Solutions for Chapter 15 RD Sharma-Free PDF
FAQs on RD Sharma Class 7 Solutions Chapter 15 - Properties of Triangles (Ex 15.3) Exercise 15.3
1. What are the main properties of triangles covered in RD Sharma Class 7 Solutions for Chapter 15, Exercise 15.3?
RD Sharma Solutions for Class 7, Chapter 15, primarily focuses on fundamental properties that define the relationship between a triangle's angles and sides. The key concepts you will use to solve problems in Exercise 15.3 include:
- Angle Sum Property: The sum of the interior angles of any triangle is always 180°.
- Exterior Angle Property: The measure of an exterior angle of a triangle is equal to the sum of its two opposite interior angles.
- Triangle Inequality Property: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
- Relationship between Angles and Opposite Sides: The side opposite the larger angle is longer than the side opposite the smaller angle.
2. How is the Angle Sum Property of a triangle used to find a missing angle in Exercise 15.3 problems?
The Angle Sum Property is a core method used in this exercise. According to this property, the sum of all three interior angles of any triangle (e.g., ∠A, ∠B, and ∠C) is exactly 180°. To find a missing angle, you simply subtract the sum of the two known angles from 180°.
For example, if a triangle has angles measuring 65° and 45°, the third angle is calculated as: 180° - (65° + 45°) = 180° - 110° = 70°.
3. How do you apply the Triangle Inequality Property to check if a triangle can be formed with given side lengths?
The Triangle Inequality Property provides a clear test to determine if three given lengths can form a triangle. For any triangle with side lengths a, b, and c, the following three conditions must all be true:
- a + b > c
- b + c > a
- a + c > b
If even one of these conditions is false, a triangle cannot be constructed with those side lengths. For example, sides of 3 cm, 4 cm, and 8 cm cannot form a triangle because 3 + 4 is not greater than 8.
4. What is the step-by-step method to solve a problem using the Exterior Angle Property?
To solve a problem using the Exterior Angle Property, follow these steps:
- Identify the exterior angle: This is the angle formed by extending one of the triangle's sides.
- Identify the interior opposite angles: These are the two angles inside the triangle that are not adjacent to the exterior angle.
- Apply the property: Set the measure of the exterior angle equal to the sum of the measures of the two interior opposite angles.
- Solve for the unknown: Use the resulting equation to find the value of the unknown angle or variable.
5. Can a triangle have two right angles? Explain your answer using the Angle Sum Property.
No, a triangle cannot have two right angles. The Angle Sum Property states that the sum of a triangle's three angles must be 180°. If a triangle had two right angles (90° each), their sum would already be 90° + 90° = 180°. This would mean the third angle must be 0°, which is impossible for a two-dimensional shape like a triangle.
6. Why is the exterior angle of a triangle always equal to the sum of its two interior opposite angles?
This property is a logical consequence of the Angle Sum Property. Consider a triangle with angles ∠1, ∠2, and ∠3. We know ∠1 + ∠2 + ∠3 = 180°. Now, if we extend the side to create an exterior angle (let's call it ∠4) adjacent to ∠3, these two angles form a linear pair, meaning ∠3 + ∠4 = 180°. Since both sums equal 180°, we can say ∠1 + ∠2 + ∠3 = ∠3 + ∠4. By subtracting ∠3 from both sides, we are left with ∠1 + ∠2 = ∠4, proving the property.
7. What is the key difference between the Angle Sum Property and the Triangle Inequality Property?
The fundamental difference lies in what they describe:
- The Angle Sum Property deals exclusively with the angles of a triangle, stating their sum must be 180°.
- The Triangle Inequality Property deals exclusively with the lengths of the sides of a triangle, stating that the sum of any two sides must be greater than the third.
One property governs the angular relationships, while the other governs the dimensional constraints of the sides.
8. How does knowing the relationship between a triangle's angles and its opposite sides help solve problems?
This relationship allows you to compare the lengths of a triangle's sides without knowing their actual measurements. The rule is: the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle. In problems where you are given the angles (e.g., 50°, 60°, 70°), you can immediately determine the order of the side lengths from shortest to longest just by looking at the angles opposite to them.
