Let the smallest angle be x. Then the second angle is 2x and the largest angle is 2x - 30. The sum of angles in a triangle is 180 degrees. Therefore, x + 2x + (2x - 30) = 180. Solving this gives x = 30 degrees.

According to the triangle inequality theorem, the length of the third side must be less than the sum and greater than the difference of the other two sides. Therefore, the third side must be greater than 3 cm and less than 17 cm.

The diameter of a circle is indeed twice the radius, making this statement true.

The sum of the interior angles of a polygon is given by the formula (n-2) * 180 degrees, where n is the number of sides. Therefore, as the number of sides increases, the sum of the interior angles also increases.

Using the property that the sum of angles in a triangle is 180 degrees, angle C can be calculated as 180 - (30 + 60) = 90 degrees.

Since two angles are equal (45 degrees each), triangle DEF is an isosceles triangle.

Since all sides are of different lengths, triangle XYZ is a scalene triangle.

Since all sides are of different lengths, triangle XYZ is a scalene triangle.

The area of a circle is given by A = πr². Thus, A = π(7)² = 49π ≈ 154 cm².

The area can be calculated as length × width = 4 × 3 = 12 square units.

Area of a rhombus = (1/2) × d1 × d2 = (1/2) × 10 × 24 = 120 cm².

The area of a triangle is given by (1/2) * base * height. Here, base = 4 and height = 3, so area = (1/2) * 4 * 3 = 6.

The area of a triangle is given by the formula (1/2) * base * height. Thus, Area = (1/2) * 10 * 5 = 25 cm².

Using the distance formula, d = √[(x2 - x1)² + (y2 - y1)²] = √[(4 - 1)² + (5 - 1)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.

The minimum number of sides a polygon can have is 3, which is a triangle.

Chords can vary in length and can be longer, equal to, or shorter than the radius.

The radius is defined as half of the diameter.

The sum of the interior angles of any quadrilateral is 360 degrees.

Surface area = 6 * (side)² = 6 * (5)² = 6 * 25 = 150 cm².

Volume = πr²h = 3.14 * (3)² * 10 = 3.14 * 9 * 10 = 282.6 cm³.

A regular polygon is defined as a polygon where all sides and angles are equal, making option 2 the correct description.

The standard equation of a circle with center (0, 0) and radius r is x^2 + y^2 = r^2. Here, r = 5, so r^2 = 25.

Parallel lines have the same slope. The slope of y = -3x + 4 is -3, so any line with the same slope, like y = -3x + 1, is parallel.

An isosceles triangle is defined as having at least two sides that are equal in length.

A regular polygon is defined as a polygon with all sides and angles equal, and it cannot be concave; it must be convex.

A circle is defined as a round shape with no corners.

In a square, all angles are right angles (90 degrees), not acute.

Polygons are defined as shapes made up of straight line segments, so having curved sides disqualifies a shape from being a polygon.

Not all triangles have all angles acute; for example, a right triangle has one angle that is 90 degrees.

Complementary angles are two angles whose sum is 90 degrees. All the given pairs meet this criterion.