Question

if hi = hk = 39 and m∠hjk = 56°, what is m∠ijk?
m∠ijk = \boxed{}°

Explanation:

Step1: Identify Triangle Properties

Triangles \( \triangle HIJ \) and \( \triangle HKJ \) are right triangles (right angles at \( I \) and \( K \)) with \( HI = HK = 39 \) and \( HJ \) common. So, \( \triangle HIJ \cong \triangle HKJ \) (HL congruence). Thus, \( \angle HJI = \angle HJK \).

Step2: Find \( \angle HJI \)

Given \( m\angle HJK = 56^\circ \), so \( m\angle HJI = 56^\circ \).

Step3: Calculate \( m\angle IJK \)

\( \angle IJK = \angle HJI + \angle HJK \). Substitute values: \( 56^\circ + 56^\circ = 112^\circ \)? Wait, no—wait, \( \angle IJK \) is the sum? Wait, no, wait: Wait, \( HI \perp IJ \), \( HK \perp JK \), \( HI = HK \), \( HJ \) is common. So \( \triangle HIJ \cong \triangle HKJ \), so \( \angle HJI = \angle HJK = 56^\circ \)? Wait, no, wait the right angles: \( \angle HIJ = 90^\circ \), \( \angle HKJ = 90^\circ \). So \( \triangle HIJ \) and \( \triangle HKJ \) are right triangles with hypotenuse \( HJ \) and leg \( HI = HK \), so they are congruent. Thus, \( \angle HJI = \angle HJK \). Wait, but \( \angle IJK \) is \( \angle HJI + \angle HJK \)? Wait, no, looking at the diagram: \( J \) is the vertex, \( I \) and \( K \) are on the perpendiculars. Wait, maybe \( \angle IJK \) is the angle at \( J \) between \( IJ \) and \( JK \). Wait, no, let's re-examine. Wait, \( HI \) is perpendicular to \( IJ \), \( HK \) is perpendicular to \( JK \), \( HI = HK \), \( HJ \) is the angle bisector? Wait, no, congruent triangles: \( \triangle HIJ \cong \triangle HKJ \), so \( \angle HJI = \angle HJK = 56^\circ \), so \( \angle IJK = \angle HJI + \angle HJK = 56 + 56 = 112 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, no—wait, the right angles are at \( I \) and \( K \), so \( IJ \perp HI \), \( JK \perp HK \), \( HI = HK \), \( HJ \) common. So \( \triangle HIJ \cong \triangle HKJ \), so \( \angle HJI = \angle HJK \). Then \( \angle IJK = \angle HJI + \angle HJK = 56 + 56 = 112^\circ \)? Wait, but let's check again. Wait, maybe the angle is \( 180 - 2\times(90 - 56) \)? No, no. Wait, in right triangle \( \triangle HKJ \), \( \angle HJK = 56^\circ \), so \( \angle JHK = 90 - 56 = 34^\circ \). Then in \( \triangle HIJ \), \( \angle JHI = 34^\circ \), so \( \angle HJI = 56^\circ \). Then \( \angle IJK = \angle HJI + \angle HJK = 56 + 56 = 112^\circ \). Yes, that makes sense.

Answer:

\( 112 \)