To find the measure of $\angle HJK$, let's analyze the information provided:

1. Given Data:

   • $HI = HK = 16$
   • $m\angle HJI = 48^\circ$

2. Understanding the Problem:

   • The points $H, I,$ and $K$ form an isosceles triangle $HIK$ because $HI = HK = 16$.
   • $J$ is a point on the base $IK$ of the isosceles triangle.
   • We need to find $m\angle HJK$.

Step-by-Step Solution:

Since $HI = HK$, $\triangle HIK$ is an isosceles triangle with $H$ as the vertex. In an isosceles triangle, the base angles are equal. Let $m\angle HIK = m\angle HJI = x$.

$x = 48^\circ \quad (\text{since } m\angle HJI = 48^\circ)$

Now, in the isosceles triangle $HIK$:

$m\angle HIK + m\angle IHK + m\angle HKI = 180^\circ$

Since $m\angle IHK = m\angle HKI$ (base angles of the isosceles triangle),

$48^\circ + 2m\angle IHK = 180^\circ$

Solving for $m\angle IHK$:

$2m\angle IHK = 180^\circ - 48^\circ = 132^\circ$ $m\angle IHK = \frac{132^\circ}{2} = 66^\circ$

Since $m\angle HJK$ forms a linear pair with $m\angle IHK$ (angles on a straight line add up to $180^\circ$):

$m\angle HJK + m\angle IHK = 180^\circ$

Substitute $m\angle IHK = 66^\circ$:

$m\angle HJK + 66^\circ = 180^\circ$

Solve for $m\angle HJK$:

$m\angle HJK = 180^\circ - 66^\circ = 114^\circ$

Answer:

$m\angle HJK = 114^\circ$

Would you like a more detailed breakdown of the steps or have any questions?

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Here are some related questions you might find interesting:

1. If $HI = 16$ and $HJ = 10$, what is the length of $JK$?
2. What is the perimeter of triangle $HIK$?
3. If $m\angle HIK = x$, what is $x$?
4. Can the triangle $HIK$ be a right triangle? Why or why not?
5. If $\triangle HIK$ is an isosceles triangle with $HI = HK$, what is $m\angle H$?

Tip: In any isosceles triangle, the base angles are always equal, which can help simplify calculations when angles are involved.