Applying the triangle inequality theorem and the exterior angle theorem, we have:
1. m∠1 > m∠3
2. MI > IX
3. 3 < MX < 27
4. m∠4 = 115°
What is the Triangle Inequality Theorem?
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side of that triangle.
What is the Relationship of Sides to Interior Angles in a Triangle?
The longest side is opposite to the largest interior angle while the shortest side of the triangle is always directly opposite the shortest interior angle in that triangle.
What is the Exterior Angle Theorem?
The exterior angle theorem states that the sum of the opposite angles to an exterior angle in a triangle equals the measure of that exterior angle.
1. The angles of a triangle are relative to the length of their opposite sides. Therefore, if MI which is relative to ∠1 is 15 and IX which is relative to ∠3 is 12, it implies that m∠1 is greater than m∠3.
2. Given, m∠1 = 65° and is relative to side MI, and m∠3 = 50° and is relative to side IX, therefore, side MI is longer or greater than side IX.
3. Given, MI = 15; IX = 12, based on the triangle inequality theorem, the measure of MX would be determined as shown below:
15 - 12 < MX < 15 + 12
3 < MX < 27
4. Given, m∠2 = 65°; m∠3 = 50°. Based on the exterior angle theorem, we have:
m∠4 = m∠2 + m∠3
Substitute
m∠4 = 65 + 50
m∠4 = 115°
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