Understanding Quadrilaterals- 2 - Free MCQ Practice Test with solutions,

The solution explains why the adjacent angles of a parallelogram are supplementary:

• In a parallelogram, adjacent angles are on the same side of a transversal line.
• Such angles are known as co-interior angles, and they add up to 180 degrees.
• In this question: One angle is 50 degrees, the adjacent angle will be 130 degrees because 180 - 50 = 130.

Therefore, the adjacent angles are supplementary.

x + 50^ο = 180^ο [ as sum of interior angles between parallel lines is 180 ]
x = 130^ο
x = z [ as corresponding angles are equal ]
so z = 130^ο

3. Rhombus

Explanation: A rhombus is a quadrilateral that has the following properties:

• All four sides are of equal length.
• It has two pairs of adjacent sides that are equal.
• The diagonals of a rhombus intersect at right angles (90 degrees) and bisect each other.

This makes the rhombus the correct answer.

• IN Trapezium: Only one pair of opposite sides is parallel. It does not guarantee equal adjacent sides or perpendicular diagonals
• A rectangle has equal opposite sides and diagonals that are equal but do not intersect at right angles.
• A circle is not a quadrilateral, so it doesn't apply to this question.

Given a rhombus RICE where
RE = 13, Ol = 5 OC = 12
In a rhombus,
Diagonals bisect each other

Also,
All sides of a rhombus are equal.
RI = RE
x = 5
∴ x = 5, y = 12 and z =13
 

Given a rhombus RICE where
RE = 13, Ol = 5 OC = 12
In a rhombus,
Diagonals bisect each other

Also,
All sides of a rhombus are equal.
RI = RE
z = 13
∴ x = 5, y = 12 and z = 13

So x+y+z = 5+12+13 = 30

• Definition: A regular polygon is a polygon that has all sides of equal length and all angles are equal.


• Hexagon: A hexagon is a polygon with 6 sides. Each side of a regular hexagon is of equal length, and each interior angle measures 120 degrees.


• Properties:
  
  
  
  • A regular hexagon can be divided into 6 equilateral triangles.
  
  
  • The sum of interior angles in a hexagon is 720 degrees.
  
  
  • A hexagon has 9 diagonals.
  
  
  
  


• Examples: Some examples of objects that have a hexagonal shape are honeycombs, snowflakes, and some nuts and bolts.

• Exterior Angles: The exterior angles of a polygon are the angles formed when one side of the polygon is extended outwards.
• Sum of Exterior Angles: The sum of the exterior angles of any polygon is always 360 degrees.
• Pentagon: A pentagon has 5 sides, so the sum of its exterior angles will be 360 degrees.

Correct Answer: Option D: 360°

• Definition: A diagonal is a line segment that connects two non-adjacent vertices in a polygon.


• Rectangle: A rectangle is a quadrilateral with opposite sides that are equal in length and all angles are right angles.


• Calculating Diagonals: In a rectangle, the number of diagonals can be calculated using the formula: n(n-3)/2, where n is the number of sides of the polygon.


• For a Rectangle: A rectangle has 4 sides, so substituting n=4 into the formula gives us: 4(4-3)/2 = 4(1)/2 = 4/2 = 2.


• Answer: Therefore, a rectangle has 2 diagonals.

AC and BD bisect each other means the diagonals intersect at their common midpoint.

If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram; this is a standard characterisation.

A rectangle and a square are special types of parallelogram and their diagonals do bisect each other, but the converse need not hold: diagonals bisecting each other does not imply right angles or equal adjacent sides.

Therefore ABCD is a parallelogram. (Option D)

• A rhombus has all sides equal in length.


• Opposite angles are equal in measure.


• Diagonals bisect each other at right angles.


• Diagonals are perpendicular bisectors of each other.

• A kite has two pairs of adjacent sides that are equal in length.


• Diagonals are perpendicular to each other.


• One diagonal bisects the other at a right angle.

• A rhombus has all the properties of a parallelogram.


• When a rhombus has adjacent sides equal in length, it also exhibits the properties of a kite.

Therefore, a shape that has all the properties of a parallelogram and also those of a kite is a rhombus. The answer is option A: rhombus.

y = 100^ο [opposite angle of parallelogram are equal]
So, correct answer is option A.

• Polygon: A polygon is a closed figure made up of line segments. It can have any number of sides, including triangles, quadrilaterals, pentagons, and so on.


• Quadrilateral: A quadrilateral is a polygon with four sides and four angles. It is a specific type of polygon, not a general closed curve made up of line segments.


• Hexagon: A hexagon is a polygon with six sides and six angles. Like the quadrilateral, it is a specific type of polygon and not a general closed curve made up of line segments.


• None of these: This option does not apply as the correct answer is A - polygon.

• A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

• In a parallelogram, the opposite sides are parallel, which means they will never intersect.


• Additionally, the opposite sides of a parallelogram are equal in length.

• Aside from having opposite sides that are parallel, a parallelogram also has opposite angles that are equal.


• The consecutive angles of a parallelogram are supplementary, meaning they add up to 180 degrees.

• While a rhombus also has opposite sides that are parallel, a rhombus has all sides that are equal in length, unlike a parallelogram.


• A quadrilateral is a broader term that encompasses all four-sided figures, including parallelograms, rhombuses, rectangles, and squares.

Step 1: Property of a parallelogram

The figure looks like a parallelogram (opposite sides parallel).
In a parallelogram:

• Opposite angles are equal.

• Adjacent angles are supplementary (their sum is 180°).

Step 2: Relation between ∠B and ∠C

Since AB is parallel to DC, angles at B and C are adjacent.
So,
∠B + ∠C = 180°

That means,
100° + x = 180°

Step 3: Solve for x

x = 180° - 100°
x = 80°

- Equal: Opposite sides of a parallelogram have the same length.
- Parallel: Opposite sides of a parallelogram run parallel to each other.

These properties ensure that the figure maintains its shape and symmetry, characteristic of parallelograms. Therefore, the correct answer is C: Equal and parallel.

A triangle has zero diagonals.Diagonals must be created across vertices in a polygon, but the vertices must not be adjacent to one another. A triangle has only adjacent vertices.

We have angle sum of pentagon=(5-2)*180 =540. Since it is a regular pentagon all angles are equal. So the sum is 5x. So each angle measures  x = 540/5 = 108º