Video Transcript
Two triangles have two
corresponding angles and one corresponding side that are equal. Are the two triangles
congruent?
In this question, we need to
consider two triangles with two angles and a side that are congruent. As an example, we could start with
any triangle, such as one like this. If we consider two angles and the
side in between them, then if we draw another triangle that has to have congruent
corresponding parts, it would start by looking something like this. The corresponding side will be the
same length. And the two angles we have are
restricted to being the same size as in the first triangle.
In this way, there is only one way
to complete the second triangle, and it will be congruent to the first. In fact, the angle-side-angle
congruency criterion tells us that two triangles are congruent if two angles and the
side drawn between their vertices in one triangle are congruent to the corresponding
parts in the other. This is equivalent to having two
pairs of corresponding angles and the included pair of sides congruent. But notice that we werenโt told
that the side has to be included. So would the triangles be congruent
if we chose a different side?
Letโs go back to this first
triangle and think about the angles in the triangle. We can recall that the angle
measures in a triangle sum to 180 degrees. So the measure of angle ๐ด is equal
to 180 degrees subtract the measure of angle ๐ต plus the measure of angle ๐ถ. And because angles ๐ต and ๐ธ are
congruent and ๐ถ and ๐น are congruent, then the measure of angle ๐ท is also equal to
180 degrees minus the measure of angle ๐ต plus the measure of angle ๐ถ. And that means that given we have
two pairs of angles congruent in two triangles, then the third pair of angles in the
triangles will also be congruent. So letโs consider the problem.
If we have two triangles that have
two pairs of congruent angles and a nonincluded side congruent, we know that because
the angle measures all add up to 180 degrees, then this will be equivalent to having
two angles and the included side congruent. And the triangles will be congruent
by the ASA criterion. The same would be true for the
triangles ๐ด๐ต๐ถ and ๐ท๐ธ๐น if we used the sides ๐ด๐ถ and ๐ท๐น instead. In fact, this can be referred to as
the angle-angle-side, or AAS, congruence criterion. Knowing that two pairs of
corresponding angles are congruent and the nonincluded sides in each triangle are
congruent would prove that the triangles themselves are congruent.
We can therefore give the answer to
the question as yes, since because we can work out the third angle when given two
angles in a triangle, then when two triangles have two corresponding angles and one
corresponding side congruent, the triangles will be congruent.
