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

Answer: A

The third angle is 100°.

The angles of a triangle add up to 180°:

\[ 180^\circ - (50^\circ + 30^\circ) = 100^\circ \]

80° is the sum of the two given angles, not the third one, and 40° and 60° do not make the total 180°.

Answer: C

The correct match is (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i).

The flat face of a blackboard represents part of a plane. A line extends endlessly in both directions, so it has no definite length. A ray starts at one end point and goes on in one direction. A line segment has two end points and a fixed length.

Options 1 and 2 give the ray two end points or the line one end point, and option 4 makes the blackboard a segment.

Answer: C

The value of \(7y + 5\) is 26.

The diagonals of a parallelogram bisect each other, so OP = OR and OQ = OS:

\[ \begin{aligned} x + 7 &= 20 \Rightarrow x = 13 \\ x + y &= 16 \Rightarrow y = 3 \\ 7y + 5 &= 21 + 5 = 26 \end{aligned} \]

21 is 7y without the 5. Equating OP with OQ instead of with OR, as if all half-diagonals were equal, is the usual slip behind the other values.

Answer: B

The value is 19.

Euler's formula for a polyhedron gives the number of vertices:

\[ \begin{aligned} F + V - E &= 2 \\ 7 + x - 15 &= 2 \Rightarrow x = 10 \\ 2F + 3E - 4x &= 14 + 45 - 40 = 19 \end{aligned} \]

A wrong value of x, for example from writing F + E - V = 2, leads to the other options.

Answer: A

The value of \(4x + 6^{\circ}\) is 154°.

Interior angles on the same side of a transversal cutting two parallel lines are supplementary:

\[ \begin{aligned} (2x + 15) + (3x - 20) &= 180 \\ 5x - 5 &= 180 \Rightarrow x = 37 \\ 4x + 6 &= 148 + 6 = 154 \end{aligned} \]

148° is 4x without adding the 6. Taking the two angles as equal, as for alternate angles, gives x = 35 and 146°.

Answer: B

The value of \(3x + 25^{\circ}\) is 85°.

Since AM = CM, ∠ACM = ∠A = x, and since BM = CM, ∠BCM = ∠B = 70°. So ∠C = x + 70°. Using the angle sum of triangle ABC:

\[ \begin{aligned} x + 70 + (x + 70) &= 180 \\ 2x &= 40 \Rightarrow x = 20 \\ 3x + 25 &= 60 + 25 = 85 \end{aligned} \]

The shortcut: M is equidistant from A, B and C, so it is the midpoint of the hypotenuse and ∠C = 90°, giving x = 20 directly.

Answer: D

∠ADC is 100°.

\[ \begin{aligned} \angle C &= 180 - 50 - 70 = 60 \\ \angle ACD &= 60 / 2 = 30 \\ \angle ADC &= 180 - 50 - 30 = 100 \end{aligned} \]

In triangle ADC the angles at A and C are 50° and 30°, so the third is 100°. 30° is ∠ACD itself, and 70° and 50° are the given angles, not ∠ADC.

Answer: A

The statement that is not true is sum of two acute angles can be equal to a straight angle.

An acute angle is less than 90°, so two acute angles together are always less than 180°. A straight angle is exactly 180°, so two acute angles can never make one.

The other statements are possible. Two acute angles can make a right angle (40° + 50° = 90°) or an obtuse angle (60° + 70° = 130°). An acute and an obtuse angle can make a straight angle (60° + 120° = 180°), as the angles in a linear pair often do.

Answer: D

The letter with exactly one line of symmetry is V.

V can be folded along one vertical line through its point so that the two halves match. No horizontal or other line does this.

O (as a circle) has infinitely many lines of symmetry, and even an oval O has two. X has two lines of symmetry, one vertical and one horizontal (more if drawn with equal arms at right angles). S has no line of symmetry at all; it has rotational symmetry of order 2 instead.

Answer: C

The sides 14, 50, 48 form a right triangle.

By Pythagoras, the square of the longest side must equal the sum of the squares of the other two.

\[ \begin{aligned} 14^2 + 48^2 &= 196 + 2304 \\ &= 2500 = 50^2 \end{aligned} \]

The others fail: \(16^2 + 30^2 = 1156\), not \(32^2 = 1024\); \(12^2 + 44^2 = 2080\), not \(50^2 = 2500\); and \(12^2 + 18^2 = 468\), not \(20^2 = 400\). A quick check is that 14, 48, 50 is just 2 times the familiar triple 7, 24, 25.