Advanced Angles Challenge

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📐 Part 1: Complex Angle Calculations

1. In triangle ABC, angle A = 3x + 15°, angle B = 2x - 10°, and angle C = x + 25°. Find the value of x and all three angles.

2. Two parallel lines are cut by a transversal. If one interior angle is (4y + 20)° and its corresponding angle is (6y - 40)°, find y and both angle measures.

3. In a regular polygon, each exterior angle measures 24°. How many sides does the polygon have, and what is the measure of each interior angle?

4. A quadrilateral has angles measuring (3a + 10)°, (2a - 5)°, (4a + 15)°, and (a + 40)°. Find the value of a and determine if this could be a parallelogram.

🔍 Part 2: Advanced Angle Relationships

5. In the diagram below, lines PQ and RS are parallel, and line MN intersects both. If angle 1 = (5x - 30)° and angle 2 = (3x + 50)° are alternate interior angles, prove that the lines are parallel by showing x = 20.

6. A triangle has one angle that is 40° larger than the smallest angle, and a third angle that is twice the smallest angle. Set up and solve an equation to find all three angles.

7. In triangle XYZ, an exterior angle at vertex Y measures 135°. If angle X = (2k + 15)° and angle Z = (3k - 10)°, find k and verify your answer using the exterior angle theorem.

🎯 Part 3: Challenge Problems

8. Two intersecting circles create four angles at their intersection point. If three of these angles measure (2m + 30)°, (3m - 15)°, and (m + 45)°, find m and all four angle measures. Explain why your answer makes geometric sense.

9. In a cyclic quadrilateral ABCD, opposite angles are supplementary. If angle A = (4p - 20)°, angle B = (2p + 40)°, and angle C = (3p + 10)°, find p and angle D. Prove that your quadrilateral satisfies the cyclic property.