Question

Angles f and g are complementary angles. Angles g and h are supplementary angles. The degree measure of each angle is a whole number. What is the smallest possible measure of angle h.
A)1*
B)89*
C)91*
D)179*

Answer 1

The correct answer is:  [C]:  " m∠H  = 91 °  "  .
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Explanation:
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Note:
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m∠F + m∠G = 90 ;
{since 2 complementary angles, by definition, add up to 90°.}.

m∠G + m∠H = 180 ; 
{since 2 supplementary angles, by definition, add up to 180°.}
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We are asked to find the smallest value of the " m∠H "
    (among the given answer choices): 
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Note:
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m∠G = 180 −  m∠H ; 

m∠H = 180 −  m∠G ;

m∠F  = 90  −  m∠G ;

m∠G = 90  −  m∠F ;

m∠G = 180  − m∠H ;

m∠H = 180  − m∠G .
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The question is:
        "What is the smallest value of " m∠H "; in whole number, among the answer choices given?" ; 
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Note:  Consider each of the answer choices given:
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 Choice: [A]:  " m∠H = 1 " ;  

{Note, This value is SMALLEST value among ALL the answer choices.}.  

If " m∠H = 1" ;  then " m∠G = 179 ", since:  " m∠G + m∠H = 180".  
 
Then, could "m∠F + m∠G = 90" ??  NO!  Because, if "m∠G = 179" ;
then m∠F would have to equal a "negative number" to get:  

 " m∠F + m∠G = 90 "  ; 

So, "Choice [A]:  " m∠H = 1 " ; is incorrect.
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 Choice  [B]:  " m∠H = 89 " ;

    If "m∠H = 89" ;  then "m∠G = 91", since:  "m∠G + m∠H = 180".  

 {Note:  "180 − m∠H = m∠G " ;  → "180 − 89 = m∠G ;
                                                    → "m∠G = 91" .}.

Then,  could "m∠F + m∠G = 90" ??  NO!   Because, if "m∠G = 91" ; then "m∠F " would have to equal a "negative number" to get:  "m∠F + m∠G = 90" ;

So; "Choice: [B]:  "m∠H = 89" ; is incorrect.
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 Choice  [C]:  " m∠H = 91 " ;

    If "m∠H = 91" ;  then "m∠G = 89", since:  "m∠G + m∠H = 180".  

 {Note:  "180 − m∠H = m∠G " ;  → "180 − 91 = m∠G ;  → "m∠G = 89" .}.

Then,  could "m∠F + m∠G = 90" ??  YES!   Because, if "m∠G = 89" ; then
 "m∠F" COULD equal "1" ;  and in such a case;  "m∠F + m∠G = 1 + 89 = 90."

So;  Choice:  [C]:  " m∠H = 91 " ; is a possible correct answer.
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Let us try the last answer choice:
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Choice  [D]:  " m∠H = 179 " ;

    If "m∠H = 179" ;  then "m∠G = 1", since:  "m∠G + m∠H = 180".  

 {Note:  "180 − m∠H = m∠G " ;  → "180 − 179 = m∠G ;  → "m∠G = 1" .}.

Then,  would "m∠F + m∠G = 90" ??  Yes!   Because, if "m∠G = 1" ; then "m∠F"  would equal "89";

  {Note:   "m∠F + m∠G = 89 + 1 = 90 " .  

 {Note:  "90  − m∠G = m∠F " ;  → "90 − 1 = m∠F " ; 
                                                  →  m∠F  = 89° . }.

→  So; "Choice:  [D]:  " m∠H = 179 " ; is a possible correct answer.
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Note:  The question asks:
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"What is the smallest possible measure of "angle H"  {" m∠H "} ?
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The 2 (TWO) possible correct answers are: 
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  Choice [C]:  " m∠H "= 91 " ;

and Choice [D]:  " m∠H = 179 " .
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The smallest possible " m∠H"  is:
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 Answer choice:  [C]:  " m∠H  = 91° " .
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