Angles - Typepad



INTERVENTION

LESSONS

FOR GHSGT

MATH OBJECTIVE 31

Angles

Angles

Lesson 1 (Interior Angles)

GHSGT Objectives:

31 Draws and measures angles; determines the number of degrees in the interior angles of geometric figures, such as right and straight angles, circles, triangles, and quadrilaterals: and classify angles (right, acute, obtuse, complementary, supplementary) and triangles (right , acute, obtuse, scalene, isosceles, and equilateral).

Warm-up / Activator

Use a straightedge to draw a triangle on a sheet of paper. Cut the triangle out. Tear off the three corners of the triangle as illustrated below. Now piece the three vertices together and tape them in place (Hint: the straightedge might be useful as a guide).

[pic]

Share your results with the class. Did everyone get the same results? Why or why not?

What do you observe about the three vertices of the triangle that you drew?

What do you observe about the sum of the measure of the three angels in a triangle?

Mini Lesson

It is widely thought that the Babylonians were first to enclose messages inscribed on clay tablets in clay envelopes baked hard around their contents. Presently, envelopes are used to carry mail and are made by cutting and folding a piece of paper at different angles. The folds of an envelope form four triangles on its surface. Use a protractor to accurately find the measure of each of the numbered angles in the envelope shown below.

[pic]

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

j. [pic] k. [pic] l. [pic]

What do you notice about the measure of the three angles of each of the triangles found in the envelope? Try and draw a triangle that does not conform to the conclusion that you reached about the three angles of a triangle. Is such a triangle possible? Is the sum of the interior angles of a triangle always the same value?

Can we use the routine from the warm up to investigate the sum of the interior angles of a quadrilateral?

Use a straightedge to draw a quadrilateral on a sheet of paper. Cut the quadrilateral out. Tear off the four corners of the quadrilateral and piece the four vertices together and tape them in place.

[pic]

Share your results with the class. Did everyone get the same results? Why or why not?

What do you observe about the four vertices of the quadrilateral that you drew?

What can you conclude about the sum of the interior angles of a quadrilateral?

Now find the unknown angle measure in each triangle or quadrilateral.

m. [pic] n.[pic]

o. p. [pic]

q. [pic] r. [pic]

Closure

Have students share their strategies and solutions with the class. Ask questions to assess student understanding of concepts. Have students summarize what they learned today.

Notes to the Teacher

Supply examples of envelopes so that the students may see the triangles formed as mentioned in the lesson. Students may have difficulty finding the sum of 180º for the three angles of a triangle if they are not able to use the protractor accurately. Encourage students to make the connection between the measure of 180º for a straight angle and 360º for a circle as they complete the investigation with the triangle and quadrilateral respectively.

Angles

Lesson 2 (Identifying and Classifying Angles)

GHSGT Objectives:

31 Draws and measures angles; determines the number of degrees in the interior angles of geometric figures, such as right and straight angles, circles, triangles, and quadrilaterals: and classify angles (right, acute, obtuse, complementary, supplementary) and triangles (right , acute, obtuse, scalene, isosceles, and equilateral).

Warm-up / Activator

There are four main types of angles that we use in geometry: acute angles, right angles, obtuse angles, and straight angles. Angles are measured in degrees using a protractor. Angles are all around us. Do you see any angles in your classroom? Look around the classroom and record examples of each type of angle. Share your results with the class.

Mini Lesson

Engage students in a discussion to define what an angle is and each of the four main types of angles.

[pic]

Some students may need to review the definitions. The InterMath Dictionary is an online resource providing definitions, diagrams, and/or examples for mathematical terms and can be used to assist students in their review of what an angle is, the types of angles, as well as the unit and tool used to measure angles.

Title: InterMath Dictionary

URL:

Ask leading questions to assess student understanding of what an angle is, the types of angles, estimating the measure of an angle, and the use of the protractor to draw and measure angles.

Now let’s examine some angle pairs. If the sum of the measures of two angles is 90º, then the angles are complementary angles. If the sum of the measures of two angles is 180º, then the angles are supplementary angles.

NOTE: complementary and supplementary angles may or may not be adjacent.

[pic] [pic]

[pic] [pic]

Work Session (Allow students to work in pairs)

Angle measure can be estimated mentally by comparing them with angles that we know or we can measure them directly. Write the word that best describes the type of angle indicated by the arrow and explain how you know your word best describes the angle, then estimate the degree measure for each angle.

[pic]

a. _________________________ b. _________________________

[pic] [pic]

c. _________________________ d. _________________________

[pic] [pic]

e. _________________________ f. _________________________

Use your protractor to accurately draw angles with the indicated measure.

g. 25° h. 138° i. 90º

j. 70° k. 180° l. 162°

Refer to the angles illustrated below to complete the following exercises.

|[pic] |

m. Which angles are right angles?

n. Which angles are straight angles?

o. Explain why [pic] is not a right angle. What type of angle is it?

p. Explain why [pic] is not a right angle. What type of angle is it?

q. Does [pic]have any complements? If so, name them.

r. Does [pic] have any supplements? If so, name them

s. Is [pic] complementary to [pic]? If so, explain why. If not, state an angle which is complementary to[pic].

t. What is m[pic]2 in the figure below?

[pic]

u. Find the value of x and the measure of the unknown angle.

[pic]

Closing

Have students share their answers with the class. Allow time for discussion of varied answers and explanations. Ask leading questions to assess student ability to correctly classify angles and to address any misconceptions. Have students summarize the concepts they learned today.

Angles

Lesson 3 (Circles)

GHSGT Objectives:

31 Draws and measures angles; determines the number of degrees in the interior angles of geometric figures, such as right and straight angles, circles, triangles, and quadrilaterals: and classify angles (right, acute, obtuse, complementary, supplementary) and triangles (right , acute, obtuse, scalene, isosceles, and equilateral).

Warm-up / Activator

In August 2000 the U.S. Census Bureau found that there were approximately 48,720,000 school-aged children in the U. S. According to a Census Bureau survey 57% of the school-aged children had access to a computer at home and at school, 23% had access only at school, 10% had access only at home, and 10% had not access at all.

How would you find the number of school-aged children in each of the four categories identified by the survey? How might the data collected by the Census Bureau regarding access to computers among school-aged children be represented graphically? Discuss the advantages and limitations of each

Mini Lesson

Use a compass to draw a circle on a sheet of construction paper. [pic] Try and use as much of the construction paper as possible, then cut the circle out. Fold the circle in half, and without opening the circle, fold the circle in half again. Now unfold the circle and use a pencil to trace along the fold lines. What do you notice about the fold lines? Share your ideas with the class.

Recall the following definitions: an acute angle has a measure between 0º and 90º; an obtuse angle has a measure between 90º and 180º; a reflex angle has a measure between 180º and 360º; a right angle has a measure of 90º; a straight angle has a measure of 180º; and a full angle has a measure of 360º.

T

H M Label your circle as shown and determine

the measure of each of the following angles

(counter clockwise rotation).

S

a. [pic] b. [pic]

c. [pic] d. [pic]

Explain how you determined each angle measure. Which of the aforementioned terms apply to the angles listed above? What fraction of the entire circle does each angle represent?

Work Session

Use the data presented in the warm-up to draw a circle graph. Discuss the plan you will use to find the measure of the central angles needed for each sector of the circle graph. Fill in the table below relative to the circle graph.

|Type of Access to Computers |Percent |Number of students |Measure of Central Angle |

| | | | |

| | | | |

| | | | |

| | | | |

Closure

Allow students to share their circle graphs with the class. Have students collect their own data and graph it using a circle graph or find relevant data and use it to draw a circle graph. Have students summarize what they learned today.

Notes to the Teacher

As students complete the warm-up activity be sure they observe that almost half of the circle graph represents students with access to a computer at school and at home and about a quarter of circle graph represents students with access to a computer only at school. Therefore, the sectors of the circle representing these categories should have angle measures close to 180º and 90º respectively. They will use this knowledge later in the lesson when they draw a circle graph.

Listen for students to describe the intersection of the fold lines as the center of the circle, and the fold lines as perpendicular lines/axes which divide the circle into 4 quadrants. This may help students notice that a right angle is a quarter of a full rotation and that a straight angle is half of a full rotation. Although right angles and straight angles are common to students, reflex angle and full angle may be new terminology for some students. The lesson may be extended by allowing students to research the origin of 360º as the measure of a full rotation about a point.

Angles

Lesson 4 (Classifying Triangles)

GHSGT Objectives:

31 Draws and measures angles; determines the number of degrees in the interior angles of geometric figures, such as right and straight angles, circles, triangles, and quadrilaterals: and classify angles (right, acute, obtuse, complementary, supplementary) and triangles (right , acute, obtuse, scalene, isosceles, and equilateral).

Warm-up / Activator

At your seat draw examples of acute, right and obtuse angles. Explain how these angles can be changed into triangles. Share your ideas with the class by illustrating your strategy on the board for the class to examine (4-5 volunteers). What do you think we should call these triangles?

Are there any other types of triangles? If so, discuss with the class and illustrate them on the board as well.

Mini Lesson

There are two ways to classify triangles: according to the measure of the angles and according to the lengths of the sides.

An acute triangle has three acute angles [pic]

An obtuse triangle has one obtuse angle [pic]

A right triangle has a right angle. [pic]

Note: The hypotenuse is the side opposite the right angle and the legs are the sides that meet at the right angle.

A scalene triangle has no two sides equal in length. [pic]

An isosceles triangle has two sides equal in length. [pic]

An equilateral triangle has three sides equal in length. [pic]

Every triangle has two classifications, one by the measure of its angles and one by the lengths of its sides. The angle classifications are mutually exclusive. However, the side classifications are not because equilateral triangles are also isosceles.

Work Session (Allow students to work in pairs)

Use a protractor to measure each angle and a ruler to measure each side of the triangles drawn below. Classify each triangle according to its angles and sides.

a. [pic] b.[pic]

_________________________ _________________________

c. [pic] d. [pic]

_________________________ _________________________

Record the angle measures and lengths of the sides for each of the previously viewed triangles from least to greatest. What do you notice? Share your observation with the class. Does this observation hold for all triangles? Try and draw a triangle that does not conform to your observation.

Is it possible for an equilateral triangle to be obtuse? Explain your answer.

Is it possible for a triangle to have two right angles? Justify your response.

Now consider each of the following descriptions. If the triangle is possible, draw it. If not, explain why it is not possible.

e. An isosceles obtuse triangle f. An equilateral right triangle

g. A scalene right triangle h. An isosceles acute triangle

Find the measure of the interior angles of each triangle given the measures of the three interior angles. Classify each triangle.

i. 2x, (3x – 10)º, 110º - x j. x, (x + 25)º, (x – 25)º

Finally, the ratio of the angles in a certain triangle is 2:3:4. Find the measure of the three angles, classify the triangle and then draw it. Compare your triangle with a partner. Discuss any similarities or differences.

Closing

Have students summarize what they learned today. Ask them to find and write about several real-world examples of the triangles defined in this lesson.

Notes to the Teacher

Students may need assistance reaching the conclusion that in any triangle if one side is longer than another side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side. It may be necessary to formally present and explain the Side Angle Inequality.

GHSGT Assessment (Standard #31)

Name_________________________

1. What is the measure of the solid line angle depicted by the following figure?

[pic]

a. 90º b. 180º

c. 225º d. 270º

2. Which of the following letters represents the vertex in the following picture?

[pic]

a. D and E b. F and G

c. G only d. H only

3. What is m[pic]B in the following figure if angle m[pic]A = 135°?

[pic]

a. 40º b. 45º

c. 50º d. 225º

4. Angle 1 is a complement of angle 2. If m[pic]1 = (14x+8)° and m[pic]2 = (8x-6)°, what is the value of x and of m[pic]1?

a. x = 4, m∠[pic]1 = 26° b. x = 4, m∠[pic]1 = 64° _

c. x = 113.3, m∠[pic]1 = 121.3° d. x = 113.3, m∠[pic]1 = 58.7°

5. In pentagon CDEFG m[pic]C = m[pic]E and m[pic]D = m[pic]G. What is m[pic]F?

[pic]

a. 45° b. 90°

c. 135° d. 180°

6. What is m[pic]1?

[pic]

a. 142° b. 218°

c. 82° d. 112°

7. If m[pic]DGE = 52 and [pic]FGE is a right angle, what is the measure of [pic]DGF?

[pic]

a. 38º b. 45º

c. 142º d. 71º

8. Which of the following best describes[pic]AGF?

[pic]

a. right b. straight

c. obtuse d. acute

9. If m[pic]DGE = 52 and [pic]FGE is a right angle, what is the measure of [pic]AGF?

[pic]

a. 38º b. 45º

c. 142º d. 71º

10. Which of the following best describes[pic]AGC?

[pic]

a. right b. straight

c. obtuse d. acute

11. An angle is double its complement. Find the angle.

a. 45º b. 60º

c. 90º d. 30º

12. Name a pair of angles that are adjacent and complementary.

[pic]

a. [pic]LON and[pic]QOL b. [pic]NOM and[pic]POM

c. [pic]NOM and[pic]LON d. [pic]QOP and[pic]NOP

13. Name a pair of angles that are adjacent and supplementary.

[pic]

a. [pic]LON and[pic]QOL b. [pic]NOM and[pic]POM

c. [pic]NOM and[pic]LON d. [pic]QOP and[pic]NOP

14. Determine the measure of an angle that is supplementary to [pic]SJA if m[pic]SJA = 67.

a. 113º b. 67º

c. 33º d. 83º

15. If m[pic]1 = 2x and m[pic]2 = 4x. Find the value of x if [pic]1 and [pic]2 are complementary.

a. 90º b. 60º

c. 30º d. 15º

16. Find an angle that is supplementary to [pic]AGB.

[pic]

a. [pic]FGE b. [pic]BGC

c. [pic]AGF d. [pic]BGD

17. Find m[pic]C in ∆ ABC if m[pic]A = (3x+10)° m[pic]B = (5x-5)°.

[pic]

a. 105º b. 15º

c. 55º d. 70º

18. Classify [pic] [pic]

a. acute, scalene b. acute, isosceles

c. acute, equilateral d. right, scalene

19. Which does not describe [pic] [pic]

a. obtuse b. acute

c. isosceles d. equilateral

20. Given isosceles triangle ABC find the measure of the base. [pic]

a. 15 b. 11

c. 4 d. 9

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1 8 7 6

11

12 9 10

2 5

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