Determining Angle Bisectors
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6 For each of the following figures, say if the semi-line $[Ox)$ is the bisector of the angle $\widehat{AOB}$, and justify.
Figure 1
Figure 2
Figure 3: $\widehat{AOC} = 40^\circ$ and $\widehat{AOB} = 80^\circ$
Figure 4: $\widehat{DOA} = 20^\circ$ and $\widehat{BOC} = 35^\circ$
This question includes visual content: The image shows four geometric figures labeled Figure 1 through Figure 4. Figure 1: Ray Ox lies inside angle AOB; arc markings show angles AOx and BOx are not equal. Figure 2: Triangle OAB where Ox passes through the midpoint of AB; AB is perpendicular to AO. Figure 3: Shows an angle AOB with ray Ox (passing through C) inside it, with text stating $\widehat{AOC} = 40^\circ$ and $\widehat{AOB} = 80^\circ$. Figure 4: Shows rays Oz, Ox, Ot, Ou starting from O. A right angle is marked at O. Text states $\widehat{DOA} = 20^\circ$ and $\widehat{BOC} = 35^\circ$. Points D, A, C, B are labeled on rays u, t, x, z respectively.
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Step by Step Written Solution
In this exercise, we will determine if the ray O x is the angle bisector of angle A O B for four different figures. Remember, an angle bisector is a ray that divides an angle into two equal adjacent angles.
Angle Bisector Analysis
Let's look at Figure 1. We can see two arcs indicating the measures of the smaller angles. However, the arcs are different sizes, and there are no markings to suggest they are equal.
Figure 1
Since the visual markings show that the upper angle is clearly not equal to the lower angle, O x is not the bisector.
Now for Figure 2. This shows a triangle O A B with O x passing through the side A B. Notice the hash marks on A x and x B.
Figure 2
While O x is a median because it bisects the side A B, the triangle is not isosceles with O A equal to O B. Therefore, in a right-angled scalene triangle like this, the median is not the angle bisector.
Moving to Figure 3. We are given the measure of angle A O C is 40 degrees and angle A O B is 80 degrees. Since C lies on the ray O x, we look at the relation.
Figure 3
If angle A O B is 80 degrees and the ray O x divides it such that one part is 40 degrees, then the other part, angle C O B, must also be 80 minus 40, which is 40 degrees.
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