Without being able to see the picture, I can't definitively answer which statements about the sides or angles are true. However, I can provide guidance on how to analyze the problem.
For Question 1:
You need to analyze the relationships between the segments BD, BC, AC, CB, AD, and CD based on the geometry of the figure.
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If segments BD and BC are parts of a triangle or a geometric figure where they could be equal (such as being two sides of an isosceles triangle sharing the same base), then BD = BC might be true.
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If AC and CB are segments that form a straight line or are two sides of a triangle, you need to check if they are equal. This might be true if AC is a segment that divides CB equally.
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If AD and CB are involved in a situation where they are equal due to angles formed or congruent triangles, then AD = CB could be true.
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Lastly, check if CD is congruent to AD. If there is a geometric principle (like symmetry or properties of isosceles triangles) indicating they are equal, then that statement can be true.
For Question 2, the theorem you may apply will depend on the relationships you find in Question 1:
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Use the Vertical Angle Theorem if the angles opposite each other when two lines intersect are involved and if you're proving sides based on those angles.
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Use the Alternate Interior Angle Theorem if you are dealing with two parallel lines and a transversal, showing that angles (and therefore segments) are congruent.
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The Perpendicular Bisector Theorem would apply if you need to prove a segment is bisected evenly.
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The Triangle Inequality Theorem is relevant when proving relationships about the lengths of sides in a triangle.
Based on the figure and the relationships you identify, you should be able to determine the correct statement and the theorem that proves it. If you provide the specifics of the figure or the relationships shown, I could offer more targeted help!