trapezoids and parallelograms common core geometry homework answers

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Common Core: High School - Geometry : Rotations and Reflections of Rectangles, Parallelograms, Trapezoids, and Regular Polygons: CCSS.Math.Content.HSG-CO.A.3

Study concepts, example questions & explanations for common core: high school - geometry, all common core: high school - geometry resources, example questions, example question #1 : rotations and reflections of rectangles, parallelograms, trapezoids, and regular polygons: ccss.math.content.hsg co.a.3.

Determine whether the statement is true or false:

The following image can be divided multiple ways to result in a reflected image.

Looking at the statement and the image given, it is seen that the object is a trapezoid.

A trapezoid has two bases of differing lengths and two side pieces of equal length in this particular case.

For an object to be divided into images that can be reflected onto one another, the images must be identically mirrored.

To identify the possible solutions, draw the lines of symmetry.

Only one line of symmetry could be drawn to allow for two images of the trapezoid to be reflected over that line. The trapezoid cannot be divided in any other way that will result in a reflected image.

Therefore, the original statement is false.

Example Question #24 : Congruence

Given a hexagon, how many lines of symmetry exist?

Recall that a hexagon is a six sided figure. Now recall that a line of symmetry creates two mirrored, identical figures which is also known as a reflection.

Knowing these two pieces of information draw the image of the hexagon and the lines of symmetry

Counting up these lines results 6 lines of symmetry.

Therefore, the answer is 6.

Example Question #25 : Congruence

A cylinder is composed of two circular bases. For one base to be carried onto the other, what geometric transformation must occur?

Rotation and Translation

All of the answers are correct.

Translation

Since a cylinder is composed of two, identical circular bases that are separated by the height of the cylinder, the transformation that must occur to have one carried onto the other is a translation. Recall that a translation is the sliding of an object without changing its size or shape. Therefore, for the one base to be carried onto the other, translation is the geometric transformation that must occur.

Example Question #4 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

If the rectangle is reflected across the x-axis, what is the resulting image?

None of the other answers.

In order to create the resulting image of a reflection first recall what a reflection is.

Reflection: To flip the orientation of an object over a specific line or function.

In this specific situation the line of reflection is the x-axis.

The original image is,

This is the correct answer.

Example Question #26 : Congruence

Given a trapezoidal based prism how can one base be carried onto the other?

None of the answers.

Since a trapezoidal based prism is composed of two, identical trapezoidal bases that are separated by the height of the prism, the transformation that must occur to have one base carried onto the other is a translation. Recall that a translation is the sliding of an object without changing its size or shape. Therefore, for the one base to be carried onto the other, translation is the geometric transformation that must occur.

Reflections and rotations never result in the same image.

To determine whether the statement is true or false, identify any example that would make the statement false.

"Reflections and rotations never result in the same image."

Imagine a square that exists in quadrant two. When this image is reflected across the y-axis it is still a square that is now in quadrant one. When the square from quadrant two is rotated around the origin, it results in the same image in quadrant one. Therefore, it is possible for a reflection and rotation to result in the same image.

Thus, the statement "Reflections and rotations never result in the same image." is false.

Example Question #7 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Looking at the original image and making one 90 degree rotation around the origin results in the following.

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Example Question #8 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

Looking at the original image and making one rotation around the origin results in the following.

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. The visual representation for this rotation is as follows.

Example Question #9 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

Example Question #10 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

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Geometry 6-6 Complete Lesson: Trapezoids and Kites

• Quadrilateral
• Perpendicular diagonals
• Congruent opposite sides
• Supplementary consecutive angles

Chapter 6, Lesson 6: Trapezoids and Kites

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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.

[FREE] Quadrilateral Check for Understanding Quiz (Grade 2 to 5)

Use this quiz to check your grade 2 to 5 students’ understanding of Quadrilaterals. 15+ questions with answers covering a range of 2nd, 3rd, 4th, and 5th grade quadrilateral topics to identify areas of strength and support!

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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• → 6th Grade
• → Area, Perimeter & Circumference

Area of Parallelograms, Trapezoids, and Rectangles Lesson Plan

Get the lesson materials.

Area of Parallelograms, Trapezoids, Rectangles Guided Notes Doodles | Worksheet

Ever wondered how to teach the area of parallelograms, trapezoids, and rectangles in an engaging way to your middle school students?

In this lesson plan, students will learn about finding the area of these polygons and their real-life applications. Through artistic, interactive guided notes, check for understanding questions, a practice coloring worksheet, and a maze activity, students will gain a comprehensive understanding of finding the area of parallelograms, trapezoids, and rectangles.

• Standards : CCSS 6.G.A.1 , CCSS 7.G.B.6
• Topic : Area, Perimeter & Circumference
• Grades : 6th Grade , 7th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Calculate the area of parallelograms, trapezoids, and rectangles using the appropriate formulas

Explain the real-life applications of area involving parallelograms, trapezoids, and rectangles

Prerequisites

Before this lesson, students should be familiar with:

How to differentiate between parallelograms, rectangles, and trapezoids

Knowledge of basic multiplication and addition skills for whole numbers

Understanding of basic algebraic concepts, such as substituting values into formulas

Colored pencils or markers

Guided notes

Key Vocabulary

Parallelogram

Introduction

As a hook, ask students why it is important to calculate the area of different polygons, specifically quadrilaterals, in real-life situations. Refer to the last page of the guided notes as well as the FAQs below for ideas.

Use the first page of the guided notes to introduce the formulas for finding the area of rectangles, parallelograms, and trapezoids. Walk through the key points of finding the length and width of a rectangle and using the formula A = l * w to calculate its area. Then, walk through the key points of identifying the base and height of a parallelogram and using the formula A = b * h to find its area. Then, walk through the key points of identifying the bases, height, and average of the bases of a trapezoid, and using the formula A = (b1 + b2) * h / 2 to calculate its area. For each polygon, students will take notes on the formula and also practice an example for a rectangle, parallelogram, and trapezoid.

Based on student responses, reteach concepts that students need extra help with. If your class has a wide range of proficiency levels, you can pull out students for reteaching, and have more advanced students begin work on the practice exercises.

Have students practice finding the area of parallelograms, trapezoids, and rectangles using the practice sheet on page 2. This activity allows students to apply the concepts they learned in the guided notes and practice calculating the area of different polygons.

Walk around the classroom to answer any student questions and provide individual support as needed. Fast finishers can dive into the color by number and maze activity for extra practice. This maze requires students to calculate the area of different shapes and find the correct path to reach the end. You can assign it as homework for the remainder of the class to reinforce their understanding of finding the area of parallelograms, trapezoids, and rectangles.

Real-Life Application

Use the last page of the guided notes "real life applications" to bring the class back together, and introduce the concept of real-life applications of calculating the area of different polygons, specifically quadrilaterals. Explain to students that understanding how to find the area of shapes can be useful in many real-life situations.

Ask students if they can think of any real-life situations where knowing how to calculate area would be important. Encourage students to share their ideas and write them on the board. Some examples may include:

Carpeting a room: Knowing the area of a room can help determine how much carpet is needed.

Painting a wall: Calculating the area of a wall can help determine how much paint is needed.

Gardening: Finding the area of a garden bed can help determine how much soil or mulch is needed.

Building a fence: Understanding the area can help determine how much material is needed to build a fence.

Refer to the Frequently Asked Questions (FAQ) section for additional ideas on how to teach real-life applications of finding the area of polygons.

Additional Self-Checking Digital Practice

If you’re looking for digital practice for area of parallelograms, trapezoids, and rectangles, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

Here is an activity to explore:

Area of Rectangles, Parallelograms, Trapezoids and Composite Figures Pixel Art

Additional Print Practice

A fun, no-prep way to practice area of parallelograms, trapezoids, and rectangles is Doodle Math — they’re a fresh take on color by number or color by code. It includes multiple levels of practice, perfect for a review day or sub plan.

Here is one activity to try:

Area of Rectangles, Parallelograms, Trapezoids, Triangles, and Composite Figures

What is the formula for finding the area of a parallelogram? Open

To find the area of a parallelogram, you can use the formula: Area = base × height . The base is the length of one side of the parallelogram, and the height is the perpendicular distance between the base and the opposite side.

How do I calculate the area of a trapezoid? Open

To calculate the area of a trapezoid, you can use the formula: Area = ½ × (base1 + base2) × height . The bases are the parallel sides of the trapezoid, and the height is the perpendicular distance between the bases.

What is the difference between a parallelogram and a rectangle? Open

A parallelogram is a quadrilateral with opposite sides that are parallel, while a rectangle is a type of parallelogram that has four right angles. In other words, all rectangles are parallelograms, but not all parallelograms are rectangles.

How can I find the area of a rectangle? Open

To find the area of a rectangle, you can use the formula: Area = length × width . The length is the longer side of the rectangle, and the width is the shorter side.

Are there any real-life applications for finding the area of quadrilaterals? Open

Yes, there are many real-life applications for finding the area of quadrilaterals. Here are a few examples:

Calculating the amount of paint needed to cover the walls of a room.

Determining the amount of fabric needed to make a rectangular tablecloth.

Estimating the size of a field for planting crops.

How can I use guided notes to teach the area of parallelograms, trapezoids, and rectangles? Open

Guided notes can be an effective teaching tool for the area of parallelograms, trapezoids, and rectangles. Here are some tips on how to use them:

Start by providing a brief overview of the topic and the formulas.

Fill in the guided notes together as a class, explaining any key concepts or steps.

Encourage students to actively participate by asking and answering questions.

Use examples and visual aids to reinforce understanding.

Provide opportunities for students to practice applying the formulas with guided practice worksheets.

What other resources can I use to teach the area of quadrilaterals? Open

In addition to guided notes, there are many other resources you can use to teach the area of quadrilaterals. Here are a few examples:

Interactive online activities or games that allow students to practice calculating the area.

Real-life examples or scenarios that require students to find the area of quadrilaterals.

Manipulatives or physical models that allow students to visualize and explore the concept of area.

How can I assess students' understanding of the area of parallelograms, trapezoids, and rectangles? Open

There are several ways to assess students' understanding of the area of parallelograms, trapezoids, and rectangles. Here are some suggestions:

Give students quizzes or tests that require them to calculate the area of various quadrilaterals.

Assign homework problems that involve finding the area of quadrilaterals.

Have students complete real-life application tasks or projects that require them to apply their knowledge of area.

Use formative assessments, such as exit tickets or quick checks for understanding, to gauge students' comprehension throughout the lesson.

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Exercise 23 Page   394

If a quadrilateral is a kite , then its diagonals are perpendicular .

Check the answer

m ∠ 1 = 90, m ∠ 2 = 90, m ∠ 3 = 90, m ∠ 4 = 90, m ∠ 5 = 46, m ∠ 6 = 34, m ∠ 7 = 56, m ∠ 8 = 44, m ∠ 9 = 56, m ∠ 10 = 44

Practice exercises

Asses your skill.

Let's find the measures of the numbered angles one at a time.

Measure of ∠ 1, ∠ 2, ∠ 3 and ∠ 4

Before we begin, let's name the vertices of our kite . We will call it quadrilateral ABCD.

If a quadrilateral is a kite, then its diagonals are perpendicular . Therefore m ∠ 1 = 90, m ∠ 2 = 90, m ∠ 3 = 90 and m ∠ 4 = 90.

Measure of ∠ 5 and ∠ 6

m ∠ 5 = 46 m ∠ 6 = 34

Measure of ∠ 7

m ∠ 3= 90, m ∠ 6= 34

LHS-124=RHS-124

Measure of ∠ 8

m ∠ 4= 90, m ∠ 5= 46

LHS-136=RHS-136

Measure of ∠ 10

Measure of ∠ 9.