Question

order the angle measures ( mangle v ), ( mangle g ), and ( mangle h ) from least to greatest.
(note that the figure is not drawn to scale.)
triangle with vertices v, h, g; sides: vh = 12, vg = 11, hg = 8
( mangle square < mangle square < mangle square )

Explanation:

Step1: Recall Triangle Angle-Side Relationship

In a triangle, the larger the side length, the larger the angle opposite to it. So we first identify the sides opposite each angle.

• Side opposite \( \angle V \): \( WG = 8 \)
• Side opposite \( \angle G \): \( WV = 12 \)
• Side opposite \( \angle W \): \( VG = 11 \)

Step2: Order the Sides by Length

Order the side lengths from least to greatest: \( 8 < 11 < 12 \).

Step3: Relate Sides to Opposite Angles

Since the side opposite \( \angle V \) is \( 8 \), opposite \( \angle W \) is \( 11 \), and opposite \( \angle G \) is \( 12 \), the angles opposite these sides will follow the same order. So the angles from least to greatest are \( \angle V \), \( \angle W \), \( \angle G \) (wait, no: wait, side opposite \( \angle V \) is \( WG = 8 \), so \( \angle V \) is opposite 8; side opposite \( \angle G \) is \( WV = 12 \), so \( \angle G \) is opposite 12; side opposite \( \angle W \) is \( VG = 11 \), so \( \angle W \) is opposite 11. So the order of sides: \( 8 (opposite \angle V) < 11 (opposite \angle W) < 12 (opposite \angle G) \). Therefore, the angles: \( m\angle V < m\angle W < m\angle G \). Wait, wait, let's correct:

Wait, in triangle \( VWG \):

• Side \( WG = 8 \): opposite angle \( \angle V \)
• Side \( WV = 12 \): opposite angle \( \angle G \)
• Side \( VG = 11 \): opposite angle \( \angle W \)

So side lengths: \( WG = 8 \), \( VG = 11 \), \( WV = 12 \). So \( 8 < 11 < 12 \). Therefore, the angles opposite these sides: \( \angle V \) (opposite 8), \( \angle W \) (opposite 11), \( \angle G \) (opposite 12). So the angle measures from least to greatest: \( m\angle V < m\angle W < m\angle G \).

Wait, but the question is to order \( m\angle V \), \( m\angle G \), and \( m\angle W \). So the correct order is \( m\angle V < m\angle W < m\angle G \).

Answer:

\( m\angle V < m\angle W < m\angle G \) (so filling the boxes: first box \( V \), second \( W \), third \( G \))