Video Transcript
In the following figure, 𝐵𝐷
equals 𝐶𝐸, the measure of angle 𝐴𝐷𝐵 equals the measure of angle 𝐴𝐸𝐶 equals
90 degrees, and the measure of angle 𝐴𝐶𝐵 equals the measure of angle 𝐴𝐵𝐶. If the measure of angle 𝐷𝐴𝐵
equals 34 degrees, find the measure of angle 𝐴𝐶𝐸.
It’s always worth beginning a
question like this by carefully reading the given information and checking if these
pieces of information are represented on the diagram.
Firstly, we have that 𝐵𝐷 equals
𝐶𝐸. So, these two line segments are
congruent. Next, both the angles 𝐴𝐷𝐵 and
𝐴𝐸𝐶 have measures of 90 degrees. And we have two more congruent
angles, as the measures of angles 𝐴𝐶𝐵 and 𝐴𝐵𝐶 are equal. Then, given that the measure of
angle 𝐷𝐴𝐵 is 34 degrees, we need to find the measure of angle 𝐴𝐶𝐸, which is in
the left triangle at the bottom of the diagram. So, let’s consider what some of
this information might tell us.
We can return to the two congruent
angles, 𝐴𝐶𝐵 and 𝐴𝐵𝐶. The isosceles triangle theorem
states that if two angles of a triangle are congruent, then the sides opposite those
angles are congruent. These opposite sides on the diagram
can be given as the line segments 𝐴𝐶 and 𝐴𝐵. So, we know that these sides are
congruent and, in fact, that triangle 𝐴𝐵𝐶 is an isosceles triangle.
We can now consider the triangles
𝐴𝐸𝐶 and 𝐴𝐷𝐵. If these triangles were congruent,
that would make finding the unknown angle much simpler. But are they congruent? Well, we do have two congruent
angles in these triangles, since angles 𝐴𝐷𝐵 and 𝐴𝐸𝐶 are both right angles. Then, we also worked out that sides
𝐴𝐶 and 𝐴𝐵 are congruent. And as these are right triangles,
we can also identify that these line segments are both the hypotenuse in their
respective triangles. And finally we have another
congruent pair of sides in each triangle, the line segments 𝐵𝐷 and 𝐶𝐸. And so we can write that the
triangles 𝐴𝐸𝐶 and 𝐴𝐷𝐵 are congruent by the RHS, or right
angle-hypotenuse-side, congruency criterion.
Now, we can identify that angles
𝐸𝐴𝐶 and 𝐷𝐴𝐵 are corresponding. And so their measures will be
equal, at 34 degrees each. We can then find the angle measure
of 𝐴𝐶𝐸 by using the property that the interior angle measures of a triangle sum
to 180 degrees. We can write that the three angle
measures in triangle 𝐴𝐸𝐶 must sum to 180 degrees.
And filling in the information that
the measure of angle 𝐸𝐴𝐶 is 34 degrees and the measure of angle 𝐴𝐸𝐶 is 90
degrees and simplifying, we find that 124 degrees plus the measure of angle 𝐴𝐶𝐸
is 180 degrees. Then, subtracting 124 degrees from
both sides, we have that the measure of angle 𝐴𝐶𝐸 is 56 degrees.
Therefore, we were able to
calculate the measure of this unknown angle by identifying that triangle 𝐴𝐵𝐶 is
an isosceles triangle and then proving that the two remaining triangles are
congruent.
