Question

ILL GIVE BRAINLIST 16) Find m

Answer 1

The answer is:  " $m$∠$TPQ$ = 124°  " .

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Step-by-step explanation:

We are asked to find: "m" (the "measurement");

specifically, " $m$∠$TPQ$ "  ;

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 →  which, from the diagram given:

         is represented by "(11x + 14)" .

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 → If we can solve for the value of "x" ; then we can solve for:

   "(11x + 14)" ;  i.e. " $m$∠$TPQ$ " .

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Let us consider " ∡$QPR$ " ;

    → which is the "supplementary angle" to:  " ∡$TPQ$ " :

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That is:  " $m$∠$TPQ$ + $m$∠$QPR$  =  180° " .

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Note:  By definition, "supplementary angles" add up to: "180° " ;

                              → even if multiple angles are involved.

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Note:  "∡$TPQ$ " and " ∡$QPR$ " —

                        together — form a "straight line" ;

                        →  which means that the 2 (two) angles are                    

                            "supplementary" ; and:

                        →  which means that the:  sum of the measurements of                        

                             the  " 2 (two) angles " —equal:  " 180° .

   {Note:  by "forming a straight line" ;  for this purpose, this criterion also is satisfied by forming a "straight line on a line segment" —even if that "line segment" is actually:

   1)  an actual "line segment" ;  or:  

   2)  a portion of a "line segment" ;

   3)  a "line segment" that is actual part of a "true geometry line" ; or

                                                                    "[geometric] ray".}

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So: We have:

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→  " $m$∠$TPQ$  +  $m$∠$QPR$  = 180 " ;

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Given:  " $m$∠$TPQ$ = (11x + 14) " ;

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Plug in this value for:  " $m$∠$TPQ$ " ;

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→  " (11x + 14)   +  $m$∠$QPR$  = 180 " ;   Solve for: " $m$∠$QPR$ " :

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→  Subtract:  "(11x + 14)" ; from Each Side of the equation:

    to isolate:  " $m$∠$QPR$ " ;  on one side of the equation;

     & to solve for:  " $m$∠$QPR$ " ; in terms of "x" :

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→  " (11x + 14)   +  $m$∠$QPR$  − (11x + 14) = 180 − (11x + 14) " ;

Note:  On the "left-hand side" of the equation:

 The:  "(11x + 14)" 's  cancel out to "0" ;

    {since:  (11x + 14)  −  (11x + 14) = 0 ;

                → {i.e. any value, minus that same value, equals: "zero".}.

 →  And we have:  " $m$∠$QPR$ =  180 − (11x + 14) " ;

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Note:  " 180 − (11x + 14) " ;

  ↔  Treat as:

         " 180 − 1 (11x + 14) " ;

             → {since multiplying by "1" results in the same value.}.

Consider the following portion:

        "  − 1 (11x + 14) " ;

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Note the "distributive property" of multiplication:

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            →  $a(b +c) = ab + ac$  ;

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Likewise:  " − 1 (11x + 14)  = (-1*11x) + (-1 *14) ;

                                    = (-11x)  + ( -14) '  

                                    =  - 11x − 14 ;  

        {Since:  "Adding a negative" results in the same value as:

                    "Subtracting a positive."}.

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Now, bring down the "180" ; and rewrite the expression:

     →  " 180 − 11x − 14  " ;

          →  Combine the "like terms" :

                 + 180 − 14  =  + 166 ;

Rewrite the expression as:

     →  " 166 − 11x " ;

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Now, we can rewrite the entire equation:

  " $m$∠$QPR$  =  166 − 11x " ;

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Now, consider the triangle:  ΔQRP ;

with its 3 (three) sides—as shown in the image attached:

Note:

By definition:

All triangles:

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1) have 3 (three) sides;

2) have 3 (three) angles;  and:

3) have angles in which the sum of the measurements of those angles add up to 180°.

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So:  For ΔQRP ; which is shown in the image attached:

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Let us consider the measurements of Each of the 3 (three) angles of that triangle:

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1)  m∡Q = "(5x + 18)" ; (given);

2)  m∡R  = " 56 " ; (given) ;

3)  m∡P —[within the triangle] = "(166 − 11x)" ; (calculated above}.

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We want to find the value for "x" ;

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So:  since all triangles, by definition; have 3 (three) angles with measurements that add up to 180° ;

 → Let us add up the measurements of each of the 3 (three) angles of:

  ΔQRP ;  and make an equation by setting this sum "equal to:  180 ."

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   →  " m∡Q  + m∡R + m∡P = 180 " ;

   →  (5x + 18) + 56) + (166 - 11x) = 180 ;

   →  5x + 18 + 56 + 166 − 11x = 180 ;

On the "left-hand side" of the equation:

 Combine the "like terms" to simplify further:

         +5x − 11x  =  − 6x ;

         + 18 + 56 + 155 = 240 ;

And rewrite the equation:

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  - 6x + 240 − 240 = 180 − 240 ;

to get:

  - 6x   =  - 60 ;  

Now, divide Each side of the equation by:  "( -6)" ;

 to isolate: "x" on one side of the equation;

    & to solve for "x" ;

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   - 6x / 6    =  - 60 / -6  ;

to get:

     "  x  = 10 " .

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Now, the question asks for:

" the measurement for angle TPQ ";

  → {that is;   " $m$∠$TPQ$ " } ;

  →  which is:  " (11x + 14) " ;

Since we know that:  " x = 10 " ;

 We can plug in our "10" as our value for "x" ; and solve accordingly:

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 →  " $m$∠$TPQ$ = (11x + 14) =  (11*10) + 14 = 110 + 14 = 124 .

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The answer is:  " $m$∠$TPQ$  =  124°  " .    

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Hope this answer—and explanation—is helpful!

  Best wishes!

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