Multiple Choice
Given a right triangle where angle is one of the non-right angles, the side opposite has length , the side adjacent to has length , and the hypotenuse has length , what is the value of ?
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
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Given that and , what is ?
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Verified step by step guidance1
Identify the points and angles involved: We have points B, G, C, and F, and the angles \( m\angle BGC = 6x - 13 \) degrees and \( m\angle CGF = 4x + 3 \) degrees. We want to find \( m\angle BGF \).
Recognize that points B, G, C, and F are arranged such that \( \angle BGC \), \( \angle CGF \), and \( \angle BGF \) are angles around point G. The sum of these three angles around point G is 180 degrees if they lie on a straight line or 360 degrees if they form a full circle. Here, since \( B, G, C, F \) are points on a plane and the problem suggests these three angles are adjacent, we consider the straight angle case where the sum is 180 degrees.
Set up the equation expressing that the sum of the three angles around point G is 180 degrees:
\( m\angle BGC + m\angle CGF + m\angle BGF = 180^\circ \)
Substitute the given expressions for \( m\angle BGC \) and \( m\angle CGF \) into the equation:
\( (6x - 13) + (4x + 3) + m\angle BGF = 180^\circ \)
Solve for \( m\angle BGF \) by isolating it on one side:
\( m\angle BGF = 180^\circ - (6x - 13) - (4x + 3) \)
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