Question

list the side lengths of $\triangle abc$ in order from shortest to longest.
$a$ angle: $z + 72^\circ$
$b$ angle: $46z$
$c$ angle: $z + 60^\circ$
$\square < \square < \square$

Explanation:

Step1: Find the value of \( z \)

The sum of the interior angles of a triangle is \( 180^\circ \). So we have the equation:
\[
(z + 72^\circ) + 46z + (z + 60^\circ) = 180^\circ
\]
Combine like terms:
\[
z + 72 + 46z + z + 60 = 180
\]
\[
48z + 132 = 180
\]
Subtract 132 from both sides:
\[
48z = 180 - 132
\]
\[
48z = 48
\]
Divide both sides by 48:
\[
z = 1
\]

Step2: Calculate each angle

• Angle \( A \): \( z + 72^\circ = 1 + 72^\circ = 73^\circ \)
• Angle \( B \): \( 46z = 46 \times 1 = 46^\circ \)
• Angle \( C \): \( z + 60^\circ = 1 + 60^\circ = 61^\circ \)

Step3: Order the angles from smallest to largest

The angles are \( 46^\circ \) (Angle \( B \)), \( 61^\circ \) (Angle \( C \)), \( 73^\circ \) (Angle \( A \)).

Step4: Relate angles to sides (Larger angle opposite longer side)

In a triangle, the longer side is opposite the larger angle. So:

• Side opposite Angle \( B \) ( \( 46^\circ \)) is \( AC \)
• Side opposite Angle \( C \) ( \( 61^\circ \)) is \( AB \)
• Side opposite Angle \( A \) ( \( 73^\circ \)) is \( BC \)

So the order of sides from shortest to longest is \( AC \), \( AB \), \( BC \) (or using the angle - side relationship: side opposite smaller angle is shorter. So since \( \angle B < \angle C < \angle A \), the sides opposite are \( AC < AB < BC \))

Answer:

\( AC < AB < BC \) (or depending on the labeling of the triangle's sides, if we consider the sides as \( BC \), \( AC \), \( AB \) might be mislabeled, but following the angle - side: angle \( B = 46^\circ \), angle \( C = 61^\circ \), angle \( A = 73^\circ \), so sides opposite: \( AC \) (opposite \( B \)), \( AB \) (opposite \( C \)), \( BC \) (opposite \( A \)) so \( AC < AB < BC \))