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Difficulty: MediumLines and Angles

Four rays, OA⃗\vec{OA}, OB⃗\vec{OB}, OC⃗\vec{OC}, and OD⃗\vec{OD}, radiate from a common point OO in consecutive clockwise order such that OA⃗⊥OC⃗\vec{OA} \perp \vec{OC} and OB⃗⊥OD⃗\vec{OB} \perp \vec{OD}. If the measure of angle ∠AOD\angle AOD is 3.53.5 times the measure of angle ∠BOC\angle BOC, what is the measure, in degrees, of angle ∠AOB\angle AOB?

  1. A
    36∘36^\circ
  2. B
    40∘40^\circ
  3. 50∘50^\circAnswer
  4. D
    65∘65^\circ
  5. E
    140∘140^\circ

Answer

50∘50^\circ
Because OA⃗⊥OC⃗\vec{OA} \perp \vec{OC} and OB⃗⊥OD⃗\vec{OB} \perp \vec{OD}, we know ∠AOB+∠BOC=90∘\angle AOB + \angle BOC = 90^\circ and ∠BOC+∠COD=90∘\angle BOC + \angle COD = 90^\circ, which implies ∠AOB=∠COD\angle AOB = \angle COD. The total angle ∠AOD=∠AOB+∠BOD=∠AOB+90∘\angle AOD = \angle AOB + \angle BOD = \angle AOB + 90^\circ. Using the given condition ∠AOD=3.5×∠BOC\angle AOD = 3.5 \times \angle BOC, we substitute ∠BOC=90∘−∠AOB\angle BOC = 90^\circ - \angle AOB to get ∠AOB+90∘=3.5(90∘−∠AOB)\angle AOB + 90^\circ = 3.5(90^\circ - \angle AOB), which solves to ∠AOB=50∘\angle AOB = 50^\circ.

Step-by-Step Solution

1
Set up angle variable definitions and perpendicular relationships.
Let ∠AOB=x\angle AOB = x, ∠BOC=y\angle BOC = y, and ∠COD=z\angle COD = z. Since OA⃗⊥OC⃗\vec{OA} \perp \vec{OC}, we have x+y=90∘x + y = 90^\circ. Since OB⃗⊥OD⃗\vec{OB} \perp \vec{OD}, we have y+z=90∘y + z = 90^\circ.
Perpendicular rays form right angles measuring 90∘90^\circ.
2
Deduce the relationship between xx, yy, and zz, and express ∠AOD\angle AOD in terms of xx.
Subtracting yy from both equations gives x=90∘−yx = 90^\circ - y and z=90∘−yz = 90^\circ - y, so x=zx = z. Thus, ∠AOD=x+y+z=x+90∘\angle AOD = x + y + z = x + 90^\circ.
Adjacent angles sharing a vertex add up to form the overall combined angle.
3
Formulate and solve the equation based on the given ratio.
We are given ∠AOD=3.5×∠BOC\angle AOD = 3.5 \times \angle BOC, so x+90∘=3.5yx + 90^\circ = 3.5y. Substituting y=90∘−xy = 90^\circ - x yields x+90∘=3.5(90∘−x)  ⟹  x+90∘=315∘−3.5x  ⟹  4.5x=225∘  ⟹  x=50∘x + 90^\circ = 3.5(90^\circ - x) \implies x + 90^\circ = 315^\circ - 3.5x \implies 4.5x = 225^\circ \implies x = 50^\circ.
Substitution creates a single linear equation in terms of x=∠AOBx = \angle AOB.

Key Concept

Perpendicular Ray Systems and Angle Addition
Estimated Time:1m 30s
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Topics

Lines and Angles

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