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

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°.

Answer: B

The value is 110°.

Since P lies on the line AB, angles QPA and QPB form a linear pair and add up to 180°:

\[ \begin{aligned} 7x + 5x &= 180^\circ \\ x &= 15^\circ \\ 8x - 10^\circ &= 120^\circ - 10^\circ = 110^\circ \end{aligned} \]

118° and 102° come from adding or subtracting the wrong amount, and 140° uses x = 18.75°, which would follow only if the angles added up to 225°.

Answer: B

The value is 60°.

Since PS = QS = RS, S is the midpoint of PQ and is equidistant from P, Q and R, so R lies on the circle with diameter PQ. The angle in a semicircle is a right angle, so ∠PRQ = 90°. (Equivalently, triangles PSR and QSR are isosceles, and their base angles add up to half of 180°.)

\[ \begin{aligned} \angle PQR &= 180^\circ - 90^\circ - 15^\circ = 75^\circ \\ 2\angle PQR - \angle PRQ &= 150^\circ - 90^\circ = 60^\circ \end{aligned} \]

75° is ∠PQR itself, and 50° and 45° do not follow once ∠PRQ is recognised as a right angle.

Answer: A

A triangle with all sides of different lengths is a scalene triangle.

Triangles are classified by their sides as scalene (no two sides equal, so all three angles are different too), isosceles (at least two sides equal, with the angles opposite them equal) and equilateral (all three sides equal, each angle 60°).

A right triangle is a classification by angle, one angle being 90°; it can itself be scalene or isosceles, so it is not the name for 'all sides different'. Equilateral and isosceles triangles both have equal sides, which contradicts the description.

Answer: D

A line segment has a definite length.

A line segment is the part of a line between two end points, so it can be measured, for example AB = 5 cm.

A line extends without end in both directions, so it has no definite length. A ray has one end point but extends without end in the other direction, so it cannot be measured either. A plane is a flat surface extending without end in all directions; it has no length at all. Only the segment, bounded at both ends, has a measurable length.

Answer: C

The letter H has two lines of symmetry.

H can be folded along a vertical line through its middle and along a horizontal line through its crossbar, and in each case the two halves match exactly. So it has one vertical and one horizontal line of symmetry.

E and B (in the usual printed form) have only a horizontal line of symmetry, and U has only a vertical one. Each of these has a single line, so only H has two. Other capital letters with two lines of symmetry are I, O and X.

Answer: C

The value of \((2x + 10^\circ)\) is 116°.

The interior angles of a pentagon add up to \((5 - 2) \times 180^\circ = 540^\circ\).

\[ \begin{aligned} 10x + (9 + 1 - 1 + 5 - 4)^\circ &= 540^\circ \\ 10x + 10^\circ &= 540^\circ \\ x &= 53^\circ \end{aligned} \]

So \(2x + 10^\circ = 106^\circ + 10^\circ = 116^\circ\). The other three values would each need a different x, and none of those makes the five angles add up to 540°.

Answer: C

The value of \((4x + 30^\circ)\) is 110°.

Corresponding angles formed by a transversal across parallel lines are equal:

\[ \begin{aligned} 3x - 20^\circ &= 5x - 60^\circ \\ 2x &= 40^\circ \\ x &= 20^\circ \end{aligned} \]

So \(4x + 30^\circ = 80^\circ + 30^\circ = 110^\circ\). Treating the angles as supplementary instead (adding to 180°) gives x = 32.5° and a different value.