Question

3.
$m\angle mjk = ______$
$m\angle mjl = ______$
$m\angle jlk = ______$
$m\angle kml = ______$
$m\angle mnl = ______$

Explanation:

Step1: Find \(m\angle MJK\)

Since \(JM\perp MK\) (property of rectangle), \(m\angle MJK = 90^{\circ}\)

Step2: Find \(m\angle MJL\)

Given \(m\angle KJL=27^{\circ}\), and \(m\angle MJL=m\angle MJK - m\angle KJL\)
\(m\angle MJL = 90^{\circ}- 27^{\circ}=63^{\circ}\)

Step3: Find \(m\angle JLK\)

In rectangle \(MJLK\), \(JK\parallel ML\), so \(\angle JLK=\angle KJL\) (alternate - interior angles)
\(m\angle JLK = 27^{\circ}\)

Step4: Find \(m\angle KML\)

\(\angle KML\) and \(\angle MJL\) are equal (because \(JN = MN\) and \(KN = LN\), \(\triangle JNK\cong\triangle MNL\) (by \(SAS\) congruence, \(JK = ML\), \(JN = MN\), \(KN = LN\)), so \(m\angle KML=m\angle MJL = 63^{\circ}\)

Step5: Find \(m\angle MNL\)

In \(\triangle MNL\), using the angle - sum property of a triangle (\(m\angle NML + m\angle NLM+m\angle MNL=180^{\circ}\)), and \(m\angle NLM = m\angle JLK = 27^{\circ}\), \(m\angle NML=m\angle KML = 63^{\circ}\)
\(m\angle MNL=180^{\circ}-(63^{\circ}+27^{\circ}) = 90^{\circ}\)

Answer:

\(m\angle MJK = 90^{\circ}\)
\(m\angle MJL = 63^{\circ}\)
\(m\angle JLK = 27^{\circ}\)
\(m\angle KML = 63^{\circ}\)
\(m\angle MNL = 90^{\circ}\)