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

Answer: B

The correct statement is \(\Delta ABC \cong \Delta DCB\), by SSS.

Match the sides: AB = 5 = DC, BC = 4 = CB and AC = 8 = DB. All three pairs of sides are equal, so the triangles are congruent by SSS, with the correspondence A ↔ D, B ↔ C and C ↔ B.

Writing \(\Delta DBC\) would pair AB with DB (5 with 8), which is wrong, so the order of letters matters. SAS is not the reason, since no angle is given; the congruence comes from the three sides alone.

Answer: B

The correct pair is ∠A = 110°, ∠C = 110°.

A quadrilateral in which one pair of opposite sides is both parallel and equal (AB ∥ DC and AB = CD) is a parallelogram.

In a parallelogram, opposite angles are equal and adjacent angles add up to 180°. So ∠D = ∠B = 70°, and ∠A = ∠C = 180° − 70° = 110°.

The other options put 70° at A or C, or 110° at D, which would make an angle adjacent to B equal to B, or the angle opposite B different from B.

Answer: B

The congruence is ΔABC ≅ ΔFED.

In a congruence statement the vertices must be listed in corresponding order. Find which vertex of ΔDEF matches each vertex of ΔABC from the equal sides.

AB = EF and CA = FD share the vertex A on one side and F on the other, so A corresponds to F. AB = EF and BC = DE share B and E, so B corresponds to E. That leaves C corresponding to D. Reading in order A, B, C gives F, E, D, and the triangles are congruent by SSS.

ΔDEF would need AB = DE; ΔEFD would need A to match E; ΔDFE would need AB = DF. None fits the given pairs of sides.

Answer: D

The sides 65 cm, 72 cm and 97 cm can form a right angled triangle.

A triangle is right angled exactly when the square of the longest side equals the sum of the squares of the other two (the converse of Pythagoras' theorem).

\[ \begin{aligned} 65^{2} + 72^{2} &= 4225 + 5184 \\ &= 9409 = 97^{2} \end{aligned} \]

The others fail: \(20^{2} + 21^{2} = 841 = 29^{2}\), not \(31^{2}\); \(35^{2} + 77^{2} = 7154\), not \(88^{2} = 7744\); and 15, 32, 57 cannot even form a triangle, since \(15 + 32 \lt 57\).

Answer: C

The polyhedron has 15 edges.

Euler's formula connects the faces F, vertices V and edges E of any convex polyhedron.

\[ \begin{aligned} F + V - E &= 2 \\ 7 + 10 - E &= 2 \\ E &= 15 \end{aligned} \]

A pentagonal prism is an example: 7 faces (two pentagons and five rectangles), 10 vertices and 15 edges.

17 is \(F + V\) without subtracting 2. With 13 or 14 edges, \(F + V - E\) would be 4 or 3, not 2, so no polyhedron has those counts.

Answer: C

The true statement is the difference between ∠ABC and ∠ACB is 35°.

∠BAD is the exterior angle at A (CA produced to D), so ∠BAC = 180° − 125° = 55°. ∠EBC is the exterior angle at B (AB produced to E), so ∠ABC = 180° − 100° = 80°. The angle sum then gives ∠ACB = 180° − 55° − 80° = 45°.

So ∠ABC − ∠ACB = 80° − 45° = 35°.

The others are false. The angles 55°, 80° and 45° are all different, so the triangle is not isosceles. AB is opposite the smallest angle (45° at C) and BC is opposite 55°, so AB is shorter than BC, not longer. ∠BAC − ∠ACB is 10°, not 20°.

Answer: B

A Square is always a regular polygon.

A regular polygon has all sides equal and all angles equal. A square has four equal sides and four right angles, so every square is regular.

An isosceles triangle has only two equal sides (unless it is equilateral), a pentagon need not have equal sides or angles, and a circle is not a polygon at all, since it has no straight sides.