## Math 6 Chapter 2 Lesson 5: Draw an angle that gives the measure

## 1. Summary of theory Tóm

Given an Ox ray, how to draw the angle \(\widehat {xOy}=m^0\) with \(0< m < 180\):

– Place the protractor so that the center of the ruler coincides with the angle O of the Ox ray and the Ox ray passes through the 0 . line^{o}

– Draw an Oy ray across the m . line^{o} of the ruler.

**Comment: **On a given half-plane with an edge containing Ox rays, one and only one Oy ray can always be drawn such that: \(\widehat {xOy} = m^\circ \).

**For example: **For X-rays. Draw the angle xOy such that \(\widehat {xOy} = {40^0}\).

– Place the protractor on a half-plane with the edge containing the Ox ray so that the center of the ruler coincides with the origin O of the Ox ray and the Ox ray passes through the 0 line.^{o} of the ruler.

– Draw an Oy ray across the 40 . line^{o} of the protractor. \(\widehat {xOy}\) is the right angle to draw.

**For example: **For Ox rays. Draw two angles xOy and xOz on the same half-plane with edge Ox rays such that \(\widehat {xOy} = {30^0},\,\widehat {xOz} = {45^0}.\) In three rays Ox, Oy, Oz which ray lies between the other two rays?

– Place the protractor on a half-plane with the edge containing the Ox ray so that the center of the ruler coincides with the origin O of the Ox ray and the Ox ray passes through the 0 line.^{o} of the ruler.

– Draw an Oy ray across the 30 . line^{o} of the protractor. \(\widehat {xOy}\) is the right angle to draw.

– Draw Oz rays through the 45 . line^{o} of the protractor. \(\widehat {xOz}\) is the right angle to draw.

– Looking at the figure, we see that the ray Oy lies between two rays Ox, Oz (because \({30^0}\, < \,{45^0}\))

**Comment: **\(\widehat {xOy} = {m^0},\,\widehat {xOz} = {n^0},\) with \({m^0}\, < \,\,{n^0} \) then the ray Oy lies between the two rays Ox and Oz.

## 2. Illustrated exercise

**Question 1: **For X-rays. Draw the angle xOy such that \(\widehat {xOy} = {64^0}\)

**Solution guide**

– Place the protractor on the half-plane with the edge containing the Ox ray so that the center of the ruler coincides with the origin O of the Ox ray and the Ox ray passes through the 0 line^{o} of the ruler.

– Draw an Oy ray passing through the 64 . line^{o} of the protractor. \(\widehat {xOy}\) is the right angle to draw.

**Verse 2: **For Ou rays. Draw two angles uOt and uOv on the same half-plane whose edge contains the ray Ou such that \(\widehat {uOt} = {39^0},\,\widehat {uOv} = {129^0}.\) In three rays Ou, Ot, Ov which ray lies between the other two rays?

**Solution guide:**

– Place the protractor on a half-plane with the edge containing the Ou ray so that the center of the ruler coincides with the O origin of the Ou ray and the Ou ray passes through the 0 line.^{o} of the ruler.

– Draw an Ot ray through the line 39^{o} of the protractor. \(\widehat {uOt}\) is the right angle to draw.

– Draw an Ov ray through the line 129^{o} of the protractor. \(\widehat {uOv}\) is the right angle to draw.

Looking at the figure, we see that the ray Ot lies between two rays Ou, Ov (because \({39^0}\, < \,{129^0}\)).

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Draw consecutive shapes according to the following expressions:

a) \(\widehat {nAx} = {180^0}\)

b) \(\widehat {mAx} = {145^0}\)

c) \(\widehat {kAx} = {60^0}\), ray Ak lies in angle xAm

d) \(\widehat {nAy} = {90^0}\), ray Ay lies in angle xAm.

**Verse 2: **Draw each picture according to each of the following expressions

a) Two complementary angles xOy and yOz, with \(\widehat {xOy} = {120^0}\)

b) Two adjacent angles mOn and nOt are complementary and complementary, with \(\widehat {nOm} = {45^0}\)

c) Given ray Ap. Draw \(\widehat {qAp} = {60^0}\)

d) Given ray Bt. Draw \(\widehat {rBt} = {90^0}\)

e) Given ray Ck. Draw \(\widehat {hCk} = {30^0}\)

**Question 3:** Draw \(\widehat {mOn} = {30^0}\). Draw the angle nOp adjacent to the angle mOn. Draw the sub-angle pOq with angle mOn and the ray Oq lies in angle nOp. What is the measure of angle nOq?

### 3.2. Multiple choice exercises Bài

**Question 1: **Let \(\widehat {AOC} = {50^0},\widehat {BOC} = {60^0}\) such that \(\widehat {AOB},\widehat {BOC}\) are adjacent. Calculate the measure of angle AOC

A. 90^{0}

B. 110^{0}

C. 120^{0}

D. 110^{0}

**Verse 2: **Given two opposite rays AM, AN, \(\widehat {MAP} = {40^0},\widehat {NAQ} = {60^0}\), ray AQ lies between two rays AN and AP. Calculate the measure of angle MAQ

A. 140^{0}

B. 110^{0}

C. 120^{0}

D. 100^{0}

**Question 3: **Let \(\widehat {aOb} = {135^0}\). Ray Oc lies in angle aOb. Know \(\widehat {aOc} = \frac{1}{2}\widehat {bOc}\). Calculate angle aOc

A. 45^{0}

B. 90^{0}

C. 60^{0}

D. 30^{0}

**Question 4:** Choose the wrong statement?

A. Complementary angles are two angles whose sum is 90^{0}

B. A flat angle is an angle whose measure is 180^{0}

C. Two adjacent angles whose sum is 180^{0}

D. A right angle is an angle whose measure is 90^{0}

**Question 5: **Given the figure below. Calculate the measure of angle tOz

A. \(\widehat {tOz} = {98^0}\)

B. \(\widehat {tOz} = {88^0}\)

C. \(\widehat {tOz} = {78^0}\)

D. \(\widehat {tOz} = {68^0}\)

## 4. Conclusion

Through this lesson, you should be able to understand the following:

- Given that on a defined half-plane with edges containing Ox rays, one and only one Oy ray can always be drawn such that \(\widehat {xOy} = m^\circ \left( {0 < m^\circ < 180^\circ } \right)\). On a given half-plane with an edge containing Ox rays, if \(\widehat {xOy}\, < \,\,\widehat {xOz}\) then the ray Oy lies between two rays Ox, Oz.
- Draw a numbered angle.

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