One angle of a triangle measures 10 degrees more than the second. The measure of the third angle is twice the sum of the first two angles. Find the measure of the angle in the triangle that has the greatest degree.
Answer (2)
Defined the angles as A , B , and C , and expressed the relationships between them: A = B + 10 and C = 2 ( A + B ) .
Used the fact that the sum of the angles in a triangle is 180 degrees: A + B + C = 180 .
Substituted the expressions for A and C into the equation and solved for B , finding B = 25 degrees.
Calculated A = 35 degrees and C = 120 degrees, and identified the largest angle as C .
The measure of the largest angle in the triangle is 120 .
Explanation
Analyze the problem and given data Let's analyze the problem. We are given a triangle with three angles, and we know the relationships between these angles. Our goal is to find the measure of the largest angle. Let's denote the three angles as A , B , and C . We know that A = B + 10 and C = 2 ( A + B ) . Also, the sum of the angles in a triangle is 180 degrees, so A + B + C = 180 .
Substitute and simplify Now, let's substitute the expressions for A and C into the equation A + B + C = 180 . We have:
( B + 10 ) + B + 2 ( B + 10 + B ) = 180
Simplifying the equation:
B + 10 + B + 2 ( 2 B + 10 ) = 180
2 B + 10 + 4 B + 20 = 180
6 B + 30 = 180
Solve for B Now, let's solve for B :
6 B = 180 − 30
6 B = 150
B = 6 150
B = 25
Solve for A Now that we have the value of B , we can find the value of A :
A = B + 10
A = 25 + 10
A = 35
Solve for C Next, we can find the value of C :
C = 2 ( A + B )
C = 2 ( 35 + 25 )
C = 2 ( 60 )
C = 120
Determine the largest angle We have found the measures of the three angles: A = 35 degrees, B = 25 degrees, and C = 120 degrees. The largest angle is C = 120 degrees.
State the final answer Therefore, the measure of the largest angle in the triangle is 120 degrees.
Examples
Understanding angles in triangles is crucial in many real-world applications. For example, architects use these principles to design stable and aesthetically pleasing structures. Imagine designing a roof for a house; the angles at which the roof panels meet are critical for ensuring the roof's structural integrity and preventing water leakage. Similarly, in navigation, understanding angles is essential for determining direction and calculating distances, whether you're sailing a boat or flying an airplane. Even in art, understanding perspective relies on the principles of angles and how they create the illusion of depth and space.
The largest angle in the triangle measures 120 degrees. This was found by defining the angles, setting up relationships, and solving a system of equations. Angle A is 35 degrees, angle B is 25 degrees, and angle C is 120 degrees, making C the greatest angle.
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