Question

select the correct angle measures and side lengths for course 1, course 2, and course 3. course 1: 60°, 60°, 60°; 5 cm, 6 cm, 6 cm course 2: 90°, 45°, 45°; 6 cm, 7 cm, 8.5 cm course 3: 30°, 60°, 90°; 3 cm, 5 cm, 5 cm course 1: 60°, 60°, 60°; 6 cm, 6 cm, 6 cm course 2: 90°, 45°, 45°; 5 cm, 7 cm, 7 cm course 3: 30°, 60°, 90°; 3 cm, 4 cm, 6 cm course 1: 60°, 60°, 60°; 5 cm, 5 cm, 5 cm course 2: 90°, 45°, 45°; 6 cm, 6 cm, 8.5 cm course 3: 30°, 60°, 90°; 3 cm, 4 cm, 5 cm course 1: 90°, 45°, 45°; 5 cm, 5 cm, 7 cm course 2: 60°, 60°, 60°; 6 cm, 6 cm, 6 cm course 3: 30°, 60°, 90°; 3 cm, 4 cm, 6 cm

Explanation:

Step1: Recall triangle angle - sum property

The sum of interior angles of a triangle is 180°.

Step2: Analyze Course 1 options

For Course 1, a triangle with angles 60°, 60°, 60° is an equilateral triangle, and all sides are equal. So side - lengths 5 cm, 5 cm, 5 cm or 6 cm, 6 cm, 6 cm are valid.

Step3: Analyze Course 2 options

For Course 2, a right - isosceles triangle has angles 90°, 45°, 45°. By the Pythagorean theorem \(a^{2}+a^{2}=c^{2}\) (where \(a\) are the legs and \(c\) is the hypotenuse). If the legs are \(a = 5\) cm, then \(c=\sqrt{5^{2}+5^{2}}=\sqrt{25 + 25}=\sqrt{50}\approx7.07\) cm. So side - lengths 5 cm, 5 cm, 7 cm are approximately correct for a 90°, 45°, 45° triangle.

Step4: Analyze Course 3 options

For Course 3, a 30 - 60 - 90 triangle has side - length ratios \(1:\sqrt{3}:2\). If the shortest side (opposite 30°) is 3 cm, the side opposite 60° is \(3\sqrt{3}\approx5.2\) cm and the hypotenuse is 6 cm. So side - lengths 3 cm, 4 cm, 6 cm are incorrect, and side - lengths 3 cm, 5 cm, 6 cm are also incorrect. The correct side - lengths for a 30 - 60 - 90 triangle with shortest side 3 cm should be 3 cm, \(3\sqrt{3}\) cm, 6 cm. But among the given options, the closest correct one for angle measures 30°, 60°, 90° is considered.

Answer:

Course 1: 60°, 60°, 60°; 6 cm, 6 cm, 6 cm
Course 2: 90°, 45°, 45°; 5 cm, 5 cm, 7 cm
Course 3: 30°, 60°, 90°; 3 cm, 5 cm, 6 cm (closest valid option for given choices)