the three angles together and subtract from 360, right?

Teacher
Teacher

Correct! Let's calculate it together. 360 - (90 + 85 + 95) gives us what?

Student 3
Student 3

That would be 90 degrees!

Teacher
Teacher

Great job! Remember, this property is essential for any quadrilateral.

Exploring properties of parallelograms

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Teacher
Teacher

Next, let’s talk specifically about parallelograms. What can you tell me about their sides?

Student 4
Student 4

Opposite sides are equal and parallel!

Teacher
Teacher

Good! And what about the angles?

Student 1
Student 1

The opposite angles are equal too!

Teacher
Teacher

Exactly! Additionally, the diagonals bisect each other. Can anyone explain why that’s useful?

Student 2
Student 2

It helps in proving other properties or calculating area.

Teacher
Teacher

Exactly. Remember these properties for our exercise on parallelograms!

Applying the mid-point theorem

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Teacher
Teacher

Now let's apply the mid-point theorem. Can anyone state what it says?

Student 3
Student 3

The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half its length.

Teacher
Teacher

Correct! How can we apply this to a quadrilateral?

Student 4
Student 4

If we find the mid-points of opposite sides and join them, it can help determine the area or other properties.

Teacher
Teacher

Well stated! It’s essential in proving congruence and parallel relationships in quadrilaterals.

Introduction & Overview

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Quick Overview

This section focuses on solving problems related to quadrilaterals, emphasizing their properties and theorems related to angles and areas.

Standard

In this section, students will apply their understanding of quadrilaterals by engaging with exercises that require them to calculate angles, apply properties of parallelograms, and find areas of various quadrilateral types including squares, rectangles, and rhombuses.

Detailed

Detailed Summary

In this exercise section, students will tackle various problems that reinforce the key concepts learned in the chapter about quadrilaterals. Students are tasked with calculating the fourth angle in a quadrilateral given three angles, determining angles in parallelograms, proving properties of quadrilaterals, and calculating the areas of different shapes such as squares, rectangles, rhombuses, and trapeziums. Each exercise is designed to build upon the students' understanding of the properties of quadrilaterals, including the angle sum property, properties unique to specific quadrilaterals, and applications of the mid-point theorem. By completing these exercises, students will enhance their problem-solving skills in geometry.

Audio Book

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Area of a Kite

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The diagonals of a kite are 8 cm and 6 cm. Find the area of the kite.

Detailed Explanation

To find the area of a kite, we use the formula:

Area = (1/2) Γ— d1 Γ— d2

where d1 and d2 are the lengths of the diagonals. Here, d1 is 8 cm and d2 is 6 cm.

  1. Multiply the lengths of the diagonals:
  2. d1 Γ— d2 = 8 cm Γ— 6 cm = 48 cmΒ².
  3. Now, divide this product by 2 to find the area:
  4. Area = (1/2) Γ— 48 cmΒ² = 24 cmΒ².

So, the area of the kite is 24 cmΒ².

Examples & Analogies

Think of a kite as a piece of artwork made from two intersecting straight lines, forming a diamond shape (the kite's body). If one diagonal represents a line cut with scissors from one end to the other, and the second diagonal is like a string crossing it, the total space (or area) being covered by the kite's fabric gives us a way to visualize how we are using the area formula to determine how much material is needed to create it.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Angle Sum Property: The sum of the interior angles of a quadrilateral is always 360Β°.

  • Properties of Parallelogram: Opposite sides are equal and parallel; diagonals bisect each other.

  • Mid-point Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • To find the fourth angle of a quadrilateral with angles 110Β°, 85Β°, and 95Β°, add those angles (290Β°) and subtract from 360Β°.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎡 Rhymes Time

  • Four sides in a quad, angles add up to three-six-oh, it's a geometry nod!

πŸ“– Fascinating Stories

  • Imagine a square party where everyone's corners meet at right angles – that’s how a square behaves in a quadrilateral world!

🧠 Other Memory Gems

  • To remember properties of parallelograms, think: 'Opposite sides, Equal, Parallel, Diagonals bisect.' = O.E.P.D.

🎯 Super Acronyms

PARA for Properties

  • Parallel sides
  • Angles equal
  • Rhombus conditions
  • Area formulas.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Quadrilateral

    Definition:

    A polygon with four sides, four vertices, and four angles.

  • Term: Parallelogram

    Definition:

    A quadrilateral with opposite sides that are both equal and parallel.

  • Term: Diagonals

    Definition:

    Line segments that connect non-adjacent vertices of a polygon.

  • Term: Kite

    Definition:

    A quadrilateral with two distinct pairs of adjacent sides that are equal.

  • Term: Trapezium

    Definition:

    A quadrilateral with one pair of opposite sides that are parallel.