Geometry of Triangles and Angles

An equilateral triangle has three equal sides. If it shares two sides with a parallelogram, those two adjacent sides of the parallelogram must be equal to the sides of the triangle, and thus equal to each other. A parallelogram with equal adjacent sides is a rhombus. Thus, the statement is true.

In an isosceles triangle, the base is the side of unequal length, while the two legs are the equal sides. By the triangle inequality theorem, any side of a triangle must be less than the sum of the other two sides. Therefore, the base cannot be greater than the sum of the two legs - it must be less than their sum. The statement is false, and the correct answer is False.

By definition in geometry, two angles are complementary if the sum of their measures is exactly 90 degrees. Therefore, the statement is true.

The sum of the interior angles of a square is always 360 degrees (90 degrees * 4), not 180 degrees. Therefore, this statement is incorrect, making it the correct option.

In geometry, the two sides of a right triangle that meet at the 90-degree angle are called the 'legs'. The side opposite the right angle is the hypotenuse. 'Arms' is not a standard geometric term for triangle sides, and 'scalene' refers to a type of triangle.

In non-Euclidean geometries, the sum of angles in a triangle differs from 180 degrees. In Riemannian geometry (spherical/elliptic geometry), the angle sum is always greater than 180 degrees - think of a triangle drawn on a sphere where the sides are great circle arcs. Lobachevsky developed hyperbolic geometry where the sum is less than 180 degrees.

A triangle with sides 3, 4, and 5 satisfies the Pythagorean theorem: 5^2 = 25 = 3^2 + 4^2 = 9 + 16 = 25. This makes it a right-angled triangle. It's not equilateral (all sides equal), not isosceles (two sides equal), and not scalene (all sides different - though technically it is scalene too, but the right angle is the defining property).

In trigonometry, cot is the abbreviation for cotangent, which is defined as the ratio of the adjacent side to the opposite side in a right triangle (cos/sin). The other options (cotton, collaborative, cotuition) are not trigonometric terms. Cot 45° = 1 because at 45°, the adjacent and opposite sides are equal.

With sides 3, 4, 5, this satisfies 3²+4²=5², so it's a right triangle by the converse of Pythagoras theorem. The angle opposite the longest side (5) is 90°.

A triangle with angles 45°, 45°, 90° has two equal angles, making it isosceles. Since one angle is 90°, it's also right-angled. Combining these properties gives a right-angled isosceles triangle.

The sum of interior angles in any triangle is always 180 degrees. This is a fundamental theorem in geometry - angle sum is constant regardless of triangle type (equilateral, isosceles, or scalene).

Let the angles of quadrilateral be 5k, 2k, 4k & k. Now, 5k + 2k + 4k + k = 360^o Then, 12k = 360^o k = 30^o 5k = 150^o

When all the three sides of a triangle are equal then it is called equilateral triangle. This is true in this statement.