Video Transcript
Which of these two figures shows
the steps for constructing a congruent angle? Figure I or figure II.
In this question, we’re given two
figures and we need to determine which of the figures demonstrates the steps of
duplicating an angle. We note that constructing means
with a compass and a straight edge.
Let’s start by saying that these
constructions are supposed to duplicate angle 𝐷𝐸𝐹. We can then recall that to
duplicate this angle, we need to start by drawing a ray in the plane, say the ray
from 𝐴 through 𝐺, where 𝐴 will be the vertex of the angle we duplicate. We can see that both constructions
have such a ray if we add a point 𝐺 to each figure as shown.
The next step in our construction
is to trace a circle centered at 𝐸 that intersects the sides of the angle we want
to duplicate at two points we will label 𝐷 prime and 𝐹 prime. After this, we need to trace a
congruent circle centered at 𝐴. In the diagrams, we only sketch an
arc of the circle to keep the construction clean. We call the point of intersection
between the ray and the circle 𝐵. The final step in the construction
is to trace a circle of radius 𝐷 prime 𝐹 prime centered at 𝐵. We can call the point of
intersection of the two circles 𝐶 as shown.
We can conclude that the angles are
congruent because triangle 𝐹 prime 𝐸𝐷 prime is congruent to triangle 𝐶𝐴𝐵 by
the side-side-side criterion. We see that this is only the case
in the first figure.
For due diligence, we can check
what the second construction gives us. We see that we have the arcs of two
congruent circles centered at 𝐶 and 𝐵. This gives us the following pairs
of lines of the same length, since they are radii of the congruent circles. If we connect 𝐴 to the point of
intersection of these circles, 𝐻, as shown, then we can note that we have two
congruent triangles. This means that all of the
corresponding angles of the two triangles must be congruent. We can then note that this means
that this gives us the angle bisector. Hence, the answer is that only
figure I shows the steps for constructing a congruent angle.
