Question

1. if jklm is a rhombus, find the following. mk = 24, jl = 20, and m∠mjl = 50°. nk =

nl =
ml =
jm =
m∠knl =
m∠kjl =
m∠mlk =
m∠jkm =
m∠jml =

Explanation:

Step1: Diagonals bisect each other

In a rhombus, diagonals bisect each other.
$NK = \frac{MK}{2} = \frac{24}{2} = 12$
$NL = \frac{JL}{2} = \frac{20}{2} = 10$

Step2: Diagonals are perpendicular

Rhombus diagonals are perpendicular, so $\triangle JNK$ is right.
$JM = \sqrt{NJ^2 + NK^2} = \sqrt{10^2 + 12^2} = 13$

Step3: All sides of rhombus equal

All sides of a rhombus are congruent.
$ML = JM = 13$

Step4: Right angle from perpendicular diagonals

Diagonals of rhombus are perpendicular.
$m\angle KNL = 90^\circ$

Step5: Alternate interior angles are equal

$JM \parallel KL$, so alternate interior angles are equal.
$m\angle KJL = m\angle MJL = 50^\circ$

Step6: Calculate $\angle JML$

Sum of angles in $\triangle MJN$ is $180^\circ$.
$m\angle JMN = 180^\circ - 90^\circ - 50^\circ = 40^\circ$, so $m\angle JML = 2 \times 40^\circ = 80^\circ$

Step7: Opposite angles of rhombus equal

Opposite angles of a rhombus are congruent.
$m\angle MLK = m\angle JML = 80^\circ$

Step8: Alternate interior angles are equal

$JM \parallel KL$, so alternate interior angles are equal.
$m\angle JKM = m\angle MKL = 50^\circ$

Answer:

$NK = 12$
$NL = 10$
$ML = 13$
$JM = 13$
$m\angle KNL = 90^\circ$
$m\angle KJL = 50^\circ$
$m\angle MLK = 80^\circ$
$m\angle JKM = 50^\circ$
$m\angle JML = 80^\circ$