If HI=HK=16 and m∠HJI=48°, what is m∠HJK?

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.

Learn how to calculate the measure of angle HJK in an isosceles triangle where HI=HK=16 and m∠HJI=48°. This problem covers key geometric concepts such as angle relationships and the properties of isosceles triangles. Suitable for students in Grades 8-10.

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