Question

1. sides: 23 mm, 20 mm, 13 mm angles: 62°, 72°, 46°

Explanation:

To determine the type of triangle (or verify angle - side relationships, perhaps for triangle classification or using the Law of Sines/Cosines), we can start by checking the sum of the angles.

Step 1: Check the sum of the interior angles of a triangle

The sum of the interior angles of any triangle should be \(180^{\circ}\).
We add the given angles: \(62^{\circ}+72^{\circ}+46^{\circ}\)

$$62 + 72+46=180^{\circ}$$

This satisfies the angle - sum property of a triangle.

If we want to check the side - angle relationships (for example, using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a,b,c\) are the side lengths and \(A,B,C\) are the opposite angles):

Let \(a = 23\space mm\), \(b = 20\space mm\), \(c = 13\space mm\) and \(A = 72^{\circ}\), \(B=62^{\circ}\), \(C = 46^{\circ}\) (we assume the largest side is opposite the largest angle, the second - largest side opposite the second - largest angle, and the smallest side opposite the smallest angle)

Step 2: Calculate \(\frac{a}{\sin A}\)

\(\sin(72^{\circ})\approx0.9511\)
\(\frac{23}{\sin(72^{\circ})}=\frac{23}{0.9511}\approx24.18\)

Step 3: Calculate \(\frac{b}{\sin B}\)

\(\sin(62^{\circ})\approx0.8829\)
\(\frac{20}{\sin(62^{\circ})}=\frac{20}{0.8829}\approx22.65\) (There is a small discrepancy, probably due to rounding of sine values or approximate side - angle matching. If we consider more precise calculations or possible measurement errors, the Law of Sines should hold approximately for a triangle)

Step 4: Calculate \(\frac{c}{\sin C}\)

\(\sin(46^{\circ})\approx0.7193\)
\(\frac{13}{\sin(46^{\circ})}=\frac{13}{0.7193}\approx18.09\) (Again, the differences are due to rounding. In a real - world or more precise mathematical context, with more accurate values of sine and side - length measurements, the ratios should be more consistent)

If the problem is to classify the triangle:

• Since all angles are less than \(90^{\circ}\) (acute angles) and all sides have different lengths (scalene, as \(23

eq20
eq13\)), the triangle is an acute scalene triangle.

If the problem was to verify the angle - side combination for a triangle, we have shown that the angle - sum property holds, and the side - angle relationships (via Law of Sines) hold approximately (considering rounding errors).

If we assume the problem is to confirm the triangle's validity (angle - sum and possible side - angle consistency):

Answer:

The given side - angle combination represents a valid triangle (since the sum of angles is \(180^{\circ}\) and the side - angle relationships follow the Law of Sines approximately). The triangle is an acute scalene triangle (all angles acute, all sides of different lengths).