1.4 measuring angles homework day 2

1-4 Measuring Angles

Here is your free content for this lesson!

Word Docs & PowerPoints

1-4 Assignment - Measuring Angles 1-4 Assignment 2 - Measuring Angles 1-4 Bell Work - Measuring Angles 1-4 Bell Work 2 - Measuring Angles 1-4 Exit Quiz - Measuring Angles 1-4 Exit Quiz 2 - Measuring Angles 1-4 Guided Notes SE -  Measuring Angles 1-4 Guide Notes 2 SE -Measuring Angles 1-4 Guided Notes TE - Measuring Angles 1-4 Guide Notes 2 TE -Measuring Angles 1-4 Lesson Plan - Measuring Angles 1-4 Online Activity - Measuring Angles 1-4 Slide Show - Measuring Angles

1-4 Assignment - Measuring Angles 1-4 Assignment 2 - Measuring Angles 1-4 Bell Work - Measuring Angles 1-4 Bell Work 2 - Measuring Angles 1-4 Exit Quiz - Measuring Angles 1-4 Exit Quiz 2 - Measuring Angles 1-4 Guided Notes SE - Measuring Angles 1-4 Guide Notes 2 SE -Measuring Angles 1-4 Guided Notes TE - Measuring Angles 1-4 Guide Notes TE -Measuring Angles 1-4 Lesson Plan - Measuring Angles 1-4 Online Activity - Measuring Angles 1-4 Slide Show - Measuring Angles

 Want access to everything? Simply click the image below to Get All of Our Lessons!

Find and Use Slopes of Lines

Teaching midpoint and distance in the coordinate plane, teaching congruent figures, triangle congruence by asa and aas, teaching perpendicular and angle bisectors, teaching bisectors in triangles, best geometry books for kids, proving theorems in geometry, how to prove lines are parallel, teaching logic in geometry, teaching angle pairs, proofs with uno cards, valentine’s day math activity – classifying quadrilaterals, christmas math worksheets, thanksgiving worksheet for geometry – happy turkey day, halloween geometry activities high school, how to teach classifying polygons, points lines and planes, tessellation project, the unit circle – hand trick, introduction to trigonometry, pi day top 5 for geometry teachers, polygons in the coordinate plane – how to use geogebra, teaching tessellations to your geometry class, 3 ways to make your life easier as a geometry teacher without more geometry worksheets, reasoning and proof, geometry games – dance dance transversal, midsegments of triangles, planning a proof, triangle congruence by sss and sas, pythagorean theorem – nfl and geometry, inequalities in one triangle, parallel lines cut by a transversal – colorful flip book notes, measuring angles putt-putt course design project, how to teach tangent lines – graphic organizer, segment addition postulate, two tips to teach volumes of prisms and cylinders, parallel lines and transversals, nets and drawings for visualizing geometry – geometry nets project, ratios and proportions – bad teacher, hybrid flipped classroom model, how to teach perimeter and area of similar figures – in jay leno’s garage, how to create a jeopardy game, 5 tips for new teachers, 8 teacher discounts you don’t want to miss this summer, your students are cheating with this math app, how to setup and use google classroom and google forms to teach geometry, special right triangles – the foundation of everything trig, how to use google forms and word clouds to help your students master their geometry vocabulary, interactive geometry teaching techniques, the magic octagon – understand congruence in terms of rigid motions, attention getters to keep your geometry class locked in, teaching strategies for your inclusion geometry class without an intervention specialist, how to teach the properties of quadrilaterals, how to make geometry interesting, how to teach circles using the common core standards, full year of geometry lesson plans, how a rock star geometry teacher uses the ipad to educate, are you a geometry teacher with no textbooks or geometry worksheets, how to deal with a class clown – classroom management strategies, geometry lessons | the game has changed | common core standards, teaching dimensions with super mario – geometry, lesson for the first day of geometry class.

• App Downloads

GEOMETRY Notes 1.4: Measure and Classify Angles

#1) Find the midpoint of . Show the midpoint on the graph, and write the name and coordinates of the point.

#2) If C is the midpoint of , plot D on the graph. What are the coordinates of D?

#3) What is the midpoint of given P(-2, 3p) and Q(4n, p)?

b) Why would it be confusing to call this ray ?

#8) Sometimes you will not see the entire rays that make the angle sides-- for example, if the angle is part of a triangle. In the applet, lengthen and shorten the segment sides of the angle-- do longer sides mean a larger angle? Name this angle in 3 different ways.

#10) Write an equation involving the angles in the applet.

New Resources

• Wallace-Simson Line, Orthopole, and Deltoid
• Flag in the wind
• Thin Slice: Using Trig Ratios to Solve for R.Triangle Sides
• Graphing Logarithmic Functions
• Is it Isosceles?

Discover Resources

• Angles_Transversal2
• TeachMeet-1Gradient
• About Axiomatic System
• Cópia de Função Quadrática

Discover Topics

• Natural Numbers
• Stochastic Process or Random Process
• Centroid or Barycenter

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

4.E: Radian Measure (Exercises)

• Last updated
• Save as PDF
• Page ID 3395

• Michael Corral
• Schoolcraft College

These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.

4.1 Exercise

For Exercises 1-5, convert the given angle to radians.

4.1.1 \(4^\circ\) 4.1.2 \(15^\circ\) 4.1.3 \(130^\circ\) 4.1.4 \(275^\circ\) 4.1.5 \(-108^\circ\) For Exercises 6-10, convert the given angle to degrees. 4.1.6 \(4 \) rad 4.1.7 \(\dfrac{\pi}{5} \) rad 4.1.8 \(\dfrac{11\pi}{9} \) rad 4.1.9 \(\dfrac{29\pi}{30} \) rad 4.1.10 \(35 \) rad 4.1.11 Put your calculator in radian mode and take the cosine of \(0 \). Whatever the answer is, take its cosine. Then take the cosine of the new answer. Keep repeating this. On most calculators after about \(50\)-\(60 \) iterations you should start to see the same answer repeating. What is that number? Try starting with a number different from \(0 \). Do you get the same answer repeating after roughly the same number of iterations as before? Try the same procedure in degree mode, starting with \(0^\circ \). Does the same thing happen? If so, does it take fewer iterations for the answer to start repeating than in radian mode, or more?

4.2 Exercise

For Exercises 1-4, find the length of the arc cut off by the given central angle \(\theta \) in a circle of radius \(r \). 4.2.1 \(\theta=0.8 \) rad, \(r=12 \) cm 4.2.2 \(\theta=171^\circ \), \(r=8 \) m 4.2.3 \(\theta=\pi \) rad, \(r=11 \) in 4.2.4 A central angle in a circle of radius \(2 \) cm cuts off an arc of length \(4.6 \) cm. What is the measure of the angle in radians? What is the measure of the angle in degrees? 4.2.5 The centers of two belt pulleys, with radii of \(3 \) inches and \(6 \) inches, respectively, are \(13 \) inches apart. Find the total length \(L \) of the belt around the pulleys. 4.2.6 In Figure 4.2.5 one end of a \(4 \) ft iron rod is attached to the center of a pulley with radius \(0.5 \) ft. The other end is attached at a \(40^\circ \) angle to a wall, at a spot \(6 \) ft above the lower end of a steel wire supporting a box. The other end of the wire comes out of the wall straight across from the top of the pulley. Find the length \(L \) of the wire from the wall to the box.

4.2.7 Figure 4.2.6 shows the same setup as in Exercise 6 but now the wire comes out of the wall \(2 \) ft above where the rod is attached. Find the length \(L \) of the wire from the wall to the box. 4.2.8 Find the total length \(L \) of the figure eight shape in Figure 4.2.7.

4.2.9 Repeat Exercise 8 but with the circle at \(A \) having a radius of \(3\) instead of \(2 \). ( Hint: Draw a circle of radius \(5 \) centered at \(A \), then draw a tangent line to that circle from \(B \).) 4.2.10 Suppose that in Figure 4.2.7 the lines do not criss-cross but instead go straight across, as in a belt pulley system. Find the total length \(L \) of the resulting shape. 4.2.11 Find the lengths of the two arcs cut off by a chord of length \(3 \) in a circle of radius \(2 \). 4.2.12 Find the perimeter of a regular dodecagon (i.e. a \(12\)-sided polygon with sides of equal length) inscribed inside a circle of radius \(\frac{1}{2} \). Compare it to the circumference of the circle.

4.3 Exercise

For Exercises 1-3, find the area of the sector for the given angle \(\theta \) and radius \(r \). 4.3.1 \(\theta = 2.1 \) rad, \(r = 1.2 \) cm 4.3.2 \(\theta = \frac{3\pi}{7} \) rad, \(r = 3.5 \) ft 4.3.3 \(\theta = 78^\circ \), \(r = 6 \) m 4.3.4 The centers of two belt pulleys, with radii of \(3 \) cm and \(6 \) cm, respectively, are \(13 \) cm apart. Find the total area \(K \) enclosed by the belt. 4.3.5 In Exercise 4 suppose that both belt pulleys have the same radius of \(6 \) cm. Find the total area \(K \) enclosed by the belt. 4.3.6 Find the area enclosed by the figure eight in Exercise 8 from Section 4.2. For Exercises 7-9, find the area of the sector for the given radius \(r \) and arc length \(s \). 4.3.7 \(r = 5 \) cm, \(s = 2 \) cm 4.3.8 \(r = a \), \(s = a\) 4.3.9 \(r = 1 \) cm, \(s = \pi \) cm For Exercises 10-12, find the area of the segment formed by a chord of length \(a \) in a circle of radius \(r \). 4.3.10 \(a = 4 \) cm, \(r = 4 \) cm 4.3.11 \(a = 1 \) cm, \(r = 5 \) cm 4.3.12 \(a = 2 \) cm, \(r = 5 \) cm 4.3.13 Find the area of the shaded region in Figure 4.3.7.

4.3.14 Find the area of the shaded region in Figure 4.3.8. ( Hint: Draw two central angles. )

4.3.15 Find the area of the shaded region in Figure 4.3.9.

4.3.16 The centers of two circles are \(4 \) cm apart, with one circle having a radius of \(3 \) cm and the other a radius of \(2 \) cm. Find the area of their intersection. 4.3.17 Three circles with radii of \(4 \) m, \(2 \) m, and \(1 \) m are externally tangent to each other. Find the area of the curved region between the circles, as in Figure 4.3.10. ( Hint: Connect the centers of the circles .)

4.3.18 Show that the total area enclosed by the loop around the three circles of radius \(r \) in Figure 4.3.11 is \(\;(\pi + 6 + \sqrt{3})\,r^2 \). 4.3.19 For a fixed central angle \(\theta \), how much does the area of its sector increase when the radius of the circle is doubled? How much does the length of its intercepted arc increase?

4.4 Exercise

For Exercises 1-6, assume that a particle moves along a circle of radius \(r \) for a period of time \(t \). Given either the arc length \(s \) or the central angle \(\theta \) swept out by the particle, find the linear and angular speed of the particle. 4.4.1 \(r=4 \) m, \(t=2 \) sec, \(\theta=3 \) rad 4.4.2 \(r=8 \) m, \(t=2 \) sec, \(\theta=3 \) rad 4.4.3 \(r=7 \) m, \(t=3.2 \) sec, \(\theta=172^\circ\) 4.4.4 \(r=1 \) m, \(t=1.6 \) sec, \(s=3 \) m 4.4.5 \(r=2 \) m, \(t=1.6 \) sec, \(s=6 \) m 4.4.6 \(r=1.5 \) ft, \(t=0.3 \) sec, \(s=4 \) in 4.4.7 An object moves at a constant linear speed of \(6 \) m/sec around a circle of radius \(3.2 \) m. How large of a central angle does it sweep out in \(1.8 \) seconds? 4.4.8 Two interlocking gears have outer radii of \(6 \) cm and \(9 \) cm, respectively. If the smaller gear rotates at \(40 \) rpm, how fast does the larger gear rotate? 4.4.9 Three interlocking gears have outer radii of \(2 \) cm, \(3 \) cm, and \(4 \) cm, respectively. If the largest gear rotates at \(16 \) rpm, how fast do the other gears rotate? 4.4.10 In Example 4.17, does equation 4.11 still hold if the radii \(r_1 \) and \(r_2 \) are replaced by the number of teeth \(N_1 \) and \(N_2 \), respectively, of the two gears as shown in Figure 4.4.2? 4.4.11 A \(78 \) rpm music record has a diameter of \(10 \) inches. What is the linear speed of a speck of dust on the outer edge of the record in inches per second? 4.4.12 The centripetal acceleration \(\alpha \) of an object moving along a circle of radius \(r \) with a linear speed \(\nu \) is defined as \(\;\alpha = \frac{\nu^2}{r} \). Show that \(\;\alpha = \omega^2 \,r \), where \(\omega \) is the angular speed.

Half a Square

2.1: Diagonals of Rectangles (5 minutes)

CCSS Standards

Building On

• HSG-SRT.B.5

Later in this lesson, students will explore isosceles right triangles and reason about them in the context of diagonals of squares. In this warm-up, students practice finding the hypotenuse of a right triangle with the Pythagorean Theorem in the context of finding the diagonal of a rectangle.

In the synthesis of a later activity, students will return to these examples to explore why the ratio of the length of the diagonal to the length of the side of a square is constant and explain why that is not the case for all rectangles, though it is the case for any family of similar rectangles (including squares).

Monitor for students who:

• leave their answers in radical form
• estimate the answer
• use a calculator to get an answer in decimal form

Student Facing

Description: <p>A four sided figure A B C D. A diagonal line is drawn between points B and D and is labeled x. Side A B is labeled 4. Angle A is marked as a right angle. Side A D is labeled 7. Angle B D A is labeled 29.7 degrees.</p>

Description: <p>A four sided figure K L M J. A diagonal line is drawn between points K and M and is labeled y. Side K J is labeled 8. Angle J is marked as a right angle. Side J M is labeled 14. Angle K M J is labeled 29.7 degrees.</p>

 Calculate the values of \(x\) and \(y\) .

Student Response

For access, consult one of our IM Certified Partners .

Anticipated Misconceptions

If students are struggling, ask them what shapes they see. (Rectangles and right triangles.) If students are still stuck, ask them how they would find \(x\) or \(y\) if there were only a right triangle. (Using the Pythagorean Theorem.)

Activity Synthesis

Focus discussion on an appropriate level of precision for the length of \(x\) . Invite students to share their answers in different forms. If no student estimated, ask them approximately how big \(x\) is based on finding that \(x^2 = 65 \text{ or } x=\sqrt{65}\) . (I know  \(x\) is a little greater than 8, because \(8^2=64\) .) Compare that with the results students got if they used a calculator to find a decimal approximation of \(x\) . Note that recording all the digits displayed on the calculator isn't helpful. We will use the convention of rounding side lengths to the nearest tenth, but sometimes knowing the exact answer of \(\sqrt{65}\) is helpful.

2.2: Decomposing Squares (15 minutes)

Building Towards

• HSG-SRT.C.6

Routines and Materials

Instructional Routines

• MLR8: Discussion Supports

Required Materials

• Copies of blackline master
• Protractors
• Scientific calculators

Optional activity

In this activity, students compute the diagonals of some squares and measure the diagonals of others. From the data they collect, they reason that for a square with side length \(s\) , the length of the diagonal is about \(1.4s\) . They also calculate the diagonal of a unit square exactly, using the Pythagorean Theorem, allowing them to connect the decimal approximation \(1.4\) to the square root of 2.

Students will need a set of squares to measure. They could use various sizes of origami paper, sticky notes, or other convenient objects. If squares are not available, there is a blackline master provided with squares students can use instead.

The goal of this activity is not to simplify radicals or make explicit the connection that, for example, \(\sqrt{50} \) can be rewritten as  \( \sqrt{25 \boldcdot 2}= 5\sqrt2\) . 

• use a calculator to determine that the diagonal of a unit square is about 1.4
• leave the diagonal of the unit square as exactly \(\sqrt2\)
• make a table to record what they learned about squares and their diagonals

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Arrange students in groups of 2. Distribute squares of several different sizes to each group (either the blackline master or other convenient squares).

• Draw a square with side lengths of 1 cm. Estimate the length of the diagonal. Then calculate the length of the diagonal.
• Measure the side length and diagonal length of several squares, in centimeters. Compute the ratio of side to diagonal length for each.
• Make a conjecture.

If students are struggling to organize their thinking, suggest that they make a table. Help students brainstorm what would be good columns to organize their measurements and calculations, such as “side length,” “diagonal length,” and “ratio of diagonal to side.”

Ask students what patterns they noticed and what conjectures they made. Invite students who organized their thinking using a table to display their work for all to see. If no students made a table, create one as a class, displayed for all to see. Include students who approximated the diagonal length of the unit square using a calculator, and students who left it as \(\sqrt2\) . Leave the table visible for all to see during the next activity as well.

If this conjecture is not mentioned by students, point it out in the table and then ask students to explain:

• The diagonal length of a square seems to be the side length multiplied by 1.4 each time. (All squares are similar to the unit square by a scale factor of the side length, so it makes sense that all the diagonal lengths are multiples of the diagonal length of the unit square.)

Ask students if they agree that both of these things are true:

• The diagonal of a square with side length \(s\) is about 1.4 times \(s\) .
• The diagonal of a square with side length \(s\) is exactly \(s\sqrt2\) .

2.3: Generalize Half Squares (15 minutes)

In the previous activity, students generalized that the diagonals of squares are related to the side length of the square by a factor of about 1.4, or exactly \(\sqrt2\) .

In this activity, students apply their generalization about the diagonals of squares to isosceles right triangles. To find the lengths of the unlabeled sides in the second figure, students will need to generalize that a right triangle with one 45 degree angle is isosceles (because it’s half a square, or because both base angles are congruent, or because it’s similar to the isosceles triangle in the first figure).

To find the unknown values in the third figure, students will have to use the ratio of diagonal length to side length to find unknown side lengths. Students may use the approximate ratio \(1.4:1\) (or \(1:0.7\) ) to find the side lengths, or they might use the exact ratio \(\sqrt2:1 \text{ or } 1:\frac{1}{\sqrt2}\) . Students are not expected or encouraged to rewrite answers of \(\frac{24}{\sqrt2}\) in another form.

Calculate the lengths of the 5 unlabeled sides.  

Are you ready for more?

Square \(ABCD\) has a diagonal length of \(x\) and side length of \(s\) . Rhombus \(EFGH\) has side length \(s\) .

• How do the diagonals of \(EFGH\) compare to the diagonals of \(ABCD\) ?
• What is the maximum possible length of a diagonal of a rhombus of side length \(s\) ?

If students are struggling, encourage them to analyze the three triangles, look for patterns, and identify the triangles as isosceles right triangles. Students can then use the patterns from the previous activity. 

Make sure all students understand that the three triangles are isosceles right triangles, and represent half of a square, and so are prepared to connect their reasoning from earlier activities to this activity.

Display this list of solutions for triangle \(PQR\) :

• \(QR=14, RP=\sqrt{392}\)
• \(QR=14, RP=14\sqrt{2}\)
• \(QR=14, RP = 19.8\)
• \(QR=14, RP = 1.4 \boldcdot 14 \)  or 19.6

Invite students to explain the reasoning behind each. Encourage students to think about which answers are most accurate, and which are most efficient. (The first two methods are equally accurate, but the first probably took more calculation, while the second could be found using scale factors quite easily. The third answer was obtained by using a calculator to evaluate either of the first two answers, which actually loses both efficiency and accuracy. The fourth answer is very efficient, as it was found using scale factors, but is a bit less accurate. For some applications, the fourth answer may be best, and it’s great for estimating.)

Lesson Synthesis

Create a class display of half a square. This display should be posted in the classroom for the remaining lessons within this unit. 

Ask students what they notice and what they wonder about the image. (It’s isosceles. It’s half a square. It’s the same triangles we’ve been looking at all class. The sides are \(s, s, \text{ and } s\sqrt2\) . Why is it important? Do we have to memorize it? Why does it have \(\sqrt2\) instead of 1.4?)

Explain to students that they don’t need to memorize this information since they have other strategies to calculate missing side lengths and hypotenuses in isosceles right triangles. 

If applicable, tell students that this type of image might be provided as a reference on standardized tests. Ask students how they might decide what types of problems to use it on, and how they would solve those problems. Here are three sample problems to discuss:

• The distance between bases on a baseball field is 90 feet. If the 2nd base player stands on second base and throws the ball to the catcher standing at home plate, how far is the throw? (If students are not familiar with the context of baseball, provide a diagram.)
• The hypotenuse of an isosceles right triangle is 10 cm. How long is the side?
• A rectangle has sides of 3 cm and 4 cm. How long is the diagonal of the rectangle?

Students may say that the throw divides the square in half so it matches the image. An isosceles right triangle also matches the image. Half of this rectangle isn’t similar to half a square, so they would need a different method such as the Pythagorean Theorem. Note that they could use the Pythagorean Theorem to solve all three problems, since that is where the labels on the image come from.

2.4: Cool-down - Another Half Square (5 minutes)

Student lesson summary.

Drawing the diagonal of a square decomposes the square into 2 congruent triangles. They are right isosceles triangles with acute angles of 45 degrees. These congruent angles make all right isosceles triangles similar by the Angle-Angle Triangle Similarity Theorem.

Consider an isosceles right triangle with legs 1 unit long where \(c\) is the length of the hypotenuse. By the Pythagorean Theorem, we can say \(1^2+1^2=c^2\) so \(c=\sqrt2\) . The hypotenuse of an isosceles right triangle with legs 1 unit long is \(\sqrt2\)  units long.

Now, consider an isosceles right triangle with legs  \(x\)  units long. By the Angle-Angle Triangle Similarity Theorem, the triangle is similar to the isosceles right triangle with side lengths of 1, 1, and \(\sqrt2\)  units. A scale factor of  \(x\)  takes the triangle with leg length of 1 to the triangle with leg length of  \(x\) . Therefore, the hypotenuse of the isosceles right triangle with legs \(x\)  units long is  \(x\sqrt2\)  units long. 

Description: <p>Triangle. An angle is marked as a right angle. The other two angles are labeled as 45 degrees each. The sides opposite the 45 degree angles are labeled x. The side opposite the right angle is labeled x times square root of 2.</p>

In triangle \(ABC, x=6\) so \(AC\)  is 6 units long and \(BC\) is  \(6\sqrt2\)  units long. 

In triangle \(DEF, x\sqrt2=12\) so \(x=\frac{12}{\sqrt2}\) , which means both \(EF\) and \(DF\) are  \(\frac{12}{\sqrt2}\)  units long.

Home > CCG > Chapter 2 > Lesson 2.1.4 > Problem 2-41

Find all missing angles in the diagrams below.  

What do you know about vertical angles? Alternate interior angles? What about adjacent angles that form a straight line?

Fill in the missing values.

© 2022 CPM Educational Program. All rights reserved.

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&