Question

if $overline{hi}congoverline{hk}$ and $mangle hji = 54^{circ}$, what is $mangle ijk?
mangle ijk=square^{circ}$

Explanation:

Step1: Identify congruent right - triangles

Since $\overline{HI}\cong\overline{HK}$ and $\angle HKI = \angle HJI=90^{\circ}$, and $HJ$ is common to right - triangles $\triangle HJI$ and $\triangle HKJ$. By the Hypotenuse - Leg (HL) congruence criterion for right - triangles, $\triangle HJI\cong\triangle HKJ$.

Step2: Find the measure of $\angle HJK$

Since $\triangle HJI\cong\triangle HKJ$, $\angle HJI=\angle HJK = 54^{\circ}$.

Step3: Calculate $\angle IJK$

$\angle IJK=\angle IJH+\angle HJK$. Given $\angle IJH = 54^{\circ}$ and $\angle HJK = 54^{\circ}$, then $\angle IJK=54^{\circ}+54^{\circ}=108^{\circ}$.

Answer:

$108$