Question

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list the angle measures of δijk in order from smallest to largest. assume that a is a positive number.

m∠□ < m∠□ < m∠□

Explanation:

Step1: Recall the triangle side-angle relationship

In a triangle, the larger the side length, the larger the opposite angle. So we first identify the lengths of the sides and their opposite angles.

• Side \( IK = 30a \), opposite angle \( \angle J \)
• Side \( IJ = 33a \), opposite angle \( \angle K \)
• Side \( JK = 35a \), opposite angle \( \angle I \)

Step2: Compare the side lengths

Given \( a>0 \), we compare the coefficients: \( 30a < 33a < 35a \), so \( IK < IJ < JK \).

Step3: Determine the order of angles

By the side - angle relationship, the angle opposite the shorter side is smaller. So:

• Opposite \( IK \) (shortest side) is \( \angle J \), so \( \angle J \) is the smallest.
• Opposite \( IJ \) (middle - length side) is \( \angle K \), so \( \angle K \) is the middle - sized angle.
• Opposite \( JK \) (longest side) is \( \angle I \), so \( \angle I \) is the largest.

Answer:

\( m\angle J < m\angle K < m\angle I \)