Quadrilaterals - Olympiad Level MCQ, Class 9 Mathematics - Class 9 MCQ

In geometry, if the diagonals of a quadrilateral bisect each other, the figure is necessarily a parallelogram. This property holds because the midpoint intersection of the diagonals implies that:

• Opposite sides are both equal and parallel.

While squares, rectangles, and rhombuses also have this property, they are specific types of parallelograms. Thus, the most general and correct answer is parallelogram.

In parallelogram ABCD, the following properties hold:

• Opposite angles are equal: ∠A = ∠C
• Consecutive angles are supplementary: ∠A + ∠B = 180°

The angle bisectors of ∠A and ∠D divide these angles into two equal parts. Since ∠A and ∠D are supplementary, their halves sum to:

• 90°

In triangle AOD, the remaining angle at O must also be 90° because the total sum of angles in a triangle is:

• 180°

Therefore, ∠AOD measures 90°.

In a parallelogram, opposite angles are equal. Therefore, the following holds true:

• ∠A is equal to ∠C
• The difference between ∠A and ∠C is zero degrees

This means that:

• Angle A = Angle C
• Angle A - Angle C = 0°

Given a rhombus where one diagonal is equal to the side length, we denote the side as s and the diagonals as d₁ and d₂. Assuming d₁ = s, the diagonals bisect each other at right angles, forming four right-angled triangles.

Using the Pythagorean theorem in one of these triangles:

(s/2)² + (d₂/2)² = s²

Solving this gives:

d₂ = s√3

Using trigonometric relationships, the angles of the rhombus can be found using:

tan(θ/2) = (d₁/2)/(d₂/2) = (s/2)/((s√3)/2) = 1/√3

This implies:

θ/2 = 30°, so θ = 60°. The other angle is 180° - 60° = 120°.

Thus, the angles of the rhombus are 60° and 120°.

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1. Understanding Rhombus Properties:

• All sides are equal.
• Opposite angles are equal.
• Diagonals bisect each other at right angles and also bisect the vertex angles.

2. Given Angles:

• Angle at B (angle ABC) is 100 degrees.
• Therefore, angle at D (angle ADC) is also 100 degrees since opposite angles in a rhombus are equal.
• Angles at A and C are each 80 degrees because the sum of adjacent angles in a parallelogram is 180 degrees.

3. Diagonals Bisecting Angles:

• Diagonal AC bisects angles at A and C into two 40-degree angles each.
• Diagonal BD bisects angles at B and D into two 50-degree angles each.

4. Triangle ABD Analysis:

• In triangle ABD, sides AB and AD are equal (since ABCD is a rhombus).
• Therefore, triangle ABD is isosceles with angle ABD = angle ADB.

5. Calculating Angles in Triangle ABD:

• The angle at vertex A (angle BAD) is 80 degrees.
• Let the base angles angle ABD and angle ADB each be x.
• Sum of angles in triangle ABD: x + x + 80 = 180.
• Solving for x: 2x = 100 hence x = 50.
• Thus, the measure of angle ADB is 50 degrees.

In a convex quadrilateral, the sum of all interior angles is 360°. Given the angles:

• ∠P = 100°
• ∠R = 100°
• ∠S = 75°

We calculate ∠Q as follows:

• ∠Q = 360° - (∠P + ∠R + ∠S)
• ∠Q = 360° - (100° + 100° + 75°)
• ∠Q = 360° - 275°
• ∠Q = 85°

Thus, the measure of ∠Q is 85 degrees.

Let the angles be 3x, 4x, 5x, and 6x. Since the sum of angles in a quadrilateral is 360°, we can set up the following equation:

• 3x + 4x + 5x + 6x = 360°
• This simplifies to 18x = 360°
• Solving for x gives x = 20

The angles are therefore:

• 3x = 3 × 20 = 60°
• 4x = 4 × 20 = 80°
• 5x = 5 × 20 = 100°
• 6x = 6 × 20 = 120°

The difference between the greatest angle (120°) and the smallest angle (60°) is:

• 120° - 60° = 60°

In any quadrilateral, the sum of all four interior angles is always 360 degrees.

Given that ∠A + ∠C = 180°, we can find ∠B + ∠D by subtracting this from the total sum:

• ∠B + ∠D = 360° - (∠A + ∠C)
• ∠B + ∠D = 360° - 180°
• ∠B + ∠D = 180°

Thus, the correct answer is 180 degrees.

Option A: True. Each diagonal in a quadrilateral splits it into two triangles.

Option B: True. This follows from the polygon inequality theorem for quadrilaterals.

Option C: True. The maximum number of obtuse angles in a quadrilateral is three.

Option D: False. A quadrilateral has only two diagonals, not four.

Thus, the false statement is D.

The sum of the interior angles in a quadrilateral is always 360°. Given the angles x°, (x – 10)°, (x + 30)°, and 2x°, we can set up the equation:

x + (x - 10) + (x + 30) + 2x = 360

Simplifying this, we get:

5x + 20 = 360

Subtracting 20 from both sides:

5x = 340

Dividing by 5:

x = 68°

Now substituting x back into each angle:

• First angle: 68°
• Second angle: 68° - 10° = 58°
• Third angle: 68° + 30° = 98°
• Fourth angle: 2 × 68° = 136°

The greatest angle is 136°, which corresponds to option A.

In a parallelogram, consecutive angles are supplementary, meaning they add up to 180°.

Given that ∠A = 80°, we can find ∠B as follows:

• ∠A + ∠B = 180°
• 80° + ∠B = 180°
• ∠B = 100°

Thus, the measure of ∠B is 100 degrees.

In trapezium ABCD, since AB is parallel to CD and AD is a transversal, the consecutive angles ∠A and ∠D are supplementary. Therefore, we have:

• ∠A + ∠D = 180°

Given that ∠A = 50°, we can find ∠D as follows:

• ∠D = 180° - 50° = 130°

Thus, the measure of ∠D is 130°, which corresponds to option B.

In square ABCD, diagonals AC and BD intersect at point O, which is their midpoint. The sides AO and BO are equal because the diagonals of a square are congruent and bisect each other equally.

Since the angle between the diagonals at their intersection in a square is 90 degrees (as the diagonals are perpendicular to each other), triangle AOB has two equal sides (AO = BO) and a right angle at O, making it an isosceles right-angled triangle.

For a general parallelogram:

• The diagonals are not equal (property A).
• The diagonals are not perpendicular to each other (property B).
• The diagonals do not divide the figure into four congruent triangles (property C).

Therefore, all of these properties (A, B, and C) are not true for a general parallelogram. Hence, the correct answer is D.

In a rhombus, the diagonals bisect each other at right angles. Given the following dimensions:

• AB = 10 cm
• BD = 16 cm

Half of BD is:

• 8 cm

Using the Pythagorean theorem in triangle AOB (where AO is half of AC), we can set up the following equation:

AO² + BO² = AB²

Substituting the known values:

AO² + 8² = 10²

AO² + 64 = 100

AO² = 36

Thus, AO = 6 cm.

Therefore, AC can be calculated as:

AC = 2 * AO = 12 cm.

Let the lengths of the adjacent sides be x and y.

• 1. The perimeter of a parallelogram is given by: 2(x + y) = 180 cm. Simplifying, we get: x + y = 90 cm.
• 2. It is also given that one side exceeds the other by 10 cm. Therefore: y = x + 10.
• 3. Substitute y = x + 10 into x + y = 90: x + (x + 10) = 90. Simplifying gives: 2x + 10 = 90. Thus, 2x = 80, leading to x = 40 cm.
• 4. Substituting x = 40 into y = x + 10: y = 40 + 10 = 50 cm.

Thus, the adjacent sides of the parallelogram are 40 cm and 50 cm.

Understand Parallelogram Properties: Opposite angles are equal: ∠ DAB = ∠ BCD. Given ∠ DAB = 75°, thus ∠ BCD = 75°.

Analyse Triangle DBC: Consider the triangle formed by diagonal BD: △ DBC. Known angles in △ DBC:

• ∠ DBC = 60° (given)
• ∠ BCD = 75° (from parallelogram property)

Apply Triangle Angle Sum: The sum of angles in a triangle is 180°. Calculate ∠ CDB:

• ∠ CDB = 180° - ∠ DBC - ∠ BCD
• ∠ CDB = 180° - 60° - 75° = 45°

Thus, the measure of angle ∠ CDB is 45°.