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, we can use the Law of Sines (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)) or check the sum of angles.

Step 1: Check the sum of angles

The sum of the interior angles of a triangle is \(180^{\circ}\). Let's add the given angles: \(62^{\circ}+72^{\circ}+46^{\circ}=180^{\circ}\). So the angle sum property of a triangle is satisfied.

Step 2: Apply the Law of Sines (optional, for angle - side correspondence)

Let \(a = 23\space mm\), \(A=72^{\circ}\); \(b = 20\space mm\), \(B = 62^{\circ}\); \(c=13\space mm\), \(C = 46^{\circ}\)

Calculate \(\frac{a}{\sin A}=\frac{23}{\sin72^{\circ}}\approx\frac{23}{0.9511}\approx24.18\)

\(\frac{b}{\sin B}=\frac{20}{\sin62^{\circ}}\approx\frac{20}{0.8829}\approx22.65\) (There is a small difference due to rounding. If we use more precise values of sines, the ratios will be closer. The slight discrepancy can also be due to measurement errors in the given side and angle values)

\(\frac{c}{\sin C}=\frac{13}{\sin46^{\circ}}\approx\frac{13}{0.7193}\approx18.09\) (Again, rounding affects the result. With more precise calculations, we can get a better approximation of the ratio)

If we assume that the triangle is a scalene triangle (since all sides \(23\space mm\), \(20\space mm\), \(13\space mm\) are of different lengths and all angles \(62^{\circ}\), \(72^{\circ}\), \(46^{\circ}\) are of different measures), we can confirm the angle - side relationship: the longest side (\(23\space mm\)) is opposite the largest angle (\(72^{\circ}\)), the middle - length side (\(20\space mm\)) is opposite the middle - sized angle (\(62^{\circ}\)) and the shortest side (\(13\space mm\)) is opposite the smallest angle (\(46^{\circ}\)), which is consistent with the properties of a triangle (longer side opposite larger angle).

If the question was to classify the triangle:

• By side: Scalene (all sides of different lengths: \(23

eq20
eq13\))

• By angle: Acute (all angles are less than \(90^{\circ}\): \(62^{\circ}<90^{\circ}\), \(72^{\circ}<90^{\circ}\), \(46^{\circ}<90^{\circ}\))

If the question was to verify the triangle - angle sum or side - angle relationship, we have shown that the angle sum is \(180^{\circ}\) and the side - angle relationship (longer side opposite larger angle) holds approximately.

Answer:

To determine the type of triangle or verify angle - side relationships, we can use the Law of Sines (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)) or check the sum of angles.

Step 1: Check the sum of angles

The sum of the interior angles of a triangle is \(180^{\circ}\). Let's add the given angles: \(62^{\circ}+72^{\circ}+46^{\circ}=180^{\circ}\). So the angle sum property of a triangle is satisfied.

Step 2: Apply the Law of Sines (optional, for angle - side correspondence)

Let \(a = 23\space mm\), \(A=72^{\circ}\); \(b = 20\space mm\), \(B = 62^{\circ}\); \(c=13\space mm\), \(C = 46^{\circ}\)

Calculate \(\frac{a}{\sin A}=\frac{23}{\sin72^{\circ}}\approx\frac{23}{0.9511}\approx24.18\)

\(\frac{b}{\sin B}=\frac{20}{\sin62^{\circ}}\approx\frac{20}{0.8829}\approx22.65\) (There is a small difference due to rounding. If we use more precise values of sines, the ratios will be closer. The slight discrepancy can also be due to measurement errors in the given side and angle values)

\(\frac{c}{\sin C}=\frac{13}{\sin46^{\circ}}\approx\frac{13}{0.7193}\approx18.09\) (Again, rounding affects the result. With more precise calculations, we can get a better approximation of the ratio)

If we assume that the triangle is a scalene triangle (since all sides \(23\space mm\), \(20\space mm\), \(13\space mm\) are of different lengths and all angles \(62^{\circ}\), \(72^{\circ}\), \(46^{\circ}\) are of different measures), we can confirm the angle - side relationship: the longest side (\(23\space mm\)) is opposite the largest angle (\(72^{\circ}\)), the middle - length side (\(20\space mm\)) is opposite the middle - sized angle (\(62^{\circ}\)) and the shortest side (\(13\space mm\)) is opposite the smallest angle (\(46^{\circ}\)), which is consistent with the properties of a triangle (longer side opposite larger angle).

If the question was to classify the triangle:

• By side: Scalene (all sides of different lengths: \(23

eq20
eq13\))

• By angle: Acute (all angles are less than \(90^{\circ}\): \(62^{\circ}<90^{\circ}\), \(72^{\circ}<90^{\circ}\), \(46^{\circ}<90^{\circ}\))

If the question was to verify the triangle - angle sum or side - angle relationship, we have shown that the angle sum is \(180^{\circ}\) and the side - angle relationship (longer side opposite larger angle) holds approximately.