Question

1. order the angles of △ghi from smallest to largest.

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the side length, the larger the angle opposite to it. This is based on the principle that in a triangle, the angle opposite the longer side is greater. Mathematically, if in \(\triangle ABC\), \(a\) is opposite \(\angle A\), \(b\) is opposite \(\angle B\), and \(c\) is opposite \(\angle C\), then if \(a > b>c\), \(\angle A>\angle B >\angle C\).

Step2: Identify the side lengths of \(\triangle GHI\)

From the diagram, the side lengths are: \(GH = 4\space cm\), \(HI=5\space cm\), and \(GI = 6\space cm\).

Step3: Identify the angles opposite to each side

• The angle opposite \(GH\) is \(\angle I\) (since \(GH\) is between \(G\) and \(H\), the angle opposite is at \(I\)).
• The angle opposite \(HI\) is \(\angle G\) (since \(HI\) is between \(H\) and \(I\), the angle opposite is at \(G\)).
• The angle opposite \(GI\) is \(\angle H\) (since \(GI\) is between \(G\) and \(I\), the angle opposite is at \(H\)).

Step4: Order the sides by length

We have \(GH=4\space cm< HI = 5\space cm

Step5: Order the angles based on the opposite sides

Since the side opposite \(\angle I\) (\(GH\)) is the shortest, \(\angle I\) is the smallest. The side opposite \(\angle G\) (\(HI\)) is of medium length, so \(\angle G\) is of medium size. The side opposite \(\angle H\) (\(GI\)) is the longest, so \(\angle H\) is the largest. So the order of angles from smallest to largest is \(\angle I<\angle G <\angle H\).

Answer:

\(\angle I\), \(\angle G\), \(\angle H\)