Find the measure of each marked angle. See Example 2.

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Identify the given angles and any relationships between them, such as complementary, supplementary, or vertical angles.

Use the properties of the angles to set up equations. For example, if two angles are complementary, their sum is 90 degrees; if they are supplementary, their sum is 180 degrees.

Solve the equations to find the measures of the unknown angles. This may involve algebraic manipulation such as adding, subtracting, or dividing both sides of the equation.

Check your solutions by substituting the values back into the original relationships to ensure they satisfy the conditions given in the problem.

If applicable, use trigonometric identities or ratios to find any additional angle measures or to verify your solutions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angle Relationships

Understanding angle relationships is crucial in trigonometry, as it involves recognizing how angles interact within geometric figures. This includes concepts such as complementary angles (summing to 90 degrees), supplementary angles (summing to 180 degrees), and vertical angles (which are equal). These relationships help in setting up equations to find unknown angle measures.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. The primary ratios—sine, cosine, and tangent—are defined as the ratios of the lengths of the sides of a right triangle. For example, sine is the ratio of the opposite side to the hypotenuse. These ratios are essential for solving problems involving angles and can be used to find unknown angles when side lengths are known.

Geometric Constructions

Geometric constructions involve creating figures using a compass and straightedge, which can help visualize and solve problems related to angles. Understanding how to construct angles and triangles accurately is vital for determining angle measures. This skill is often applied in conjunction with angle relationships and trigonometric ratios to find the measures of marked angles in various geometric contexts.

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