Question

1. list the sides of $\triangle fgh$ in order from least to greatest if $m\angle f = (5x + 6)\degree$, $m\angle g = (12x - 4)\degree$, and $m\angle h = (4x + 31)\degree$.

1. list the sides of $\triangle stu$ in order from least to greatest if $m\angle s = (7x - 36)\degree$, $m\angle t = (5x - 1)\degree$, and $m\angle u = (x + 9)\degree$.

1. list the angles of $\triangle abc$ in order from least to greatest if $ab = 6x - 35$, $bc = 4x + 11$, $ac = x + 29$, and the perimeter of $\triangle abc = 192$.

1. list the angles of $\triangle wxy$ in order from least to greatest if $wx = 9x - 7$, $xy = 12x - 1$, $wy = 3x + 4$, and the perimeter of $\triangle wxy = 68$.

Explanation:

Problem 5

Step1: Sum angles to 180°

$(5x+6)+(12x-4)+(4x+31)=180$

Step2: Simplify to solve for x

$21x + 33 = 180$
$21x = 147$
$x=7$

Step3: Calculate each angle

$m\angle F=(5(7)+6)=41^\circ$
$m\angle G=(12(7)-4)=80^\circ$
$m\angle H=(4(7)+31)=59^\circ$

Step4: Match sides to opposite angles

Side opposite $\angle F$: $GH$; opposite $\angle G$: $FH$; opposite $\angle H$: $FG$
Order angles: $41^\circ < 59^\circ < 80^\circ$, so sides follow.

Step1: Sum angles to 180°

$(7x-36)+(5x-1)+(x+9)=180$

Step2: Simplify to solve for x

$13x - 28 = 180$
$13x=208$
$x=16$

Step3: Calculate each angle

$m\angle S=(7(16)-36)=76^\circ$
$m\angle T=(5(16)-1)=79^\circ$
$m\angle U=(16+9)=25^\circ$

Step4: Match sides to opposite angles

Side opposite $\angle S$: $TU$; opposite $\angle T$: $SU$; opposite $\angle U$: $ST$
Order angles: $25^\circ < 76^\circ < 79^\circ$, so sides follow.

Step1: Use perimeter to solve x

$(6x-35)+(4x+11)+(x+29)=192$

Step2: Simplify and solve for x

$11x + 5 = 192$
$11x=187$
$x=17$

Step3: Calculate each side length

$AB=(6(17)-35)=67$
$BC=(4(17)+11)=79$
$AC=(17+29)=46$

Step4: Match angles to opposite sides

Angle opposite $AB$: $\angle C$; opposite $BC$: $\angle A$; opposite $AC$: $\angle B$
Order sides: $46 < 67 < 79$, so angles follow.

Answer:

$GH < FG < FH$

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Problem 6