All questions of Understanding Quadrilaterals for Class 8 Exam

Each polygon have 360• as it's exterior angel but it's Interior angle is depend on it's sides, so the answer is D

• 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.

• A square also has diagonals that intersect at right angles, but all four sides are equal rather than just two pairs of adjacent sides.
• 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.

The angle near to angle z we will name it a. z+a =180 (linear pairs). a=50(vertically opposite angle). ...............a+z=180, so,. 50+z =180(a=50). ............ z=180-50 ............. z=130............... z=x (corresponding angle) so if z=130 then x=130 .............

D)

A noun and a verb.

Opposite angle are equal

Explanation:
A rhombus is a quadrilateral with all sides equal in length. It is also known as a diamond shape. The diagonals of a rhombus are the line segments that connect opposite vertices of the rhombus.

The diagonals of a rhombus have some important properties:

1. The diagonals of a rhombus bisect each other at right angles. This means that the point where the diagonals intersect is the midpoint of each diagonal, and the diagonals form four right angles.

2. The diagonals of a rhombus are equal in length. This is because a rhombus has four sides of equal length, and the diagonals connect opposite vertices, which are also equidistant from the center of the rhombus.

3. The diagonals of a rhombus are perpendicular bisectors of each other. This means that each diagonal divides the other diagonal into two equal parts, and each diagonal is perpendicular to the other diagonal.

Therefore, we can conclude that the correct answer is option C, which states that the diagonals of a rhombus are perpendicular bisectors of each other.

Sum of all angles of a quadrilateral is 360 degree . so we will add all the angles of quadrilateral and subtract it from 360 to get x. there is a linear pair on left side and exterior angle is 90 hence the other angle will be 90 by linear pair property. hence 70+60+90+x=360 220+x=360 x=360-220 x=140 hence C is the right answer.

A rhombus has all the properties of a parallelogram and also that of a kite. Let's break down the properties of each shape to understand why a rhombus fits the description.

**Properties of a Parallelogram:**
1. Opposite sides are parallel: In a parallelogram, opposite sides are always parallel. This means that if you extend the sides of a parallelogram, they will never intersect.

2. Opposite sides are congruent: The lengths of opposite sides in a parallelogram are equal. This implies that if you measure the length of one side and compare it to the length of the opposite side, they will be the same.

3. Opposite angles are congruent: The angles formed by the intersection of the sides in a parallelogram are equal. If you measure one angle and compare it to the angle opposite it, they will have the same measure.

4. Consecutive angles are supplementary: The sum of two consecutive angles in a parallelogram is always 180 degrees. For example, if angle A and angle B are consecutive angles in a parallelogram, then angle A + angle B = 180 degrees.

**Properties of a Kite:**
1. Two pairs of adjacent sides are congruent: In a kite, two consecutive sides are equal in length. This means that if you measure the length of one side and compare it to the length of the side next to it, they will be the same.

2. One pair of opposite angles is congruent: The angles formed by the intersection of the sides in a kite are equal. If you measure one angle and compare it to the angle opposite it, they will have the same measure.

3. Diagonals are perpendicular: The diagonals of a kite are perpendicular to each other. This means that if you draw the diagonals of a kite, they will intersect at a right angle.

Based on these properties, a rhombus fits the description of having all the properties of a parallelogram and also that of a kite.

In a rhombus:
- Opposite sides are parallel (parallelogram property).
- Opposite sides are congruent (parallelogram property).
- Opposite angles are congruent (parallelogram property).
- Consecutive angles are supplementary (parallelogram property).
- Two pairs of adjacent sides are congruent (kite property).
- One pair of opposite angles is congruent (kite property).
- Diagonals are perpendicular (kite property).

Therefore, a rhombus is the correct answer as it encompasses all the properties of both a parallelogram and a kite.

(A) option (the curve made up of line segments).

This is the answer because the angle sum of a quadrilateral is 360 degree.
Both the polygons here are quadrilateral.
Therefore, the angle sum of them must be equal.

Explanation:

A diagonal is a line segment that connects two nonadjacent vertices of a polygon. In the case of a triangle, a diagonal would connect two vertices that are not adjacent to each other.

A triangle has three sides and three vertices. Let's label the vertices as A, B, and C.

To count the number of diagonals, we need to connect each vertex to every other vertex that is not adjacent to it.

Connecting vertex A:
- A cannot be connected to itself, so we eliminate one possibility.
- A can be connected to B, which forms one diagonal.
- A can be connected to C, which forms another diagonal.

Connecting vertex B:
- B cannot be connected to itself, so we eliminate one possibility.
- B can be connected to A, which forms one diagonal.
- B can be connected to C, which forms another diagonal.

Connecting vertex C:
- C cannot be connected to itself, so we eliminate one possibility.
- C can be connected to A, which forms one diagonal.
- C can be connected to B, which forms another diagonal.

Total number of diagonals:
Adding up all the possibilities, we have a total of 2 + 2 + 2 = 6 diagonals.

However, we need to remember that a diagonal must connect two nonadjacent vertices. In a triangle, all the vertices are adjacent to each other. Therefore, no diagonals can be formed within a triangle.

Hence, the correct answer is option C: 0 diagonals.

Understanding Adjacent Angles in a Parallelogram
Adjacent angles in a parallelogram have specific properties that stem from its geometric characteristics. A parallelogram is defined as a quadrilateral with opposite sides that are parallel and equal in length.
Key Properties of Parallelograms:
- Opposite Angles Are Equal: In a parallelogram, the angles opposite to each other are equal. For example, if one angle is 60 degrees, the opposite angle is also 60 degrees.
- Sum of Interior Angles: The sum of all interior angles in any quadrilateral, including parallelograms, is 360 degrees.
Adjacent Angles Are Supplementary:
- Definition of Supplementary Angles: Two angles are said to be supplementary if their sum equals 180 degrees.
- Adjacent Angles in Parallelograms: In a parallelogram, each pair of adjacent angles adds up to 180 degrees. For instance, if one angle measures 70 degrees, the adjacent angle will measure 110 degrees (70 + 110 = 180).
Conclusion:
Thus, the correct answer to the question about the nature of adjacent angles in a parallelogram is option 'C': they are supplementary angles. This property is essential for understanding the relationships between angles in various geometric shapes, especially in parallelograms.

BC because opposite sides of llgram are equal

A) Pentagon have 5 sides B) Hexagon have 6 sides C) Heptagon have 7 sides So the correct answer is B.

In a parallelogram, opposite sides are equal in length.

Therefore, AB = CD and AD = BC.

Given that AB = 2x + 5, CD = y + 1, AD = y + 5, and BC = 3x, we can set up two equations:

2x + 5 = y + 1 (equation 1)
y + 5 = 3x (equation 2)

From equation 1, we can isolate y:

y = 2x + 4

Substituting this value of y into equation 2, we get:

2x + 4 + 5 = 3x

Simplifying, we have:

9 = x

Now we can substitute this value of x back into the equations to find the lengths of the sides:

AB = 2(9) + 5 = 23
CD = y + 1 = (2(9) + 4) + 1 = 23
AD = y + 5 = (2(9) + 4) + 5 = 27
BC = 3(9) = 27

So, the lengths of the sides are AB = 23, CD = 23, AD = 27, and BC = 27.

Diagonals in a Rectangle

A rectangle is a four-sided polygon with opposite sides that are equal in length and four right angles. Diagonals are line segments that connect two nonadjacent vertices of a polygon. In the case of a rectangle, the diagonals connect the opposite corners or vertices of the rectangle.

Calculating the Number of Diagonals

To determine the number of diagonals in a rectangle, we need to consider the number of possible pairs of nonadjacent vertices. In a rectangle, there are two pairs of opposite vertices. Let's examine each pair separately.

1. Pair of Opposite Vertices on the Longer Side:
- The longer side of a rectangle is also known as its length.
- Let's assume the length of the rectangle is 'L'.
- The longer side has two opposite vertices.
- These two vertices can be connected by a diagonal.
- Therefore, there is 1 diagonal formed by this pair of vertices.

2. Pair of Opposite Vertices on the Shorter Side:
- The shorter side of a rectangle is also known as its width.
- Let's assume the width of the rectangle is 'W'.
- The shorter side has two opposite vertices.
- These two vertices can also be connected by a diagonal.
- Therefore, there is 1 diagonal formed by this pair of vertices.

Total Number of Diagonals

By adding up the number of diagonals formed by each pair of opposite vertices, we can determine the total number of diagonals in a rectangle.

In this case, we have:
- 1 diagonal formed by the pair of opposite vertices on the longer side.
- 1 diagonal formed by the pair of opposite vertices on the shorter side.

Therefore, a rectangle has a total of 2 diagonals.

Conclusion

To summarize, a rectangle has two diagonals. One diagonal is formed by a pair of opposite vertices on the longer side, while the other diagonal is formed by a pair of opposite vertices on the shorter side.

Definition of a Parallelogram:

A parallelogram is a quadrilateral in which opposite sides are parallel and congruent. In other words, a parallelogram is a quadrilateral with two pairs of parallel sides.

Properties of a Parallelogram:

- Opposite sides are congruent
- Opposite angles are congruent
- Consecutive angles are supplementary
- Diagonals bisect each other
- The sum of the squares of the lengths of its sides is equal to the sum of the squares of the lengths of its diagonals.

Examples of Parallelograms:

- Rectangle: A parallelogram in which all angles are right angles.
- Rhombus: A parallelogram in which all sides are congruent.
- Square: A parallelogram in which all sides are congruent and all angles are right angles.
- Trapezoid: A quadrilateral with one pair of parallel sides.

Conclusion:

A parallelogram is a special type of quadrilateral in which opposite sides are parallel. It has many properties that make it useful in geometry and other areas of mathematics. Some examples of parallelograms include rectangles, rhombuses, squares, and trapezoids.