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Answer:

The following items are the answers to each item:

1. m\angle{LAK}=90^om∠LAK=90o

2. m\angle{KAJ}=48^om∠KAJ=48o

3. m\angle{LAJ}=138^om∠LAJ=138o

4. m\angle{JAH}=42^om∠JAH=42o

5. m\angle{KAM}=132^om∠KAM=132o

6. \begin{gathered}m\stackrel{\frown}{LK}=90^o\\\end{gathered}mLK⌢=90o

7. m\stackrel{\frown}{JK}=48^omJK⌢=48o

8. m\stackrel{\frown}{LMG}=150^omLMG⌢=150o

9. m\stackrel{\frown}{JH}=42^omJH⌢=42o

10. m\stackrel{\frown}{KLM}=132^omKLM⌢=132o

Step-by-step explanation:

A Central angle is an angle whose vertex is at the center of a circle. Remember that the measurement of the arc intercepted by the central angle is the same as the degree measure of the said angle. Since a circle is a complete 360 degrees, then this means that a semicircle is 180 degrees. Also, remember that an L shape angle measures 90 degrees.

Some Angles in Circle

1. Inscribe angles.

2. Angles subtended by the diameter.

3. Tangent Angle

4. Vertical Angles

List Down the Known Items

1. m\angle{LAM}=42^om∠LAM=42o

2. \begin{gathered}m\angle{HAG}=30^o\\\end{gathered}m∠HAG=30o

3. m\angle{KAH}=90^om∠KAH=90o , since it's a right angle.

Answers and explanation of the following items:

1. Since m\angle{KAH}=90^om∠KAH=90o , then m\angle{LAK}=90^om∠LAK=90o because \overline{LH}LH is a diameter which means that \angle{LAK}∠LAK and \angle{KAH}∠KAH are supplementary, and supplementary angles have a sum of 180^o180o .

2. We already know that \begin{gathered}m\angle{LAM}=42^o\\\end{gathered}m∠LAM=42o and from number one that m\angle{LAK}=90^om∠LAK=90o , so since \overline{JM}JM is a diameter, then m\angle{KAJ}=180^o-90^o-42^o=48^om∠KAJ=180o−90o−42o=48o .

3. Note that m\angle{LAJ}=m\angle{LAK}+m\angle{KAJ}m∠LAJ=m∠LAK+m∠KAJ . Then

m\angle{LAJ}=90^o+48^o=138^om∠LAJ=90o+48o=138o .

4. \angle{LAM}∠LAM and \angle{JAH}∠JAH are vertical angles and the measurement of vertical angles are the same, so m\angle{JAH}=42^om∠JAH=42o .

5. Note that m\angle{KAM}=m\angle{LAM}+m\angle{LAK}m∠KAM=m∠LAM+m∠LAK . Then m\angle{KAM}=42^o+90^o=132^om∠KAM=42o+90o=132o .

Remember that the arc intercepted by the central angle has the same measure with the central angle.

6. \stackrel{\frown}{LK}LK⌢ corresponds to the central angle \angle{LAK}∠LAK , thus m\stackrel{\frown}{LK}=90^omLK⌢=90o .

7. \stackrel{\frown}{JK}JK⌢ corresponds to the central angle \angle{KAJ}∠KAJ , thus m\stackrel{\frown}{JK}=48^omJK⌢=48o .

8. \stackrel{\frown}{LMG}LMG⌢ corresponds to the central angle \angle{LAG}∠LAG and m\angle{LAG}=m\angle{LAM}+m\angle{MAG}m∠LAG=m∠LA