How do solve for the unknown lengths and angle measures of triangle ABC where angle $C = {90^{}}$ degrees, angle $B = 23$degrees and side $a = 24?$

Hint:In this question they are given two angels and we need to find the remaining one angel. We can find the unknown angel by using the triangle angle formula.The sum of all the angles is 180 degrees. Here they have given the length of only one side and we need to find the remaining two sides. We can solve this by applying trigonometric functions.

Complete step by step answer:
We need to first understand the requirement of the question which is finding the unknown angle and lengths of a right angled triangle. As per the question we have a triangle right angled at C i.e. ${90^ \circ }$. Let us draw a diagram according to the question;

We know that in a triangle the sum of all three angles is 180 degrees. Since we know the two angels we can find the other easily.
$\angle A + \angle B + \angle C = {180^0}$
$\Rightarrow \angle A + {23^0} + {90^0} = {180^0}$
$\Rightarrow \angle A + {113^0} = {180^0}$
$\Rightarrow \angle A = {180^0} - {113^0}$
$\therefore \angle A = {67^0}$.
This is the required angel. Here in the diagram we have assumed that the sides $AB = c,BC = a$ and $CA = b$.

Now using the trigonometric function. That is, we know that cosine is the ratio of the adjacent side to the hypotenuse side. That is
$\cos B = \dfrac{{BC}}{{AB}}$
$\Rightarrow \cos B = \dfrac{a}{c}$.
So to find the length of the side AB that is C, we have
\[c = \dfrac{a}{{\cos B}}\]
\[\Rightarrow c = \dfrac{{24}}{{\cos 23}}\]
Using scientific calculator we have \[\cos {23^0} = 0.9205\]
\[c = \dfrac{{24}}{{0.9205}}\]
\[\Rightarrow c = 26.072\]
After rounding off we have
\[ \therefore c = 26.07\,units\]

Similarly we know that tangent function is the ratio of opposite side to adjacent side. That is
$\tan B = \dfrac{{CA}}{{BC}}$
$\Rightarrow \tan B = \dfrac{b}{a}$.
$\Rightarrow b = a \times \tan B$
$\Rightarrow b = 24 \times \tan {23^0}$
Using scientific calculator we have \[\tan {23^0} = 0.4244\]
$b = 24 \times 0.4244$
$\Rightarrow b = 10.1856$
Rounding off we have,
$ \therefore b = 10.187\,units$

Note:We should note that the sum of all the interior angles of a triangle is $180$ degrees, so we can always cross check our answer if we got the correct answer or not. Here the unit of the sides length is not given. Usually it is in meters. If they have asked us to find the area of the triangle we should have all the sides in meters only.