Answer VerifiedHint: Use the angle Sum Property of Triangle which states that Sum of three interior angles of a triangle is equal to ${180^0}$.Complete step-by-step answer: According to the question, it is given that,Each angle of a triangle is less than the sum of the other two angle in a triangleNow consider First case,$ A < B + C{\text{ }}......\left( 1 \right) \\ B < C + A{\text{ }}......\left( 2 \right) \\ C < A + B{\text{ }}......\left( 3 \right) \\ $Now, by angle sum property of triangle, we can write\[ A + B + C = {180^0} \\ B + C = {180^0} - A \\ \]By putting the above value in equation (1), we get\[ A < {180^0} - A \\ 2A < {180^0} \\ A < {90^0} \\ \]Similarly, for Second case,\[ A + B + C = {180^0} \\ C + A = {180^0} - B \\ \]By putting the above value in equation (2), we get\[ B < C + A \\ B < {180^0} - B \\ 2B < {180^0} \\ B < {90^0} \\ \]Similarly, for third case,\[ A + B + C = {180^0} \\ A + B = {180^0} - C \\ \]By putting the above value in equation (3), we get\[ C < A + B \\ C < {180^0} - C \\ 2C < {180^0} \\ C < {90^0} \\ \]Hence Proved, so we found out that all angles are acute then it is proven that this triangle is acute.Note: In this question, observe the question carefully and apply the property directly. The exterior angle of a triangle is equal to the sum of its opposite interior angles. Triangle is the smallest polygon which has three sides and three interior angles.
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Finding the third angle of a triangle when you know the measurements of the other two angles is easy. All you've got to do is subtract the other angle measurements from 180° to get the measurement of the third angle. However, there are a few other ways to find the measurement of the third angle of a triangle, depending on the problem you're working with. If you want to know how to find that elusive third angle of a triangle, see Step 1 to get started.
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Co-authors: 24 Updated: April 30, 2022 Views: 241,835 Article Rating: 63% - 82 votes Categories: Geometry
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