## Math 6 Chapter 2 Lesson 4: When is angle xOy + angle yOz= angle xOz?

## 1. Summary of theory Tóm

### 1.1. When is the sum of the measures of two angles xOy and yOz equal to the measure of the angle xOz?

If the ray Oy lies between the two rays Ox and Oz then \(\widehat {xOy}\,\, + \,\widehat {yOz\,}\, = \,\widehat {xOz}\)

Conversely, if \(\widehat {xOy}\,\, + \,\widehat {yOz\,}\, = \,\widehat {xOz}\) then the ray Oy lies between the two rays Ox, Oz.

### 1.2. Two angles are adjacent, supplementary, complementary, adjacency

**– Two adjacent angles** are two angles with a common edge and the other two sides lying on opposite half-planes with adjacent edges.

**For example:** \(\widehat {xOy}\) and \(\widehat {yOz}\) are two adjacent angles, the common side is Oy.

**– Two complementary angles **are two angles whose sum is equal to \({90^0}\)

**For example: **angle \({35^0}\) and angle \({55^0}\) are complementary angles because \({35^0} +{55^0}=90^0\).

**– Two complementary angles** are two angles whose sum is equal to \({180^0}\)

**For example:** \({115^0}\) and angle \({65^0}\) are complementary angles because \({115^0}+{65^0}=180^0\)

**– Two offset corners** are two angles that are both adjacent and complementary.

**For example: **The two angles xOy and yOz in the figure are complementary because they have a common side Oy and two sides Ox and Oz are opposite rays.

## 2. Illustrated exercise

**Question 1: **Let Oz be the ray lying between two rays Ox and Oy. Knowing \(\widehat {xOy}\, = \,{a^0},\,\widehat {zOx}\, = \,{b^0}.\) Calculate \(\widehat {yOz}\)?

**Solution guide**

\(\widehat {yOz} = \widehat {xOy} – \widehat {zOx}\, = \,{a^0}\, – \,{b^0}.\)

**Verse 2: **For pictures

a) Name the pairs of angles that are adjacent to the vertex O in that figure.

b) Give the measure of the vertices O in the figure.

c) Indicate the pairs of complementary angles O đỉnh

d) Indicate the pairs of complementary angles vertex O.

**Solution guide**

a) The pairs of adjacent angles at O are: \(\widehat {mOn}\) and \(\widehat {nOw}\); \(\widehat {mOn}\) and \(\widehat {nOz}\); \(\widehat {mOn}\) and \(\widehat {nOt}\); \(\widehat {mOw}\) and \(\widehat {zOw}\); \(\widehat {mOw}\) and \(\widehat {tOw}\); \(\widehat {mOz}\) and \(\widehat {zOt}\); \(\widehat {wOn}\) and \(\widehat {zOw}\); \(\widehat {wOn}\) and \(\widehat {tOw}\); \(\widehat {wOz}\) and \(\widehat {zOt}\).

b) \(\widehat {mOt} = {180^0};\,\widehat {mO{\rm{w}}}\, = {90^0};\,\widehat {nO{\rm{w }}}\, = {60^0};\,\widehat {{\rm{w}}Oz} = {45^0}\)

c) \(\widehat {mOn}\) and \(\widehat {nOw}\); \(\widehat {wOz}\) and \(\widehat {zOt}\)

d) \(\widehat {mOn}\) and \(\widehat {nOt}\); \(\widehat {wOm}\) and \(\widehat {wOt}\); \(\widehat {mOz}\) and \(\widehat {zOt}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **In the figure, two rays OI, OK are opposite each other. Ray OI intersects line segment AB at I. Know \(\widehat {KOA}\, = \,\,{120^0},\,\widehat {BOI}\, = \,{45^0}.\) Calculate \(\widehat {KOB},\,\widehat {AOI},\,\widehat {BOA}.\)

**Verse 2: ** See the figure, how to measure only two angles and know the measure of all three angles \(\widehat {xOy}\), \(\widehat {xOz}\), \(\widehat {yOz}\)?

**Question 3:** On the line d from left to right, we take the points A, D, C, B and take the point O outside the line d. Know \(\widehat {AOD} = {30^0},\,\widehat {DOC} = {40^0},\,\widehat {AOB}\, = \,{90^0}.\) Calculate \(\widehat {AOC},\,\widehat {COB}.\)

### 3.2. Multiple choice exercises Bài

**Question 1: **Two complementary angles are two angles:

A. Has a total measurement of 180^{0}

B. Share a ray and have a total measure of 180^{0}

C. Adjacent to each other and has a total measurement of 180^{0}

D. Shares a side and has a total measure of 180^{0}

**Verse 2: **Calculate angle yOz on the figure:

A. 32^{0}

B. 70^{0}

C. 38^{0}

D. 60^{0}

**Question 3: **Let angle A and angle B be complementary angles and they have the same measure. Calculate the measure of each angle

A. \(\widehat A = {30^0},\widehat B = {60^0}\)

B. \(\widehat A = \widehat B = {40^0}\)

C. \(\widehat A = \widehat B = {45^0}\)

D. \(\widehat A = \widehat B = {55^0}\)

**Question 4: **Let angle xOy and angle yOy’ be complementary adjacent angles. Given \(\widehat {xOy} = {80^0}\), the measure of angle yOy’ is:

A. 100^{0}

B. 70^{0}

C. 80^{0}

D. 60^{0}

**Question 5: **Given the figure, know that the ray On lies between the two rays Ot and Om .

The measure of angle tOm is:

A. 105^{0}

B. 100^{0}

C. 115^{0}

D. 95^{0}

## 4. Conclusion

Through this lesson, you should understand the following main topics:

- Identify pairs of adjacent, complementary, complementary, and adjacent angles.
- Know how to apply the relation to solve simple unknown angle problems.

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