Question

hw 7: relationships in triangles
list the sides of \\( \triangle f g h \\) in order from least to greatest if \\( m \angle f = 5 x + 6, m \angle g = 12 x - 4 \\), and \\( m \angle h = 4 x + 31 \\).
\\( \square, \square, \square \\)
\\( \\# g h \\) \\( \\# f g \\) \\( \\# f h \\)

Explanation:

Step1: Find the value of \( x \)

The sum of the interior angles of a triangle is \( 180^\circ \). So we set up the equation:
\( (5x + 6) + (12x - 4) + (4x + 31) = 180 \)
Combine like terms:
\( 5x + 12x + 4x + 6 - 4 + 31 = 180 \)
\( 21x + 33 = 180 \)
Subtract 33 from both sides:
\( 21x = 180 - 33 \)
\( 21x = 147 \)
Divide both sides by 21:
\( x = \frac{147}{21} = 7 \)

Step2: Calculate each angle

• For \( \angle F \): \( 5x + 6 = 5(7) + 6 = 35 + 6 = 41^\circ \)
• For \( \angle G \): \( 12x - 4 = 12(7) - 4 = 84 - 4 = 80^\circ \)
• For \( \angle H \): \( 4x + 31 = 4(7) + 31 = 28 + 31 = 59^\circ \)

Step3: Order the angles from least to greatest

\( \angle F = 41^\circ \), \( \angle H = 59^\circ \), \( \angle G = 80^\circ \)
So the order of angles is \( \angle F < \angle H < \angle G \)

Step4: Relate angles to sides

In a triangle, the larger the angle, the longer the side opposite to it.

• Side opposite \( \angle F \) is \( GH \)
• Side opposite \( \angle H \) is \( FG \)
• Side opposite \( \angle G \) is \( FH \)

Since \( \angle F < \angle H < \angle G \), the sides opposite them will be in the same order (least to greatest): \( GH < FG < FH \)

Answer:

GH, FG, FH