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

Answer: D

\(\angle BOC\) is 65°.

O is where the bisectors of the two exterior angles at B and C meet. With \(\angle B\) and \(\angle C\) the interior angles of the triangle:

\[ \begin{aligned} \angle OBC &= \tfrac{1}{2}(180^{\circ} - B) \\ \angle OCB &= \tfrac{1}{2}(180^{\circ} - C) \end{aligned} \]

\[ \begin{aligned} \angle BOC &= 180^{\circ} - \angle OBC - \angle OCB \\ &= \tfrac{1}{2}(B + C) \\ &= \tfrac{1}{2}(180^{\circ} - 50^{\circ}) = 65^{\circ} \end{aligned} \]

This is the standard result \(\angle BOC = 90^{\circ} - \frac{A}{2}\) for exterior bisectors. 115° is \(90^{\circ} + \frac{A}{2}\), the result for the bisectors of the interior angles, which is the usual mix-up.

Answer: B

The figure is a parallelogram.

A general parallelogram looks the same after a half-turn (180°) about the point where its diagonals meet, so it has rotational symmetry of order 2. But no line through it folds one half exactly onto the other, so it has no line of symmetry.

A kite has one line of symmetry (along its main diagonal) and no rotational symmetry. A rectangle and a rhombus have both: two lines of symmetry each, as well as rotational symmetry of order 2.

Answer: D

The ratio is \(\dfrac{1}{2}\).

Let \(AD = DE = EB = a\), so \(AB = 3a\), \(DB = 2a\) and \(EB = a\). Because the angle at B is a right angle, Pythagoras' theorem applies to triangles DBC, ABC and EBC:

\[ \begin{aligned} DC^{2} - BC^{2} &= DB^{2} = 4a^{2} \\ AC^{2} - EC^{2} &= AB^{2} - EB^{2} \\ &= 9a^{2} - a^{2} = 8a^{2} \end{aligned} \]

\[ \frac{4a^{2}}{8a^{2}} = \frac{1}{2} \]

BC cancels in both differences, so the answer does not depend on the length of BC. \(\frac{4}{9}\) is \(DB^{2} : AB^{2}\), which forgets to subtract \(EB^{2}\).

Answer: A

The value of \((2x - 4^\circ)\) is 72°.

Supplementary angles add up to 180°.

\[ \begin{aligned} (3x - 30^\circ) + (2x + 20^\circ) &= 180^\circ \\ 5x - 10^\circ &= 180^\circ \\ x &= 38^\circ \\ 2x - 4^\circ &= 76^\circ - 4^\circ = 72^\circ \end{aligned} \]

76° is \(2x\) before subtracting 4°. Treating the angles as complementary (sum 90°) gives \(x = 20^\circ\) and a value of 36°, which is not offered.

Answer: C

The value is 270°.

The bisectors split the angles at P and Q in half, so in triangle POQ the angles at P and Q are 15° and 25°.

\[ \begin{aligned} \angle POQ &= 180^\circ - 15^\circ - 25^\circ = 140^\circ \\ 2\angle POQ - 10^\circ &= 280^\circ - 10^\circ = 270^\circ \end{aligned} \]

The same result follows from the rule \(\angle POQ = 90^\circ + \frac{1}{2}\angle R\), with \(\angle R = 100^\circ\). The option 150° comes from using \(\angle P + \angle Q = 80^\circ\) in place of \(\angle POQ\): \(2 \times 80^\circ - 10^\circ = 150^\circ\).

Answer: D

The set that cannot form a right triangle is 8, 16, 12.

Three lengths form a right triangle only when the square of the longest equals the sum of the squares of the other two (the converse of Pythagoras' theorem).

\[ \begin{aligned} 8^{2} + 12^{2} &= 64 + 144 = 208 \\ 16^{2} &= 256 \ne 208 \end{aligned} \]

The other sets pass: \(2.5^{2} + 6^{2} = 6.25 + 36 = 42.25 = 6.5^{2}\); \(12^{2} + 16^{2} = 400 = 20^{2}\); and \(4.5^{2} + 6^{2} = 20.25 + 36 = 56.25 = 7.5^{2}\).

Answer: C

The letter without rotational symmetry is E.

A figure has rotational symmetry if it looks the same after a rotation of less than a full turn. E only looks the same after a complete 360° turn, so it has no rotational symmetry (it has only a horizontal line of symmetry).

H and N each look the same after a half turn of 180°, so both have rotational symmetry of order 2. O, drawn as a circle or an oval, looks the same after a half turn as well. So H, O and N are all wrong answers.

Answer: B

The value of \((7x - y)\) is 8 cm.

The diagonals of a parallelogram bisect each other, so \(OP = OR\) and \(OQ = OS\).

\[ \begin{aligned} y + 7 &= 20 \Rightarrow y = 13 \\ x + y &= 16 \Rightarrow x = 3 \\ 7x - y &= 21 - 13 = 8 \end{aligned} \]

The trap is pairing the wrong segments, such as setting \(OP = OQ\); in a general parallelogram the two diagonals are not equal, so only opposite halves of the same diagonal match.

Answer: A

The value of (4x + 40°) is 160°.

Complementary angles add up to 90°:

\[ \begin{aligned} (2x - 10) + (3x - 50) &= 90 \\ 5x - 60 &= 90 \\ x &= 30 \end{aligned} \]

\[ 4x + 40 = 4(30) + 40 = 160^\circ \]

Check: the two angles are 50° and 40°, which add to 90°. Treating the angles as supplementary (180°) gives x = 48 and 232°, which is not offered. 180°, 120° and 150° come from slips in solving for x.

Answer: C

The correct relation is \(2x + 3y - 2z = 8\).

A triangular pyramid (tetrahedron) has 4 triangular faces, 4 vertices and 6 edges, so x = 4, y = 4 and z = 6. These satisfy Euler's formula F + V − E = 2.

\[ \begin{aligned} 2x + 3y - 2z &= 8 + 12 - 12 \\ &= 8 \end{aligned} \]

The others fail: \(2x + 2z - 3y = 8 + 12 - 12 = 8\), not 6; \(3x + 3y - 2z = 12 + 12 - 12 = 12\), not 6; and \(3x + 2y - 3z = 12 + 8 - 18 = 2\), not 4.