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

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.

Answer: C

The sum of the angles of a triangle is 180°.

Draw a line through one vertex parallel to the opposite side. The three angles at that vertex, which lie along the straight line, are equal to the three angles of the triangle (alternate angles), and angles on a straight line add up to 180°.

60° is one angle of an equilateral triangle, 90° is a right angle, and 360° is the angle sum of a quadrilateral.

Answer: C

The angle is 240 degrees.

One right angle is 90°.

\[ \begin{aligned} 2\tfrac{2}{3} &= \tfrac{8}{3} \\ \tfrac{8}{3} \times 90 &= 8 \times 30 = 240 \end{aligned} \]

210 is \(2\tfrac{1}{3}\) right angles and 330 is \(3\tfrac{2}{3}\), and 285 is not a whole number of thirds of a right angle at all.

Answer: C

The value of 2x + 3y is 22 (printed with a degree sign as 22°).

In a parallelogram opposite sides are parallel, so the diagonal BD makes equal alternate angles with them. AD is parallel to BC, so \(\angle CBD = \angle ADB\); AB is parallel to DC, so \(\angle ABD = \angle CDB\):

\[ \begin{aligned} 12x &= 60 \Rightarrow x = 5 \\ 7y &= 28 \Rightarrow y = 4 \\ 2x + 3y &= 10 + 12 = 22 \end{aligned} \]

Pairing the angles the other way (12x with 28 or 7y with 60) does not give whole-number answers that match any option, which is a useful check on which angles are alternate.