Geometry (Lines, Angles, Triangles, Polygons) Questions and Answers

Answer: C

Correct answer: 720 degrees

The interior angles of an \(n\)-sided polygon sum to \((n-2) \times 180^\circ\).

\[(6-2) \times 180 = 4 \times 180 = 720^\circ\]

For a regular hexagon each angle is therefore \(720 \div 6 = 120^\circ\).

Answer: C

The intersection point of the perpendicular bisectors of the sides of a triangle is the circumcenter. This point is equidistant from all three vertices of the triangle and is the center of the circle that circumscribes the triangle.

Answer: B

The intersection point of the angle bisectors of the angles of a triangle is the incenter. This point is equidistant from all three sides of the triangle and is the center of the circle that is inscribed within the triangle.

Answer: D

The diagonals of a rhombus bisect each other at right angles, cutting it into four congruent right triangles.

\[ 90^{\circ} \]

This is why the area can be written as \( \tfrac{1}{2}d_{1}d_{2} \).

Answer: B

We use the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to find the lengths of the sides.

Side 1 (between (0,0) and (4,0)): \(\sqrt{(4-0)^2} = 4\).

Side 2 (between (0,0) and (2,3)): \(\sqrt{(2-0)^2 + (3-0)^2} = \sqrt{4+9} = \sqrt{13}\).

Side 3 (between (4,0) and (2,3)): \(\sqrt{(2-4)^2 + (3-0)^2} = \sqrt{(-2)^2+3^2} = \sqrt{4+9} = \sqrt{13}\).

Since two sides have the same length (\(\sqrt{13}\)), the triangle is isosceles.

Answer: B

A line segment is one-dimensional: it has length and two endpoints, but no breadth and no thickness.

Length and endpoints are precisely what it does have, which is what makes breadth the odd one out.

Answer: C

The condition given is that the ratios of the lengths of the corresponding sides of the two triangles are equal. This is the definition of the SSS (Side-Side-Side) similarity criterion. If this condition is met, the two triangles are similar.

Answer: A

The statement that is not true is that a kite becomes a rectangle if its opposite angles are equal.

A kite has two pairs of equal adjacent sides. If both pairs of its opposite angles are equal, it becomes a parallelogram with all four sides equal, that is, a rhombus, not a rectangle. A kite becomes a rectangle only in the special case of a square.

The other statements are true: a rectangle with all sides equal is a square, a rhombus with all angles equal (90 degrees each) is a square, and a parallelogram with all angles equal is a rectangle.

Answer: B

The true statement is that its diagonals are perpendicular to each other.

A quadrilateral with two pairs of equal adjacent sides (PQ = PS and RQ = RS) is a kite. P and R are each equidistant from Q and S, so both lie on the perpendicular bisector of QS. The diagonal PR is therefore that perpendicular bisector, and the diagonals cross at right angles.

The other properties do not hold for a kite in general. Equal diagonals belong to rectangles, diagonals bisecting each other to parallelograms, and both diagonals bisecting the vertex angles to a rhombus; in a kite only PR bisects the angles at P and R.

Answer: C

The value is 210°.

In a rhombus each diagonal bisects the vertex angles, so ∠ADC = 2 × 25° = 50°. Opposite angles are equal and adjacent angles are supplementary:

\[ \begin{aligned} \angle ABC &= \angle ADC = 50^\circ \\ \angle BAD &= 180^\circ - 50^\circ = 130^\circ \\ 2\angle BAD - \angle ABC &= 260^\circ - 50^\circ = 210^\circ \end{aligned} \]

260° is 2 ∠BAD before subtracting ∠ABC, and 230° and 240° come from taking ∠ABC as 30° or 20° instead of 50°.