Angle Calculation within Multiple Semi-lines
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Calculation based on reasoning
$1^{\circ}$ Draw an angle $\widehat{xOy}$ measuring $160^{\circ}$.
$2^{\circ}$ Inside $\widehat{xOy}$, draw the semi-lines $[Ou)$ and $[Ov)$ such that each of angle $\widehat{xOu}$ and $\widehat{yOv}$ is right.
$3^{\circ}$ Calculate $\widehat{uOy}$, $\widehat{xOv}$ and $\widehat{uOv}$.
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Step by Step Written Solution
Let's work through this geometry problem together. We are asked to construct a specific configuration of angles and then calculate three missing measures.
Geometry: Angle Calculations
First, we start by drawing the main angle x O y, which is given as 160 degrees. Since it's larger than 90 degrees, it's an obtuse angle.
Next, we draw two semi-lines inside the angle. The first is O u, such that angle x O u is a right angle, meaning it measures 90 degrees.
We also draw O v inside the angle, such that angle y O v is a right angle. This means y O v is also 90 degrees.
Part one: let's calculate the measure of angle u O y. Since the semi-line O u is inside x O y, we can use the angle addition property.
1. Calculate $\widehat{uOy}$
We know that angle x O y is composed of angle x O u plus angle u O y.
Substituting the known values, we have 160 degrees equals 90 degrees plus angle u O y.
Solving for u O y, we subtract 90 from 160 to get 70 degrees.
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