Question

Use the side lengths of △ABC to order its angle measures in increasing order if the perimeter of the triangle is 60 units.(1 point)Responsesm∠C<m∠A<m∠Bm angle upper C less than m angle upper A less than m angle upper Bm∠B<m∠A<m∠Cm angle upper B less than m angle upper A less than m angle upper Cm∠A<m∠B<m∠Cm angle upper A less than m angle upper B less than m angle upper Cm∠A<m∠C<m∠B

Answer 1

In any triangle, the relationship between the lengths of the sides and the measures of the angles is given by the fact that the larger the side length, the larger the opposite angle.

Let's denote the side lengths of triangle \( \triangle ABC \) as \( a \), \( b \), and \( c \) opposite to angles \( A \), \( B \), and \( C \) respectively.

Given that the perimeter of the triangle is 60 units, we have \( a + b + c = 60 \).

To determine the order of the angles based on the side lengths, we need to consider the comparative size of \( a \), \( b \), and \( c \):

1. If \( a < b < c \), then \( m\angle A < m\angle B < m\angle C \).
2. If \( a < c < b \), then \( m\angle A < m\angle C < m\angle B \).
3. If \( b < a < c \), then \( m\angle B < m\angle A < m\angle C \).
4. If \( b < c < a \), then \( m\angle B < m\angle C < m\angle A \).
5. If \( c < a < b \), then \( m\angle C < m\angle A < m\angle B \).
6. If \( c < b < a \), then \( m\angle C < m\angle B < m\angle A \).

Without loss of generality, if we're given that \( a \), \( b \), and \( c \) are determined such that \( a < b < c \) (which we can assume for the sake of the ordering), we have:

• \( m\angle C \) (opposite the longest side \( c \)) is the largest.
• \( m\angle B \) (opposite the middle side \( b \)) is in the middle.
• \( m\angle A \) (opposite the shortest side \( a \)) is the smallest.

Therefore, the correct order in increasing measures of angles is:

\( m\angle A < m\angle B < m\angle C \).

So, the response is:

m∠A < m∠B < m∠C.