Video Transcript
Which figure shows the construction
of a congruent angle to angle 𝐴𝐵𝐷? (I) or (II)?
We need to think about which of
these figures shows a construction of an angle which has the same measure as angle
𝐴𝐵𝐷. Looking at the first figure, we
recall that angles on a straight line sum to 180 degrees, so we can calculate the
measure of angle 𝐴𝐵𝐷. It is 180 degrees minus 140
degrees, which is 40 degrees.
Now, looking at the construction on
the right of figure (I), the acute angle 𝑋𝑌𝑍 would correspond to angle 𝐴𝐵𝐷 if
it had the same measure. However, it only measures 30
degrees, so it is not congruent to angle 𝐴𝐵𝐷. Figure (I) doesn’t show a
construction of a congruent angle to angle 𝐴𝐵𝐷.
Looking at figure II then, we can
see that the construction on the right does show acute angle 𝑋𝑌𝑍, which is
congruent to angle 𝐴𝐵𝐷. It has a measure of 40 degrees. However, do we know that it was
constructed specifically to replicate angle 𝐴𝐵𝐷? Let’s take a look. We’ve cleared a bit of space to
enable us to go through a construction of an angle which is congruent to angle
𝐴𝐵𝐷.
First, we draw a horizontal line
and mark on it a point 𝑌, which will correspond to point 𝐵. Then, we place the point of our
compasses at 𝐵, open them up so our pencil rests at point 𝐴, then draw an arc with
center 𝐵 and radius 𝐵𝐴, which intersects the other leg of our angle at 𝐷. Because they are both radii on the
same arc, we know that length 𝐵𝐴 is equal to length 𝐵𝐷.
Carefully keeping our radius the
same, we can now place the point of our compasses at 𝑌 and replicate the arc on the
other diagram. We can mark the point of
intersection between the arc and the horizontal line 𝑋. It corresponds to point 𝐴. But we don’t know exactly where on
the arc we should mark point 𝑍, which corresponds to point 𝐷. One option would be to place the
point of our compasses back on point 𝐴 and open them up so the pencil lines up with
point 𝐷. This captures the length 𝐴𝐷 in
the radius of our compasses. We could then place the point of
the compasses on 𝑋 and mark point 𝑍 where this radius intersects the orange
arc. Although this would be the quickest
way to complete the construction, it isn’t quite what our figure shows.
In the figure, we have points 𝐶
and 𝐿 on the constructions. A bit of careful inspection and
measurement shows us that the distance 𝐵𝐶 is equal to distance 𝐵𝐴. This means that, after drawing arc
𝐴𝐷, the constructor also used the same radius and center 𝐵 to mark point 𝐶 to
the left of 𝐵 on the horizontal line. We can replicate that process in
our new construction. With the radius still set to length
𝐵𝐴, we can place the point of our compasses at 𝑌 and mark off point 𝐿 the same
distance 𝐵𝐶 to the left of 𝑌.
Now, we can capture length 𝐶𝐷 in
our compasses by placing the point on 𝐶 and opening up the radius until the pencil
sits on 𝐷. Then, we can carefully keep the
same radius, place the point of the compasses on 𝐿, and mark off where the radius
intersects the orange arc from 𝑋. This is point 𝑍. Finally, we can draw the ray from
𝑌 through 𝑍, and we’ve replicated the angle 𝐴𝐵𝐷 in angle 𝑋𝑌𝑍.
So, we’ve shown that figure (II)
has angle 𝑋𝑌𝑍 which is congruent to angle 𝐴𝐵𝐷 and that the markings are
consistent with a construction of a congruent angle. We also showed that figure (I)
didn’t have a congruent angle to angle 𝐴𝐵𝐷. The answer, then, is that only
figure (II) shows the construction of a congruent angle to angle 𝐴𝐵𝐷.
