NCERT Solutions Class 7 Maths Chapter 7 — Triangles & Angle Properties

Solution:

Step 1: An acute angle is an angle that measures less than 90°.

Step 2: An acute-angled triangle is defined as a triangle where all three of its interior angles are acute.

Step 3: Therefore, in an acute-angled triangle, every angle must be less than 90°.

Solution:

Step 1: An obtuse angle is an angle greater than 90°.

Step 2: If a triangle had two obtuse angles, their sum would be greater than 90° + 90° = 180°.

Step 3: However, the angle sum property of a triangle states that the sum of all three interior angles must be exactly 180°.

Step 4: Therefore, it is impossible for a triangle to have two obtuse angles, as their sum alone would exceed the total allowed sum for the triangle.

Solution:

Step 1: The sum of the interior angles in any triangle is 180°. [∠P + ∠Q + ∠R = 180°]

Step 2: Substitute the given values: 45° + 65° + ∠R = 180°.

Step 3: Calculate the sum of the known angles: 110° + ∠R = 180°.

Step 4: Solve for ∠R: ∠R = 180° - 110° = 70°.

Solution:

Step 1: The exterior angle property states that an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

Step 2: Let the exterior angle be E = 120° and one interior opposite angle be A = 70°.

Step 3: Let the other interior opposite angle be B. So, E = A + B. [120° = 70° + B]

Step 4: Solve for B: B = 120° - 70° = 50°.

Solution:

Step 1: An isosceles triangle is defined as a triangle with at least two sides of equal length. In most contexts, it means exactly two sides are equal.

Step 2: If all three sides were equal, it would be an equilateral triangle (which is a special type of isosceles triangle).

Step 3: If all three angles were equal, it would also be an equilateral triangle.

Step 4: An isosceles triangle can be acute-angled, right-angled, or obtuse-angled; it does not necessarily have a right angle.