4.7 Angles and Angle Measurement
Unit Goals
 Students learn to draw and identify points, rays, segments, angles, and lines, including parallel and perpendicular lines. Students also learn how to use a protractor to measure angles and draw angles of given measurements, and identify acute, obtuse, right, and straight angles in twodimensional figures.
Section A Goals
 Draw and identify points, lines, rays, segments, and parallel and intersecting lines in geometric figures.
 Recognize that angles are formed wherever two rays share a common endpoint and identify angles in twodimensional figures.
Section B Goals
 Recognize that angles can be measured in degrees, and can be found using addition and subtraction.
 Use a protractor to measure and draw angles, and recognize that perpendicular lines meet or cross at a right angle.
Section C Goals
 Draw and identify acute, obtuse, right, and straight angles in twodimensional figures.
 Write equations to represent angle relationships and reason about and find unknown measurements.
Section A: Points, Lines, Segments, Rays, and Angles
Problem 1
Preunit
Practicing Standards: 3.G.A.1
Draw a rectangle on the grid and label it A. Draw a triangle and label it B. Draw a hexagon and label it C.
Solution
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Problem 2
Preunit
Practicing Standards: 3.G.A.1

Is the shape a rhombus? Explain your reasoning.
 Is the shape a rectangle? Explain your reasoning.
 Is the shape a square? Explain your reasoning.
Solution
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Problem 3
 Draw 4 different lines through points on the grid. At least two of the lines should cross another line.
 Mark at least 3 different segments in your drawing.
Solution
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Problem 4

Circle the line segments that make up the letter A.
 Draw 4 rays that surround a rectangle.
 Can you find 4 different rays that surround the same rectangle?
Solution
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Problem 5
 Explain why Andre is not correct.
 Draw a line that is parallel to one of the lines in the image.
Solution
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Problem 6
 Which segments of the letter Z are a pair of parallel lines? Draw the lines.
 Sketch a line that is parallel to the third segment in the Z.
Solution
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Problem 7
 Find one angle in the figure. Draw a pair of rays to show the angle and extend them as far as you can.
 Find another angle in the figure. Draw a pair of rays to show it. Extend the rays as far as you can. (If you’d like, you can use a different colored pencil for this pair of rays.)
 Now that you have drawn some rays, do you see other angles that you didn’t see before? If you see one or more, label each one with a letter.
Solution
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Problem 8
Exploration
Here is a riddle. Can you solve it?
“I am a capital letter made of more than 1 segment with no curved parts. I have no perpendicular segments or parallel segments. What letter could I be?”
Solution
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Problem 9
Exploration
 Name or describe any shapes that you recognize in the painting.
 Do you see any parallel lines? If so, trace or circle them. (If you’d like, you can use a different colored pencil for each set of parallel lines.)
 Are there any angles in the painting? If so, mark them or describe where they are.
Solution
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Section B: The Size of Angles
Problem 1
 Write two statements that compare the size of angles A and B.
 Draw an angle C that is bigger than both angle A and angle B.
Solution
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Problem 2
 Which set of clock hands make a greater angle? Explain how you know.
 Choose one of the clocks and describe how to use the clock to draw the angle represented by the hands on the clock.
Solution
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Problem 3
This angle has a measure of \(10^\circ\).
 How many of these angles can you put together, without overlaps, to make a complete circle? Explain or show how you know.
 Explain how you can use the given angle to sketch a \(5^\circ\) angle.
Solution
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Problem 4
Solution
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Problem 5
Which of these shapes have segments that are perpendicular to one another? Trace or circle the perpendicular segments.
Solution
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Problem 6
Draw a ray. How many different \(35^\circ\) angles can you make using your ray and another ray? Explain your reasoning and make the angles.
Solution
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Problem 7
Exploration
What is the smallest angle you can draw?
 Can you draw a \(10^\circ\) angle?
 What about a \(5^\circ\) angle or a \(1^\circ\) angle?
 What is difficult about drawing a small angle?
Solution
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Problem 8
Exploration
 What are the measurements of the angles on the pentagon?
 Connect every pair of vertices of the pentagon with a line segment. What do you notice? What do you wonder?
Solution
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Problem 9
Exploration
Can you estimate or find the measurement of the angle labeled \(f\)? If so, explain or show how you know.
Solution
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Section C: Angle Analysis
Problem 1
 Draw an acute angle. Explain how you know the angle is acute.
 Extend one of the rays of your angle in the opposite direction. Explain why you’ve now created a new angle that is obtuse.
Solution
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Problem 2

The circle is divided into 12 equal parts. What is the measure of angle \(h\)?
Explain or show how you know.
 Can you put together \(20^\circ\) angles to make a circle? How many of them will it take?
Solution
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Problem 3

A circle has been cut into eighths. How many degrees is the angle labeled \(m\)? Explain or show your reasoning.

Another circle has been cut into fifths. How many degrees is the angle labeled \(p\)? Explain or show your reasoning.
Solution
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Problem 4

What is the measure of each angle \(p\)? Explain or show how you know.
 How many \(25^\circ\) angles can you fit together at one vertex without gaps or overlaps?
Solution
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Problem 5

What angles are made by the hour and minute hands on a clock at these times? Explain or show your reasoning.
 3:00
 5:00
 6:00
 How many degrees has the hour hand moved between 3:00 and 7:00? Explain or show how you know.
Solution
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Problem 6
When Jada looked up at the clock, the long hand pointed at 12. Less than an hour later, she looked up again, and the long hand of the clock had turned 210 degrees. How many minutes had passed? Explain or show your reasoning.
Solution
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Problem 7
Find the measure of each labeled angle in the drawing. Assume that:
 The angles of the triangles meeting at the point in the middle of the figure are all the same.
 The other angles of the triangles all have the same measure.
Solution
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Problem 8
In this diagram each angle on the left hand side is the same as the corresponding angle on the right hand side. Find the measure of angles \(x\), \(y\), and \(z\). Explain or show your reasoning.
Solution
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Problem 9
Exploration
Tyler wonders if the hour hand and minute hand ever point in the same direction at the same time. Can you find some times when the hour hand and minute hand point in the same direction? Explain or show your reasoning.
Solution
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Problem 10
Exploration
 Draw a rhombus with a \(50^\circ\) angle. Explain how you know your shape is a rhombus.
 Draw another rhombus with a \(50^\circ\) angle. How are your rhombuses the same? How are they different?
Solution
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Problem 11
Exploration
How many degrees does the minute hand turn in each of the following times? Show how you know.
 30 seconds
 10 seconds
 80 minutes
 2.5 hours
Solution
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Problem 12
Exploration
Here are diagrams of some pattern blocks. Each shape has some angles.

How many angles do you see inside each shape?
 a triangle
 a trapezoid
 a rhombus
 a hexagon
 Here are diagrams of each type of blocks arranged around a shared point.
Use what you know about angle measurement to find the sizes of angles \(a\)–\(f\). Show your reasoning.
Solution
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