Lesson 15 Measuring And Constructing Angles
Monitoring Progress
Write three names for the angle.
Question 1.An Angle is formed when two rays have the same endpoint or vertex.So,Here, the angle is QHence,The three names for the given angle are: Q, PQR, and RQP
Question 2.An Angle is formed when two rays have the same endpoint or vertex.So,Here, the angle is YHence,The three names for the given angle are: 1, XYZ, and ZYX
Question 3.An Angle is formed when two rays have the same endpoint or vertex.So,Here, the angle is EHence,The three names for the given angle are: 2, FED, and DEF
Use the diagram in Example 2 to find the angle measure. Then classify the angle.
Question 4.
EFH and HFG are 60° and 30° respectively
Question 10.Angle MNP is a straight angle, and \Q bisects MNP. Draw MNP and \Q. Use arcs to mark the congruent angles in your diagram. Find the angle measures of these congruent angles.Answer:It is given that angle MNP is a straight angle, and \Q bisects MNP. Draw MNP and \Q.So,The representation of the above statement is:Hence, from the above,We can conclude that the angle measure of each congruent angle is: 55.3°
Lesson 27 Perimeters And Areas Of Similar Figures
EXPLORATION 1
Work with a partner. Draw a rectangle in the coordinate plane.a. Dilate your rectangle using each indicated scale factor k. Then complete the table for the perimeter P of each rectangle. Describe the pattern.b. Compare the ratios of the perimeters to the ratios of the corresponding side lengths. What do you notice?c. Repeat part to complete the table for the area A of each rectangle. Describe the pattern.d. Compare the ratios of the areas to the ratios of the corresponding side lengths. What do you notice?e. The rectangles shown are similar. You know the perimeter and the area of the red rectangle and a pair of corresponding side lengths. How can you nd the perimeter of the blue rectangle? the area of the blue rectangle?
2.7 Lesson
Try It
Question 1.The height of Figure A is 9 feet. The height of a similar Figure B is 15 feet. What is the value of the ratio of the perimeter of A to the perimeter of B?
Answer: The ratio of the perimeter of A to B is 3/5
Explanation:We know that when two figures are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.Perimeter of figure A/Perimeter of figure B = Height of figure A/Height of figure BPerimeter of figure A/Perimeter of figure B = 9/15 = 3/5Thus the ratio of the perimeter of A to B is 3/5
Try It
Self-Assessment for Concepts & SkillsSolve each exercise. Then rate your understanding of the success criteria in your journal.
Question 3.
Reflections Homework & Practice 22
Review & Refresh
The vertices of a quadrilateral are P, Q, R, and S. Draw the figure and its image after the translation.
Question 1.7 units down
Answer:We know that to translate a figure a units horizontally and b units vertically in coordinate plane, a is added to x-coordinate and b is added to y-coordinates of the vertices.A = A’the value a and b will be positive if shift is right and vertical up and the value of a and b will be negative if shift is left and vertical down.Given,S and a = 0, b = -7P’ = P’ = P’Q’ = Q’ = Q’R’ = R’ = R’S’ = S’ = S’Thus the coordinate of the image is P’, Q’, R’, and S’
Question 2.3 units left and 2 units up
Answer:We know that to translate a figure a units horizontally and b units vertically in coordinate plane, a is added to x-coordinate and b is added to y-coordinates of the vertices.A = A’the value a and b will be positive if shift is right and vertical up and the value of a and b will be negative if shift is left and vertical down.Given,S and a = -3, b = 2P’ = P’ = P’Q’ = Q’ = Q’R’ = R’ = R’S’ = S’ = S’Thus the coordinate of the image are P’, Q’, R’ and S’
Question 3.
Question 4.
Thus the correct answer is option B.
Concepts, Skills, & Problem SolvingDESCRIBING RELATIONSHIPSDescribe the relationship between the given point and the point A in terms of reections.
IDENTIFYING A REFLECTIONTell whether the blue gure is a reection of the red gure.
Question 12.
Answer: No
Question 13.
Answer: Yes
Question 14.
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Describing Pairs Of Angles
Essential Question
How can you describe angle pair relationships and use these descriptions to find angle measures?Answer:Two adjacent angles are a linear pair when their non-common sides are opposite rays. The angles in a linear pair are supplementary angles. common side L1+22=180°. Two angles are vertical angles when their sides form two pairs of opposite rays
Exploration 1
Work with a partner: The five-pointed star has a regular pentagon at its center.
a. What do you notice about the following angle pairs?x° and y°
We can observe that c and e are the opposite anglesSo,We can say that,c° = e°
b. Find the values of the indicated variables. Do not use a protractor to measure the angles.Explain how you obtained each answer.Answer:The given figure is: SquareSo,All the angles in the square are: 90°Hence, from the above,c° = 90°, d° = 90°, and e° = 90°
Communicate Your Answer
How can you describe angle pair relationships and use these descriptions to find angle measures?Answer:Two adjacent angles are a linear pair when their non-common sides are opposite rays. The angles in a linear pair are supplementary angles. common side L1+22=180°. Two angles are vertical angles when their sides form two pairs of opposite rays
*** Thanksgiving Recipes ***
These roasted brussels sprouts get a fair amount of spice from the crushed red pepper flakes, which cuts…
We love how the cinnamon-scented streusel topping lets the juicy berries peek through.
Poach quince in rosé with a dash of cocktail bitters and a few warm spices, then assemble into a tart…
Alton Brown’s turkey brine recipe from Good Eats will give you a flavorful Thanksgiving turkey with juicy…
Bone-in turkey breasts are easy to find, and as impressive as a whole bird when you roast them in butter…
Leave the sausage, nuts, dried fruit behind in favor of this easy, vegetarian-friendly stuffing recipe…
After trying every turkey-roasting method under the sun, I’ve finally settled on this as absolutely the…
Italian sausage and a heap of Parmesan cheese lend signature flair to the easy Thanksgiving dressing…
It’s all about the layers and ruffles in this dramatic seasonal pie.
This recipe was handed down by my great-grandmother. It is a family favorite that we make every Thanksgiving….
Cutting leeks into large pieces gives them a presence equal…
This crunchy crumble is the perfect topping for an ice cream sundae, easy apple crumble, or our Build-Your-Own…
I adapted this from a 50-year-old peach crisp recipe. Although it works well with peaches, it works even better with fresh, crisp cooking apples.
A simple dessert that’s great served with ice cream.
It takes a little work, but it is worth it.
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Exercise 16 Describing Pairs Of Angles
Vocabulary and Core Concept Check
Question 1.Explain what is different between adjacent angles and vertical angles.Answer:
WHICH ONE DID DOESNT BELONG?Which one does hot belong with the other three? Explain your reasoning.Answer:The angle that is the same in more than 1 angle-pair and has a common side is called Adjacent AngleSo,From the above figure,We can observe that only the first figure has an adjacent angle whereas all the three figures dont have any adjacent anglesHence, from the above,We can conclude that the first figure does not belong with the other three
Monitoring Progress and Modeling with Mathematics
In Exercises 3-6, use the figure.
Question 3.Name a pair of adjacent complementary angles.Answer:
Name a pair of adjacent supplementary angles.Answer:The angles that have the sum of the angle measures 180° are called Supplementary anglesHence, from the figure,A pair of adjacent supplementary angles are: LJN + LJK
Question 5.Name a pair of nonadjacent complementary angles.Answer:
Name a pair of nonadjacent supplementary angles.Answer:The angles that have the sum of the angle measures 180° are called Supplementary anglesHence, from the figure,A pair of adjacent supplementary angles are: LJN + LJK and NGP + HGF
In Exercises 7 10. find the angle measure.
Question 7.1 is a complement of 2, and m1 = 23°. Find, m2.Answer:
CAB = 58° and CAD = 32°
Question 13.UVW and XYZ arc complementary angles, mUVW = °. and mXYZ = °.Answer:
c. CAD EAF.
Proofs With Parallel Lines
Exploration 1
Exploring Converses
Work with a partner: Write the converse of each conditional statement. Draw a diagram to represent the converse. Determine whether the converse is true. Justify your conclusion.CONSTRUCTING VIABLE ARGUMENTSTo be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.
a. Corresponding Angles Theorem : If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.Converse:If the pairs of corresponding angles are
congruent, then the two parallel lines are
cut by a transversal.
b. Alternate Interior Angles Theorem : If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.Converse:If the pairs of alternate interior angles are
congruent, then the two parallel lines are
cut by a transversal.
Answer:The converse of the Alternate Interior angles Theorem:The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallelSo,When we compare the actual converse and the converse according to the given statement,we can conclude that the converse we obtained from the given statement is false
are congruent, then the two parallel
lines are cut by a transversal.
are supplementary, then the two parallel lines
are cut by a transversal
Communicate Your Answer
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Big Ideas Math Book Geometry Answer Key Chapter 2 Reasoning And Proofs
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Question 13.ABSTRACT REASONINGCan you use the equation for an arithmetic sequence to write an equation for the sequence 3, 9, 27, 81. . . . ? Explain our reasoning.Answer:
Parallel And Perpendicular Lines Cumulative Assessment
Question 1.Use the steps in the construction to explain how you know that\ is the perpendicular bisector of \.Answer:Draw a line segment of any length and name that line segment as ABStep 2:Draw an arc by using a compass with above half of the length of AB by taking the center at A above ABStep 3:Draw another arc by using a compass with above half of the length of AB by taking the center at B above ABStep 4:Repeat steps 3 and 4 below ABStep 5:Draw a line segment CD by joining the arcs above and below ABStep 6:Measure the lengths of the midpoint of AB i.e., AD and DB.By measuring their lengths, we can prove that CD is the perpendicular bisector of AB
Question 2.The equation of a line is x + 2y = 10.a. Use the numbers and symbols to create the equation of a line in slope-intercept formthat passes through the point and is parallel to the given line.Answer:The given line equation is:x + 2y = 10The given point is: Now,The given equation in the slope-intercept form is:y = \x + 5We know that,The slopes of the parallel lines are the sameSo,The equation of the line that is parallel to the given line equation is:y = \x + cTo find the value of c,Substitute in the above equationSo,-5 = \ + cc = -5 + 2We can conclude that the line that is parallel to the given line equation is:y = \x 3
We can conclude that 42° and 48° are the vertical angles
Question 5.Enter a statement or reason in each blank to complete the two-column proof.Given 1 3The completed table is:
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Measuring And Constructing Segments
Essential QuestionHow can you measure and construct a line segment?Answer:The steps used to measure a line segment are:a. Pick up a scale to measure the length of a line segment.b. Identify the line segment you want to measurec. Place the tip of the ruler at the starting of the line segment
The steps used to construct a line segment are:a. Place the compass at one end of the lineb. Adjust the compass to slightly longer than half of the lines lengthc. Draw arcs above and below the lined. Keeping the same compass width, draw arcs from the other end of the linee. Place ruler where the arcs cross and draw the line segment
Exploration 1
Measuring Line Segments Using Nonstandard Units
Work with a partner.
a. Draw a line segment that has a length of 6 inches.Answer:We will use a ruler to draw a line segment and the ruler we use generally is the Centimeter rulerBut,It is given that we have to draw a line segment that has a length of 6 inchesBut, it is not possibleSo,6 inches = 15.24 cmHence,The representation of the line segment that has the length of 6 inches in terms of cm is:
c. Write conversion factors from paper clips to inches and vice versa.Answer:We can conclude that the conversion of paper clips into inches and vice-versa is:1 paperclip = 1.377 inch1 inch = 2.54 paperclip
Exploration 2
Measuring Line Segments Using Nonstandard Units
Work with a partner.
Exploration 3
Measuring Heights Using Nonstandard Units
Work with a partner.
Communicate Your Answer:
Exercise 14 Perimeter And Area In The Coordinate Plane
Question 1.The perimeter of a square with side length s is P = _________ .Answer:
Question 2.WRITINGWhat formulas can you use to find the area of a triangle in a coordinate plane?Answer:We know that,The formula you can use to find the area of a triangle in a coordinate plane is:Area of a triangle = \ × Base × Height
Monitoring Progress and Modeling with Mathematics
In Exercises 3 6, classify the polygon by the number of sides. Tell whether it is convex or concave.
Question 3.
The number of sides is: 6We know that,The figure is concave if all the interior angles are greater than 180°The figure is convex if all the interior angles are less than 180°Hence, from the above,We can conclude that the given polygon is a hexagon and it is a concave polygon as it has interior angles greater than 180°
In Exercises 7 12. find the perimeter of the polygon with the given vertices.
Question 7.G, H, J, KAnswer:
Q, R, s, TAnswer:The given vertices of a polygon are:Q , R , S , T Now,The length of QR = \=\= 4The length of RS = \=\= 4The length of ST = \=\= 4The length of TQ = \= \= 4From the lengths of all of the sides,We can say that the vertices belong to a squareSo,The perimeter of a square = 4 × Side= 4 × 4We can conclude that the perimeter of the given polygon is: 16
Question 9.U, V, WAnswer:
Question 11.
In Exercises 13 16. find the area of the polygon with the given vertices.
Question 13.E, F, GAnswer:
Question 15.W, X, Y, ZAnswer:
In Exercises 17 24, use the diagram.
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Study Skills: Analyzing Your Errors
Mathematical Practices
Question 1.Draw the portion of the diagram that you used to answer Exercise 26 on page 130.Answer:The portion of the diagram that you used to answer Exercise 26 on page 130 is:
Question 2.In Exercise 40 on page 144. explain how you started solving the problem and why you started that way.Answer:In Exercise 40 on page 144,You started solving the problem by considering the 2 lines parallel and two lines as transversalsSo,If p and q are the parallel lines, then r and s are the transversalsIf r and s are the parallel lines, then p and q are the transversals
Big Ideas Math Book Geometry Answer Key Chapter 1 Basics Of Geometry
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We can conclude that the area of the given triangle is: 200 in²
Question 13.ABSTRACT REASONINGDescribe the possible values for x and y when |x y| > 0. What does it mean when |x y| = 0 ? Can |x y| < 0? Explain your reasoning.Answer:We know that,The value of the absolute expression must be greater than or equal to 0 but not less than 0So,The values for | x y | do not existNow,The possible values of | x y | > 0 should be greater than 0 and maybe x > y and x < yThe possible values of | x y | = 0 should be only one value i.e., 0 as x and y must be equal to make the difference value 0
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