### Video Transcript

In the given figure, ππ equals
three π₯ minus seven and ππ equals two π₯ minus two. Find ππ.

We can begin by filling in the
given information about the lengths onto the diagram. Weβre not told any information
about this figure other than the fact that there is a pair of right angles and
thereβs also another pair of congruent angles. So, letβs check if these two
triangles are congruent.

Letβs start by noting that we have
two right angles, angle π½ππ and angle π½ππ. So, we have a pair of congruent
angles. We can see that angle ππ½π and
angle ππ½π from the diagram are also congruent. Weβre given the lengths for ππ
and ππ, but weβre not told that these are congruent. And weβre not given any information
about the line π½π or π½π. We can see, however, that the line
π½π is common to both triangles, which means that we have a pair of congruent
sides.

As we found two angles and a
nonincluded side, we could say that triangle π½ππ and triangle π½ππ are
congruent using the angle-angle-side or AAS rule. Letβs see if this will help us to
find the length ππ. Using the fact that these triangles
are congruent, we know that the length ππ in triangle π½ππ is congruent with the
length ππ in triangle π½ππ. And so, two π₯ minus two must be
equal to three π₯ minus seven. So, weβll need to solve to find the
value of π₯.

If we rewrite this so that we have
our higher coefficient of π₯ on the left-hand side, weβll have three π₯ minus seven
equals two π₯ minus two. Subtracting two π₯ from both sides
of the equation, weβll have π₯ minus seven equals negative two. We can then add seven to both sides
of the equation, so π₯ equals negative two plus seven. And therefore, π₯ is equal to
five. It can be easy to stop at this
point and think that weβve found the answer. However, we werenβt asked for
π₯. We were asked for the length
ππ.

We were given that ππ equals
three π₯ minus seven. So, we plug in our value of π₯
equals five into this equation, which gives us ππ equals three times five minus
seven. And as three times five is 15,
weβll have 15 minus seven, which is eight. And so, we have our answer ππ
equals eight, which we found by proving that the two triangles were congruent.