The OM and OH beams are the bisectors of the adjacent AOC and BOC

| April 3, 2021 | education

The OM and OH beams are the bisectors of the adjacent AOC and BOC angles, respectively. Find the angle between the bisectors of the angles MOA and HOB.

1.∠AOC + ∠BOC = 180 °.
Let ∠AOC = x, then ∠BOC = 180 ° – x.
Since OM and OH are bisectors ∠AOC and ∠BOC, respectively, then:
∠MOA = ∠MOC = ∠AOC / 2 = x / 2;
∠HOC = ∠HOB = ∠BOC / 2 = (180 ° – x) / 2.
2. Draw the bisectors ∠MOA and ∠HOB – rays OK and OE, respectively.
Then:
∠KOA = ∠KOM = ∠MOA / 2 = x / 2: 2 = x / 4;
∠EOH = ∠EOB = ∠HOB / 2 = (180 ° – x) / 2: 2 = (180 ° – x) / 4.
3. Thus, the angle between the bisectors ∠MOA and ∠HOB is equal to:
∠KOE = ∠KOM + ∠MOC + ∠HOC + ∠EOH = x / 4 + x / 2 + (180 ° – x) / 2 + (180 ° – x) / 4 = (bring the fractions to a common denominator 4) = (x + 2 * x + 2 * (180 ° – x) + 180 ° – x) / 4 = (x + 2 * x + 2 * 180 ° – 2 * x + 180 ° – x) / 4 = (we give similar terms in the numerator) = 540 ° / 4 = 135 °.
Answer: ∠KOE = 135 °.



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