Mathematics

Mathematics

Problem

Lines ll and mm intersect at point OO. Rays OAOA, OBOB, OCOC and ODOD are such that OAOA and OCOC are collinear but in opposite directions, and OBOB and ODOD are collinear but in opposite directions. Which of the following statements is always true? a) ∠AOB=∠BOC\angle AOB = \angle BOC b) ∠AOB+∠COD=180∘\angle AOB + \angle COD = 180^{\circ} c) ∠AOB=∠COD\angle AOB = \angle COD d) ∠AOB+∠BOC=90∘\angle AOB + \angle BOC = 90^{\circ}
Intuitive explanation

When two lines intersect at a point, they create four rays emanating from that center—and the key insight is recognizing the symmetry this creates. Since OAOA and OCOC are opposite rays on one line, and OBOB and ODOD are opposite rays on the other line, you have two intersecting lines forming four angles around point OO. The crucial relationships to identify are which angles are equal (vertical angles) and how adjacent angles relate (supplementary). Think about what must always hold true about angles formed this way, regardless of how the lines are tilted.

Intuitive explanation

When two lines intersect at a point, they create four rays emanating from that center—and the key insight is recognizing the symmetry this creates. Since OAOA and OCOC are opposite rays on one line, and OBOB and ODOD are opposite rays on the other line, you have two intersecting lines forming four angles around point OO. The crucial relationships to identify are which angles are equal (vertical angles) and how adjacent angles relate (supplementary). Think about what must always hold true about angles formed this way, regardless of how the lines are tilted.

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