Video Transcript
In the figure, triangle 𝐴𝐸𝐷 is
similar to triangle 𝐴𝐷𝐵. If the measure of angle 𝐴𝐷𝐸
equals three 𝑥 plus five degrees and the measure of angle 𝐴𝐵𝐷 equals four 𝑥
minus five degrees, what is the measure of angle 𝐴𝐷𝐸?
In this question, we’re given that
there are two similar triangles. Triangle 𝐴𝐸𝐷 is similar to
triangle 𝐴𝐷𝐵. We should remember that similar
triangles have corresponding pairs of angles congruent and corresponding pairs of
sides in the same proportion. Let’s fill in the angle
measurements that we’re given onto the diagram. The measure of angle 𝐴𝐷𝐸 is
three 𝑥 plus five degrees, and the measure of angle 𝐴𝐵𝐷 is four 𝑥 minus five
degrees.
We might be a little confused at
the fact that we’re asked for the measure of angle 𝐴𝐷𝐸, which we’re already told
is three 𝑥 plus five degrees. However, it’s reasonable to assume
that there must be some numerical value that we can give for this angle measurement
instead. Perhaps the best way to approach
this question is to look at the two triangles and see if we can work out which pairs
of angles will be corresponding and, therefore, equal.
If we look at this angle 𝐴𝐷𝐸
that we wish to calculate and we use the fact that we have this similarity statement
that triangle 𝐴𝐸𝐷 is similar to triangle 𝐴𝐷𝐵, it means that the corresponding
angle in triangle 𝐴𝐷𝐵 must be the angle 𝐴𝐵𝐷. We have been given in algebraic
terms the measurement of both of these angles. And since we know that
corresponding angles are congruent, we could therefore set up the equation of four
𝑥 minus five equals three 𝑥 plus five. Now we need to solve this equation
to find the value of 𝑥.
If we start this rearrangement by
subtracting three 𝑥 from both sides of the equation, on the left-hand side we’d
have four 𝑥 minus three 𝑥 minus five, which simplifies to 𝑥 minus five, equals
five. Finally, adding five to both sides
would give us that 𝑥 is equal to 10. It would be very easy to stop here
and think that we finished the question. After all, we found the value of
𝑥. But remember, we need to find the
measurement of angle 𝐴𝐷𝐸.
As the measurement of angle 𝐴𝐷𝐸
is equal to three 𝑥 plus five, then we simply need to plug in the value of 𝑥
equals 10 into this measurement. This gives us that the measurement
of angle 𝐴𝐷𝐸 is equal to three times 10 plus five degrees. And three times 10 is 30. Adding on five gives us 35
degrees. And so we’ve answered the question
by using the fact that similar triangles have corresponding angles equal.
