Video Transcript
𝐴𝐵𝐶𝐷 is a rectangle, and the
point 𝐶 prime is the reflection of 𝐶 in line 𝐴𝐵. If the measure of angle 𝐶𝐴𝐵
equals 71 degrees, what is the measure of angle 𝐶 prime 𝐴𝐶?
The best way to begin a question
such as this is by drawing a sketch to see what is happening. Let’s take the first piece of
information that 𝐴𝐵𝐶𝐷 is a rectangle. We can sketch out a coordinate
grid, and we can place 𝐴𝐵𝐶𝐷 anywhere on the grid with any width or length. As it is a rectangle, we know that
it has four congruent angles of measure 90 degrees each. The important thing about naming
the rectangle’s vertices is that they must follow a pattern. We can start with vertex 𝐴
anywhere on the rectangle and name the vertices in order in either a clockwise or
counterclockwise direction. But we can’t jumble up the
letters. So on the grid, we can label the
rectangle 𝐴𝐵𝐶𝐷 with its vertices in a clockwise direction like this.
In the next part of this question,
we are told that point 𝐶 prime is the reflection of 𝐶 in the line 𝐴𝐵. Now, we haven’t got 𝐶 prime drawn
already, so we’ll need to work out where it goes on the diagram. The line 𝐴𝐵 will be the mirror
line, or line of reflection, that creates the point 𝐶 prime, which means that point
𝐶 will be reflected somewhere above the rectangle. Let’s think about where that will
be.
When we perform a reflection, we
consider the perpendicular distance of that point from the mirror line. Let’s say that 𝐶 is 𝑥 length
units away from point 𝐵 on the mirror line. So the reflected point will be the
same perpendicular distance away from the mirror line on the opposite side. And so we have created the image of
𝐶, which is 𝐶 prime.
Now we have found this point, we
can look at the angle measure we are given. We are told that the measure of
angle 𝐶𝐴𝐵 is 71 degrees. But to find this angle on the
diagram, we need to draw in a new line segment, the line segment 𝐴𝐶, which is the
diagonal of the rectangle. So given this angle measure, we
need to find the measure of angle 𝐶 prime 𝐴𝐶, which means we need another line
segment, the line segment 𝐴𝐶 prime. With this, we can see that angle 𝐶
prime 𝐴𝐶 is the angle which is created between the diagonal of the rectangle and
the line segment between 𝐴 and the point 𝐶 prime.
In order to determine this angle,
let’s take a closer look at triangles 𝐴𝐵𝐶 and 𝐴𝐵𝐶 prime. We know that the line segments 𝐵𝐶
prime and 𝐵𝐶 are congruent because the point 𝐶 and its image 𝐶 prime will be the
same distance from point 𝐵 on the mirror line. Both these line segments are
perpendicular to the line 𝐴𝐵, so the measures of angles 𝐴𝐵𝐶 prime and 𝐴𝐵𝐶
are 90 degrees. And furthermore, within these
triangles, there is a common side, 𝐴𝐵, which is congruent in both triangles. So we have proved that triangles
𝐴𝐵𝐶 and 𝐴𝐵𝐶 prime are congruent by the SAS, or side-angle-side, congruence
criterion.
In these congruent triangles, we
can note that angles 𝐶 prime 𝐴𝐵 and 𝐶𝐴𝐵 are corresponding, and therefore their
measures are equal. They will both be 71 degrees. Therefore, the measure of angle 𝐶
prime 𝐴𝐶, which is comprised of these two angles, can be found by adding 71
degrees and 71 degrees, which gives us the answer that the measure of angle 𝐶 prime
𝐴𝐶 is 142 degrees.
