Video Transcript
In the given figure, use the
properties of congruent triangles to find the measure of angle 𝐵𝐶𝐷.
So, let’s begin by looking at
this figure and identifying any key information from the markings on it. Firstly, we have two pairs of
congruent line segments marked. Sides 𝐴𝐵 and 𝐴𝐷 are
congruent, and sides 𝐶𝐵 and 𝐶𝐷 are congruent. We also have the angle measure
of angle 𝐴𝐶𝐷 given as 29 degrees. Since we are told to use the
properties of congruent triangles, let’s identify two triangles we may be able
to use.
We have triangle 𝐴𝐵𝐶 and
triangle 𝐴𝐷𝐶. These two triangles share a
common side of 𝐴𝐶. So this length will be equal in
both triangles. And so, we have in fact
recognized that there are three pairs of congruent sides. We can then write that triangle
𝐴𝐵𝐶 is congruent to triangle 𝐴𝐷𝐶 by the SSS congruency criterion. So, all corresponding pairs of
sides and angles in each triangle are congruent.
We can then identify a pair of
congruent angles. Given that angles 𝐴𝐶𝐵 and
𝐴𝐶𝐷 are corresponding, these will both be 29 degrees. But we are asked to find the
measure of angle 𝐵𝐶𝐷. Since angle 𝐵𝐶𝐷 is comprised
of angles 𝐴𝐶𝐵 and 𝐴𝐶𝐷, we need to add their measures of 29 degrees each,
which gives us a final answer of 58 degrees.
In this example, we find an
unknown angle in a kite. But there is also another
property proved here which may not be immediately obvious. And that is that the longer
diagonal of a kite bisects the angles at the vertices on this diagonal. Although we used specific
values for the measures of angles 𝐴𝐶𝐵 and 𝐴𝐶𝐷, we could just have easily
had a general angle of 𝑥 degrees for each. Since a kite must have two
pairs of congruent sides, we will always be able to create two congruent
triangles within any kite. And so this angle of two 𝑥
degrees at the end of the longer diagonal in the kite will always be bisected by
this diagonal. And because of the congruency
of the triangles, we can also say the same for the angle at the other end of the
longer diagonal. It would also be bisected.
