geometry unit 4 lesson 7 homework

Unit 4 Similarity and Right Triangle Trigonometry

Learning focus.

Describe the essential features of a dilation transformation.

Lesson Summary

In this lesson, we observed the key features of a dilation transformation while figuring out how a photocopy machine enlarges an image. We learned how to locate points on a dilated image by using the center and scale factor that define a specific dilation. We observed that “the same shape, different size” relationship between the pre-image and image figures are a consequence of the way dilations are defined.

Create similar figures by dilation given the scale factor.

Prove a theorem about the midlines of a triangle using dilations.

In this lesson, we extended our understanding of similar figures. Since corresponding segments of similar figures are proportional, and dilations produce similar figures, corresponding parts of an image and its pre-image after a dilation are proportional. We also learned that corresponding line segments in a dilation are parallel. These two observations provided a tool for proving a theorem about the midlines of a triangle, a segment connecting the midpoints of two sides of a triangle.

Determine criteria for triangle similarity.

In this lesson, we examined what it means to say that two figures are similar geometrically, and we examined conditions under which two triangles will be similar. We wrote and justified several theorems for triangle similarity criteria.

Prove that a line drawn parallel to one side of a triangle that intersects the other two sides divides the other two sides proportionally.

In a previous lesson, we learned that a midline of a triangle, a line that passes through the midpoints of two of the sides, is parallel to the third side and half its length. In this lesson, we extended this theorem to include other segments that cut the sides of a triangle proportionally. We also proved a non-intuitive “side-splitting” theorem about the multiple segments formed when multiple lines parallel to a side of a triangle cut the other two sides of the triangle.

Practice using geometric reasoning in computational work.

In this lesson, we drew upon a variety of theorems to support the computational work of finding missing sides and angles. To identify which theorems to use, we had to examine the available features of the diagram. For many measurements, multiple strategies could be used. We also used the diagram, along with our computed measurements, to develop and justify a conjecture for the sum of the interior angles of any polygon, similar to the theorem we proved previously about the sum of the interior angles in a triangle.

Locate the midpoint of a segment and a point that divides the segment in a given ratio.

In this lesson, we examined strategies for dividing a line segment into two parts that fit a given ratio. One common application of this concept is to find the coordinates of the midpoint of a segment, given the coordinates of the endpoints.

Prove the Pythagorean theorem algebraically.

In today’s lesson, we learned that drawing the altitude of a right triangle from the vertex at the right angle to the hypotenuse divides the right triangle into two smaller triangles that are similar to each other and to the original right triangle. We were able to prove the Pythagorean theorem using proportionality statements about the three similar triangles.

Investigate corresponding ratios of right triangles with the same acute angle.

In this lesson, we learned about some special ratios, called trigonometric ratios, that occur in right triangles. If two right triangles have a pair of corresponding acute angles that are congruent, the right triangles will be similar. Therefore, corresponding ratios of the sides of these two right triangles will be equal. This observation is so useful when working with right triangles that have the same acute angle that values of these ratios were recorded in tables for each acute angle between 0 ° and 90 ° .

Examine properties of trigonometric expressions.

In this lesson, we examined some relationships between trigonometric ratios, such as a relationship between the sine and cosine of complementary angles. We were able to use the properties of a right triangle, including the Pythagorean theorem that describes a relationship between the lengths of the sides, to justify the observations we made today.

Solve for the missing side and angle measures in a right triangle.

In this lesson, we extended our strategies for finding unknown sides and angles in a right triangle beyond using the Pythagorean theorem and the angle sum theorem for triangles, since sometimes we don’t have enough information in terms of side lengths or angle measures to use these theorems. We found that trigonometric ratios are useful in solving for unknown sides and that inverse trigonometric relationships are useful for finding unknown angles in a right triangle. Adding these tools allows us to find all of the missing sides and angles in a right triangle given two pieces of information: two sides of the triangle or one side and an angle.

Solve application problems using trigonometry.

In this lesson, we learned about the modeling process and how to use right triangle trigonometry to model many different types of applications, even applications that didn’t naturally include right triangles. A right triangle became a tool for representing a situation so we could draw upon trigonometric ratios and inverse trigonometric relationships to answer important problems in construction, aviation, transportation, and other contexts.

Common Core Geometry Math (Worksheets, Homework, Lesson Plans)

Related Topics: Common Core Math Resources, Lesson Plans & Worksheets for all grades Common Core Math Video Lessons, Math Worksheets and Games for Geometry Common Core Math Video Lessons, Math Worksheets and Games for all grades

Looking for video lessons that will help you in your Common Core Geometry math classwork or homework? Looking for Common Core Math Worksheets and Lesson Plans that will help you prepare lessons for Geometry students?

The following lesson plans and worksheets are from the New York State Education Department Common Core-aligned educational resources. Eureka/EngageNY Math Geometry Worksheets.

These Lesson Plans and Worksheets are divided into five modules.

Geometry Homework, Lesson Plans and Worksheets

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Geometry (all content)

Unit 1: lines, unit 2: angles, unit 3: shapes, unit 4: triangles, unit 5: quadrilaterals, unit 6: coordinate plane, unit 7: area and perimeter, unit 8: volume and surface area, unit 9: pythagorean theorem, unit 10: transformations, unit 11: congruence, unit 12: similarity, unit 13: trigonometry, unit 14: circles, unit 15: analytic geometry, unit 16: geometric constructions, unit 17: miscellaneous.

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Eureka Math Grade 4 Module 4 Lesson 7 Answer Key

Engage ny eureka math 4th grade module 4 lesson 7 answer key, eureka math grade 4 module 4 lesson 7 practice sheet answer key.

Explanation: Here, the angle <WYX would be an acute angle. As In the above image, we can see that the angle is less than 90°. So the angle would be 60° or above but less 90° and the angle is called an acute angle.

Explanation: Here, the angle <CAB would be an acute angle. As In the above image, we can see that the angle is less than 90°. So the angle would be 60° or above but less 90° and the angle is called an acute angle.

Explanation: By measuring the angle DEF with a protractor we will get the angle as 120° and it is an obtuse angle as the angle is greater than 90°.

Explanation: By measuring the angle QRS with a protractor we will get the angle as 120° and it is an obtuse angle as the angle is greater than 90°.

Eureka Math Grade 4 Module 4 Lesson 7 Problem Set Answer Key

Construct angles that measure the given number of degrees. For Problems 1–4, use the ray shown as one of the rays of the angle with its endpoint as the vertex of the angle. Draw an arc to indicate the angle that was measured.

Explanation: Here, we have constructed a 65° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 65° angle.

Explanation: Here, we have constructed a 135° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 135° angle.

Question 5. 5° Answer: The 5° angle is an acute angle.

Explanation: Here, we have constructed a 5° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 5° angle.

Question 6. 175° Answer: The 175° angle is an obtuse angle.

Explanation: Here, we have constructed a 175° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 175° angle.

Question 7. 27° Answer: The 5° angle is an acute angle.

Question 8. 117° Answer: The 117° angle is an obtuse angle.

Explanation: Here, we have constructed a 117° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 117° angle.

Question 9. 48° Answer: The 48° angle is an acute angle.

Explanation: Here, we have constructed a 48° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 48° angle.

Question 10. 132° Answer: The 132° angle is an obtuse angle.

Eureka Math Grade 4 Module 4 Lesson 7 Exit Ticket Answer Key

Construct angles that measure the given number of degrees. Draw an arc to indicate the angle that was measured.

Question 1. 75° Answer: The 75° angle is an acute angle.

Explanation: Here, we have constructed a 75° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 75° angle.

Question 2. 105° Answer: The 105° angle is an obtuse angle.

Explanation: Here, we have constructed a 105° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 105° angle.

Question 3. 81° Answer: The 81° angle is an acute angle.

Explanation: Here, we have constructed a 81° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 81° angle.

Question 4. 99° Answer: The 99° angle is an obtuse angle.

Explanation: Here, we have constructed a 99° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 99° angle.

Eureka Math Grade 4 Module 4 Lesson 7 Homework Answer Key

Explanation: Here, we have constructed a 85° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 85° angle.

Explanation: Here, we have constructed a 140° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 140° angle.

Explanation: Here, we have constructed a 83° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 83° angle.

Question 5. 108° Answer: The 108° angle is an obtuse angle.

Explanation: Here, we have constructed a 108° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 108° angle.

Question 6. 72° Answer: The 72° angle is an acute angle.

Explanation: Here, we have constructed a 72° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 72° angle.

Question 7. 25° Answer: The 25° angle is an acute angle.

Explanation: Here, we have constructed a 25° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 25° angle.

Question 8. 155° Answer: The 155° angle is an obtuse angle.

Explanation: Here, we have constructed a 155° angle using a protractor and it is an obtuse angle as the angle is greater than 90°. The below image represents the 155° angle.

Question 9. 45° Answer: The 45° angle is an acute angle.

Explanation: Here, we have constructed a 45° angle using a protractor and it is an acute angle as the angle is less than 90°. The below image represents the 45° angle.

Question 10. 135° Answer: The 135° angle is an obtuse angle.

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About This Course

Welcome to the Math Medic Geometry course! Here you will find a ready-to-be-taught lesson for every day of the school year, along with expert tips and questioning techniques to help the lesson be successful. Each lesson is designed to be taught in an Experience First, Formalize Later (EFFL) approach, in which students work in small groups on an engaging activity before the teacher formalizes the learning.

Our Geometry course develops reasoning, justification, and proof skills through an in-depth study of shapes and their properties, rigid transformations and congruence, and the relationship between similarity and right triangle trigonometry. Rich opportunities for problem solving culminate in the unit on surface area and volume. This course was created using the Common Core State Standards as a guide. The standards taught in each Math Medic Geometry lesson can be found here . Additionally, we've chosen to include a unit on Statistics and Probability that can be used as a stand-alone unit at any time during high school course work. The unit overviews and learning targets for the Math Medic Geometry course can be found here .

Math Medic Help

7.4 Proportional Relationships and Percentages

In this unit, students use ratios, scale factors, unit rates (also called constants of proportionality), and proportional relationships to solve multi-step, real-world problems that involve fractions and percentages. They use long division to write fractions presented in the form \(\frac a b\) as decimals, e.g., \(\frac{11}{30} = 0.3\overline6\) . They learn to understand and use the terms “repeating decimal,” “terminating decimal,” “percent increase,” “percent decrease,” “percent error,” and “measurement error.” They represent amounts and corresponding percent rates with double number line diagrams and tables. They use these terms and representations in reasoning about situations involving sales taxes, tips, markdowns, markups, sales commissions, interest, depreciation, and scaling a picture. Students use equations to represent proportional relationships in which the constant of proportionality arises from a percentage, e.g., relationship between price paid and amount of sales tax paid.

Proportional Relationships with Fractions

• 1 Lots of Flags
• 2 Ratios and Rates With Fractions
• 3 Revisiting Proportional Relationships
• 4 Half as Much Again
• 5 Say It with Decimals

Percent Increase and Decrease

• 6 Increasing and Decreasing
• 7 One Hundred Percent
• 8 Percent Increase and Decrease with Equations
• 9 More and Less than 1%

Applying Percentages

• 10 Tax and Tip
• 11 Percentage Contexts
• 12 Finding the Percentage
• 13 Measurement Error
• 14 Percent Error
• 15 Error Intervals

Let's Put it to Work

• 16 Posing Percentage Problems