problem solving involving trapezoids

Module 11: Geometry

Using the properties of trapezoids to solve problems, learning outcomes.

• Find the area of a trapezoid given height and width of bases
• Use the area of a trapezoid to answer application questions

A trapezoid is four-sided figure, a quadrilateral , with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[/latex], and the length of the bigger base [latex]B[/latex]. The height, [latex]h[/latex], of a trapezoid is the distance between the two bases as shown in the image below.

A trapezoid has a larger base, [latex]B[/latex], and a smaller base, [latex]b[/latex]. The height [latex]h[/latex] is the distance between the bases.

The formula for the area of a trapezoid is:

[latex]{\text{Area}}_{\text{trapezoid}}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex]

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See the image below.

Splitting a trapezoid into two triangles may help you understand the formula for its area.

Properties of Trapezoids

• A trapezoid has four sides.
• Two of its sides are parallel and two sides are not.
• The area, [latex]A[/latex], of a trapezoid is [latex]\text{A}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex] .

Find the area of a trapezoid whose height is [latex]6[/latex] inches and whose bases are [latex]14[/latex] and [latex]11[/latex] inches.

If we draw a rectangle around the trapezoid that has the same big base [latex]B[/latex] and a height [latex]h[/latex], its area should be greater than that of the trapezoid. If we draw a rectangle inside the trapezoid that has the same little base [latex]b[/latex] and a height [latex]h[/latex], its area should be smaller than that of the trapezoid.

Step 7. Answer the question. The area of the trapezoid is [latex]75[/latex] square inches.

In the next video we show another example of how to use the formula to find the area of a trapezoid given the lengths of it’s height and bases.

Find the area of a trapezoid whose height is [latex]5[/latex] feet and whose bases are [latex]10.3[/latex] and [latex]13.7[/latex] feet.

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[/latex] yards and the bases are [latex]8.2[/latex] and [latex]5.6[/latex] yards. How many square yards will be available to plant?

• Question ID 146533, 146534, 146535. Authored by : Lumen Learning. License : CC BY: Attribution
• Ex: Find the Area of a Trapezoid. Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/WNo7s-XoI4w . License : CC BY: Attribution
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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Trapezoid problems.

Trapezoid problems are presented along with their with detailed solutions.

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Here you will learn about a trapezoid, including the properties of a trapezoid, how to identify a trapezoid, and how to classify a trapezoid.

Students will first learn about the trapezoid as part of geometry in 1 st grade, but they will learn about the properties of a trapezoid in 5 th grade.

What is a trapezoid?

A trapezoid is a type of quadrilateral, which is a polygon with four straight sides. A trapezoid has one pair of parallel sides which are called the bases of the trapezoid. The lengths of the bases are not congruent. The other two sides of the trapezoid are called the legs.

The legs may be different lengths, but they are not parallel to each other. The angles that share the same base are called base angles.

Properties of a trapezoid

In order for a polygon to be a trapezoid, it must have the following properties:

Four sides: A trapezoid is a four-sided polygon.

Two parallel sides: A trapezoid has two sides that are parallel to each other. These are called the “bases.”

Two non-parallel sides: The other two sides are not parallel to each other. These are often referred to as the “legs.”

Opposite angles: The angles formed by the longer base and each leg are equal to each other (congruent). The same goes for the angles formed by the shorter base and each leg.

Adjacent angles: Angles that share a side (adjacent angles) add up to 180 degrees.

Diagonals: A trapezoid has two diagonals, which are line segments connecting non-adjacent vertices. These diagonals are not equal in length.

There are three types of trapezoids:

Classification of a trapezoid

As you can see in the quadrilateral hierarchy, a trapezoid classifies as a quadrilateral because it has 4 sides. It does not classify as a parallelogram like a rectangle, rhombus, or square because, unlike those shapes, a trapezoid only has 1 pair of parallel sides instead of 2.

Common Core State Standards

How does this relate to 5 th grade math?

• Grade 5 – Geometry (5.G.B.3) Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
• Grade 5 – Geometry (5.G.B.4) Classify two-dimensional figures in a hierarchy based on properties.

How to identify a trapezoid

In order to identify a trapezoid:

Look for the characteristics of a trapezoid.

State whether or not the shape is a trapezoid.

If the shape is not a trapezoid, explain what characteristics are different.

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Trapezoid examples

Example 1: identify trapezoids.

Is this shape a trapezoid?

For a shape to be a trapezoid, it must have 4 sides and 1 pair of parallel sides.

2 State whether or not the shape is a trapezoid .

Since this shape has 4 sides and 1 pair of parallel sides, it is a trapezoid.

Example 2: identify trapezoids

This shape is not a trapezoid.

Since this shape has 4 sides but no pairs of parallel sides, it is NOT a trapezoid.

Example 3: identify types of trapezoids

Is this shape a trapezoid? If so, what type of trapezoid is it?

This shape is a trapezoid. Since it has one pair of right angles, it is a right trapezoid.

Example 4: identify types of trapezoids

Which trapezoid is a scalene trapezoid?

For a shape to be a trapezoid, it must have 4 sides and 1 pair of parallel sides. For a trapezoid to be a scalene trapezoid, it must also have no congruent side lengths or angles.

The 3 rd trapezoid is a scalene trapezoid.

The first two trapezoids are isosceles trapezoids since they have a pair of opposite sides that are congruent and two pairs of congruent angles. The last trapezoid is a right trapezoid since it has a pair of right angles.

Example 5: identify types of a trapezoid

Which trapezoid is an isosceles trapezoid?

For a shape to be a trapezoid, it must have 4 sides and 1 pair of parallel sides. For a trapezoid to be an isosceles trapezoid, it must also have a pair of opposite sides that are congruent and two pairs of congruent angles.

The 2 nd trapezoid is an isosceles trapezoid.

The first trapezoid is a scalene trapezoid since it has no congruent side lengths or angles. The last trapezoid is a right trapezoid since it has a pair of right angles.

Example 6: trapezoid classification

Marnie says this shape is a trapezoid, a quadrilateral, and a parallelogram. Is she correct?

This shape is a trapezoid.

This shape is a trapezoid. It is also a quadrilateral because it has 4 sides. It is not a parallelogram because a trapezoid only has 1 pair of parallel sides and a parallelogram has 2. Therefore, Marnie is incorrect.

Teaching tips for trapezoids

• Provide real-life examples of trapezoids to help students connect the concept to their surroundings. For instance, point out trapezoidal shapes in buildings, road signs, or furniture.
• Discuss the properties of a trapezoid. Highlight that the non-parallel sides of a trapezoid are the legs, while the parallel sides are the bases of the trapezoid. Emphasize that the bases have to be different lengths, so there is always a longer base.

For example,

• Present students with worksheets that include problem-solving tasks involving trapezoids. For example, ask them to calculate the perimeter of a trapezoid or the area of a trapezoid using given measurements. Encourage them to explain their reasoning and share their solutions with the class.

Easy mistakes to make

• Thinking that a trapezoid has at least one pair of parallel lines instead of exactly one pair Some students may think any shape with at least one pair of parallel lines can be classified as a trapezoid, but this is incorrect. To be a trapezoid, a shape must have exactly one pair of parallel lines.
• Incorrectly identifying parallel sides Identifying the parallel sides of a trapezoid can be challenging for some students. They may mistakenly choose non-parallel sides as the bases of the trapezoid. Reinforce the definition of a trapezoid as a quadrilateral with one pair of parallel sides. Encourage students to look for the sides that are always the same distance apart and never intersect when extended.

Related quadrilateral lessons

• Quadrilateral
• Types of quadrilaterals
• Parallelogram
• Square shape

Practice trapezoid questions

1. Which shape is a trapezoid?

The last shape is the only shape that has 4 sides and exactly 1 pair of parallel sides.

2. Which shape is not a trapezoid?

The first shape is not a trapezoid. It has 4 sides, but it does not have a pair of parallel sides.

3. What are the properties of a trapezoid?

4 sides, 2 pairs of parallel sides

4 sides, 2 pairs of parallel sides, 2  right angles

4 sides, 1 pair of parallel sides

4 sides, 1 pair of parallel sides, 4 right angles

To be a trapezoid, a shape must have 4 sides and 1 pair of parallel sides.

4. A trapezoid is also a …

parallelogram

quadrilateral

Since a trapezoid has 4 sides, it classifies as a quadrilateral.

5. Name the type of trapezoid.

scalene trapezoid

isosceles trapezoid

right trapezoid

equilateral trapezoid

In addition to 4 sides and 1 pair of parallel sides, an isosceles trapezoid must also have a pair of opposite sides that are congruent and two pairs of congruent angles.

6. Which trapezoid is a right trapezoid?

In addition to 4 sides and 1 pair of parallel sides, a right trapezoid must also have a pair of right angles. Trapezoid ABCD is a right trapezoid.

Trapezoid FAQs

Trapezoids are quadrilaterals ( 4 -sided shape) with 1 pair of parallel sides.

A trapezoid has 4 sides and 1 pair of parallel sides.

A trapezoid can be classified as a polygon and a quadrilateral.

To find the area, you can use the area of a trapezoid formula which is A = \cfrac{1}{2} \, (a + b) \, h. Students likely won’t be asked to find the area of the trapezoid until middle school or high school.

The median of a trapezoid is a line segment that connects the midpoints of the two non-parallel sides of a trapezoid.

A trapezoid and a trapezium are one and the same; other English-speaking countries, such as the UK, refer to a trapezoid as a trapezium.

The next lessons are

• Angles in polygons
• Surface area

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Understanding Trapezoids – Properties, Types, and Applications

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Historical Significance

Isosceles trapezoid, right trapezoid, scalene trapezoid, parallel bases, base angles, opposite angles, the sum of base angles.

• Perimeter (P)
• Midsegment Length (m)
• Diagonal Length (d)

Architecture and Construction

Engineering and structural design, finance and economics, mathematics and geometry education, graphics and computer modeling, physics and mechanics, surveying and land measurement, art and design, industrial manufacturing, pattern making and sewing.

Trapezoids are captivating geometric shapes with a unique blend of symmetry and versatility . Defined as a quadrilateral with one pair of parallel sides , trapezoids play a significant role in the world of geometry , offering a wealth of mathematical properties and applications . From architectural structures to artistic designs , trapezoids are omnipresent , showcasing their intriguing attributes and contributing to various fields. 

In this article, we will embark on a journey to unravel the mysteries of trapezoids, exploring their defining characteristics , diverse types, and captivating mathematical concepts . Join us as we delve into the enchanting realm of trapezoids, discovering their hidden wonders and appreciating their significance in geometric shapes.

A  trapezoid  is a  quadrilateral  with one pair of  parallel sides . This unique geometric shape showcases a combination of straight lines and distinct angles , making it a fascinating subject of study in the field of geometry . In a trapezoid, the non-parallel sides are called  legs , while the parallel sides are called  bases . Unlike other quadrilaterals, trapezoids do not possess equal angles or sides , allowing for a wide range of possible configurations. The symmetry and distinct properties of trapezoids contribute to their prominence in mathematics, engineering, and architectural design . Through their versatility and intriguing geometric properties, trapezoids continue to captivate the minds of scholars and enthusiasts alike. Below we present the generic diagram of the trapezoid.

Figure-1: Generic trapezoid.

The  trapezoid  has a rich historical background that dates back to ancient civilizations and the development of mathematical knowledge. The study of geometry and geometric shapes played a fundamental role in the intellectual pursuits of ancient Greek mathematicians, notably Euclid , who laid the foundations of geometric principles in his work “ Elements .”

The concept of the trapezoid was integral to the ancient Greeks’ exploration of quadrilaterals and their properties. Euclid defined the trapezoid as a quadrilateral with one pair of parallel sides . His seminal work provided a systematic approach to understanding and classifying geometric shapes, including the trapezoid, within a broader geometric framework.

Over the centuries, the study of trapezoids continued to evolve, with scholars and mathematicians exploring their properties and relationships with other shapes. In the Islamic Golden Age, scholars like Al-Khwarizmi further expanded geometric knowledge, contributing to the development of algebraic methods to solve problems involving trapezoids and other polygons.

During the Renaissance , the works of renowned mathematicians such as Descartes and Pascal brought new insights into the realm of geometry. Their advancements in coordinate systems and projective geometry led to a deeper understanding of trapezoids and their properties.

Today, the study of trapezoids and their applications extends beyond mathematics, finding relevance in diverse fields such as architecture , engineering , physics , and computer graphics . The historical journey of the trapezoid stands as a testament to humanity’s continuous exploration and fascination with geometric shapes, leaving a lasting impact on mathematical education and practical applications.

Specific characteristics and properties define several main types of trapezoids . Let’s explore them:

An  isosceles trapezoid  is a trapezoid in which the  legs (non-parallel sides) have the same length. This means that both base angles are congruent. The bases of an isosceles trapezoid are still parallel , but the legs are of equal length. Below we present the generic diagram of the isosceles trapezoid.

Figure-2: I sosceles  trapezoid.

A  right trapezoid is a trapezoid that has one right angle. This means that one of the base angles measures 90 degrees , while the opposite base angle is its supplementary angle (measuring 90 degrees as well) . Below we present the generic diagram of the right trapezoid.

Figure-3: Right trapezoid.

A  scalene trapezoid is a trapezoid in which none of the sides or angles are equal. It has no congruent sides or angles, making it the most general type of trapezoid. Below we present the generic diagram of the scalene trapezoid.

Figure-4: Scalene trapezoid.

Although a rectangle is not typically referred to as a trapezoid in general geometry, it can be seen as a special case of a trapezoid. A rectangle can be considered a trapezoid with two pairs of parallel sides and congruent base angles . Below we present the generic diagram of the rectangle trapezoid.

Figure-5: Rectangle trapezoid.

These types of trapezoids exhibit varying properties and relationships between their sides , angles , and bases . Understanding these distinctions can aid in solving geometric problems and analyzing the characteristics of specific trapezoids.

A  trapezoid  is a  geometric shape  with several distinct properties:

A trapezoid is a  quadrilateral  with four sides. It is characterized by having two sides that are  parallel  to each other and two sides that are not parallel.

The bases of a trapezium are its parallel sides . These bases are parallel to each other and are the longest sides of the trapezoid.

The legs of a trapezium are its remaining two sides. They connect the non-parallel sides and are typically shorter in length than the bases.

The angles formed between the bases and the legs are known as the  base angles . Each base angle is adjacent to one of the bases.

The angles formed between the legs of the trapezoid are called the  opposite angles . Each opposite angle is across from another opposite angle.

The sum of the base angles in a trapezoid is always equal to  180 degrees . This property holds true for all trapezoids.

The trapezoid has two diagonals , which are line segments connecting the non-adjacent vertices . These diagonals are not necessarily equal in length.

A line segment connecting the midpoints of the two non-parallel sides is known as the midsegment of a trapezium . It is parallel to the bases and is equal in length to the average of the bases.

The following formula can be used to determine a trapezoid’s area : A = ((base1 + base2) × height) / 2 , where  A  represents the area and  height  refers to the perpendicular distance between the bases.

Trapezoids are generally not symmetric, as their sides and angles can have different lengths and measures. However, certain special trapezoids, such as isosceles trapezoids , possess symmetry properties.

Understanding these properties helps in identifying and analyzing trapezoids in various mathematical problems and real-world situations.

Ralevent Formulas 

Certainly! Here are the related formulas associated with a trapezoid:

The following formula can be used to determine a trapezoid’s area :  A = ((base1 + base2) × height) / 2 , where  base1  and  base2  represent the lengths of the two parallel bases, and  height  refers to the perpendicular distance between the bases. The bases are the longer sides of the trapezoid.

Perimeter  (P)

The perimeter of a trapezoid is calculated by adding the lengths of all four sides. Since the trapezoid has two parallel sides (bases) and two non-parallel sides (legs), the perimeter can be expressed as:  P = base1 + base2 + leg1 + leg2 .

Midsegment Length  (m)

A line segment that joins the midpoints of the two non-parallel sides is known as the midsegment of a trapezium. It is parallel to the bases and equal in length to the average of the two bases. The formula to find the midsegment length is:  m = (base1 + base2) / 2 .

Diagonal Length  (d)

A trapezoid has two diagonals , which are line segments connecting non-adjacent vertices. The formula to find the length of a diagonal depends on the given information. If the diagonals and the angle between them are known, the formula is derived from the Law of Cosines :  d = √(side₁² + side₂² – 2 × side₁ × side₂ × cos(θ)) . If the lengths of the bases and the height are known, the formula for the diagonal length can be derived using the Pythagorean theorem.

These formulas provide a comprehensive understanding of the properties and measurements associated with a trapezoid . They enable the calculation of important quantities such as area , perimeter , midsegment length , and diagonal length , aiding in various mathematical and practical applications involving trapezoids.

Applications 

The  trapezoid , with its unique properties and distinctive shape, finds practical applications across various fields, showcasing its versatility and significance. Let’s delve into some notable applications of trapezoids in different domains.

Trapezoids are frequently employed in architectural designs , particularly in the construction of roofs . The shape of a trapezoid allows for efficient water drainage and provides stability , making it ideal for creating sloping or gabled roofs .

Trapezoids play a vital role in engineering and structural design , especially in the construction of bridges and trusses . The stability and load-bearing capacity of trapezoidal structures make them suitable for supporting heavy loads and distributing forces effectively.

Trapezoidal rule is utilized in financial calculations , such as estimating the area under a curve for numerical integration . It helps approximate the values of definite integrals , enabling the evaluation of financial quantities, such as option pricing or portfolio risk measures .

Trapezoids serve as important subjects of study in mathematics and geometry education . They help students understand fundamental geometric concepts such as parallel lines , angles , and quadrilaterals . Trapezoids also provide a practical context for learning about properties like bases , legs , and diagonals .

Trapezoids are utilized in computer graphics and modeling , particularly for rendering two-dimensional shapes and creating visual effects . They form the basis for various algorithms and techniques used in computer-aided design (CAD) , animation , and video game development .

Trapezoidal configurations are often encountered in mechanical systems , such as pulley arrangements , where trapezoidal belts or chains are employed for power transmission . The geometry of the trapezoid ensures smooth and efficient transfer of rotational motion .

Trapezoids find applications in land surveying and measurement , especially in irregular-shaped plots or fields . By dividing the area into trapezoids, surveyors can calculate the total land area accurately using geometric formulas .

Trapezoids inspire artistic expression and are incorporated into various design elements . In graphic design , trapezoidal shapes can create dynamic and visually appealing compositions, adding a sense of movement and balance to artworks .

Trapezoidal profiles are commonly used in industrial manufacturing processes , such as conveyor belts and timing belts . The shape ensures a secure grip and efficient transfer of motion in machinery and production lines .

Trapezoidal patterns are commonly used in garment and fabric design . They serve as the basis for creating various clothing components, such as sleeves , skirts , and pants . Trapezoidal panels allow for shaping and fitting garments to different body contours .

These are just a few examples highlighting the diverse applications of trapezoids across different fields. Their geometry and unique characteristics make them valuable in practical contexts, demonstrating the enduring relevance of this geometric shape . Understanding the applications of trapezoids in these diverse fields highlights their significance and utility. From architecture and mathematics education to computer graphics and manufacturing , the properties of trapezoids find practical implementation across a range of disciplines, contributing to innovation and problem-solving .

In a trapezoid, the lengths of the bases are  10 centimeters  and  6 centimeters , and the height is  8 centimeters . What is the area of the trapezoid?

Given: Base1 =  10 centimeters , Base2 =  6 centimeters , Height =  8 centimeters .

The area of a trapezoid is calculated using the formula

A = ((base1 + base2) × height) / 2 .

Substituting the given values, we have:

A = ((10 centimeters + 6 centimeters) × 8 centimeters) / 2

A = 56 square centimeters.

Therefore, the area of the trapezoid is 56 square centimeters.

 In a trapezoid, the lengths of the bases are  12 meters  and  8 meters , and the length of one leg is  5 meters . Find the length of the other leg .

Given: Base1 =  12 meters , Base2 =  8 meters , Leg1 =  5 meters .

To find the length of the other leg, we can use the formula for the midsegment of a trapezoid, which is:

m = (base1 + base2) / 2

m = (12 meters + 8 meters) / 2

m = 10 meters

Therefore, the length of the other leg is 10 meters.

In a trapezoid, the lengths of the bases are  16 inches  and  10 inches , and the length of one leg is  6 inches . Find the length of the other leg .

Given: Base1 =  16 inches , Base2 =  10 inches , Leg1 =  6 inches .

m = (base1 + base2) / 2 .

m = (16 inches + 10 inches) / 2

m = 13 inches

Therefore, the length of the other leg is 13 inches.

In a trapezoid given in Figure-6, f ind the length of the other diagonal .

Given: Base1 =  8 centimeters , Base2 =  12 centimeters , Diagonal1 =  10 centimeters .

To find the length of the other diagonal, we can use the formula derived from the Law of Cosines:

d = √(side₁² + side₂² – 2 × side₁ × side₂ × cos(θ))

In this case, the lengths of the bases are known, so the formula becomes:

d = √((8²) + (12²) – 2 × 8 × 12 × cos(θ))

To find the length of the other diagonal, the angle value is required.

In a trapezoid, the lengths of the bases are  15 inches  and  9 inches , and the length of one diagonal is  12 inches . Find the length of the other diagonal.

Given: Base1 =  15 inches , Base2 =  9 inches , Diagonal1 =  12 inches .

d = √((15 inches)² + (9 inches)² – 2 × 15 inches × 9 inches × cos(θ))

In a trapezoid, the lengths of the bases are  14 centimeters  and  10 centimeters , and the length of one diagonal is  8 centimeters . Find the measure of the included angle .

Given: Base1 =  14 centimeters , Base2 =  10 centimeters , Diagonal1 =  8 centimeters . 

To find the measure of the included angle, we can use the formula derived from the Law of Cosines:

angle = acos((diagonal₁² + diagonal₂² – side₁² – side₂²) / (2 × diagonal₁ × diagonal₂))

In this case, we only have the length of one diagonal, so we need additional information or the value of the second diagonal to find the measure of the included angle.

In a trapezoid, the lengths of the bases are  18 meters  and  12 meters , and the measure of one base angle is  60 degrees . Find the measure of the other base angle .

Given: Base1 =  18 meters , Base2 =  12 meters , Angle1 =  60 degrees .

Since a trapezoid has two pairs of opposite angles, the other base angle can be measured by subtracting the given angle from 180 degrees. Therefore, the measure of the other base angle is 180 degrees – 60 degrees =  120 degrees .

In a trapezoid, the lengths of the bases are  7 inches  and  9 inches , and the measure of one base angle is  45 degrees . Find the measure of the other base angle .

Given: Base1 =  7 inches , Base2 =  9 inches , Angle1 =  45 degrees .

Since a trapezoid has two pairs of opposite angles, the measure of the other base angle is also 45 degrees. Therefore, the measure of the other base angle is  45 degrees .

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PROBLEMS INVOLVING PARALLELOGRAMS, TRAPEZOIDS AND KITES

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• Q 4 A ____________is a quadrilateral with exactly one pair of parallel sides. diagonals Trapezoids isosceles trapezoids congruent 30 s
• Q 5 An ____________ is a trapezoid with congruent non-parallel sides. congruent  bases  trapezoids isosceles trapezoids 30 s
• Q 6 The ____________ of a trapezoid is equal to half the sum of the lengths of the two bases. bases complementary supplementary area 30 s
• Q 7 In a parallelogram, any two opposite sides are congruent. true false True or False 30 s
• Q 8 In a parallelogram, any two consecutive angles are congruent. false true True or False 30 s
• Q 9 In a parallelogram, any two opposite angles are supplementary. false true True or False 30 s
• Q 10 The diagonals of a parallelogram bisect each other. true false True or False 30 s

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