Video Transcript
In the following figure, π΅π·
equals πΆπΈ, the measure of angle π΄π·π΅ equals the measure of angle π΄πΈπΆ equals
90 degrees, and the measure of angle π΄πΆπ΅ equals the measure of angle π΄π΅πΆ. If the measure of angle π·π΄π΅
equals 34 degrees, find the measure of angle π΄πΆπΈ.
Itβs 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 π΅π· equals
πΆπΈ. So, these two line segments are
congruent. Next, both the angles π΄π·π΅ and
π΄πΈπΆ have measures of 90 degrees. And we have two more congruent
angles, as the measures of angles π΄πΆπ΅ and π΄π΅πΆ are equal. Then, given that the measure of
angle π·π΄π΅ is 34 degrees, we need to find the measure of angle π΄πΆπΈ, which is in
the left triangle at the bottom of the diagram. So, letβs consider what some of
this information might tell us.
We can return to the two congruent
angles, π΄πΆπ΅ and π΄π΅πΆ. 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 π΄πΆ and π΄π΅. So, we know that these sides are
congruent and, in fact, that triangle π΄π΅πΆ is an isosceles triangle.
We can now consider the triangles
π΄πΈπΆ and π΄π·π΅. 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 π΄π·π΅ and π΄πΈπΆ are both right angles. Then, we also worked out that sides
π΄πΆ and π΄π΅ 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 π΅π· and πΆπΈ. And so we can write that the
triangles π΄πΈπΆ and π΄π·π΅ are congruent by the RHS, or right
angle-hypotenuse-side, congruency criterion.
Now, we can identify that angles
πΈπ΄πΆ and π·π΄π΅ are corresponding. And so their measures will be
equal, at 34 degrees each. We can then find the angle measure
of π΄πΆπΈ 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 π΄πΈπΆ must sum to 180 degrees.
And filling in the information that
the measure of angle πΈπ΄πΆ is 34 degrees and the measure of angle π΄πΈπΆ is 90
degrees and simplifying, we find that 124 degrees plus the measure of angle π΄πΆπΈ
is 180 degrees. Then, subtracting 124 degrees from
both sides, we have that the measure of angle π΄πΆπΈ is 56 degrees.
Therefore, we were able to
calculate the measure of this unknown angle by identifying that triangle π΄π΅πΆ is
an isosceles triangle and then proving that the two remaining triangles are
congruent.