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30 September, 17:27

Quadrilateral JKLM is a rhombus. The diagonals intersect at N. If the measure of angle KJL is 2x + 5° and angle MJN = 3x - 8 find the measure of angle KLM.

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  1. 30 September, 17:30
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    The measure of angle KLM is 62°

    Step-by-step explanation:

    * Lets revise the properties of the rhombus

    - The rhombus has 4 equal sides in length

    - Every two opposite angles are equal in measure

    - The two diagonals bisect each other

    - The two diagonals perpendicular to each other

    - The two diagonals bisect the vertices angles

    * Lets solve the problem

    ∵ JKLM is a rhombus

    ∴ m∠MJK = m∠KLM ⇒ opposite angles in the rhombus

    ∵ JL and KM are diagonals in the rhombus and intersect each

    other at N

    ∴ JL bisects ∠MJK

    ∴ m∠KJL = m∠MJN

    ∵ m∠KJL = (2x + 5) °

    ∵ m∠MJN = (3x - 8) °

    ∴ 2x + 5 = 3x - 8 ⇒ subtract 2x from both sides

    ∴ 5 = x - 8 ⇒ add 8 to both sides

    ∴ 13 = x

    ∴ The value of x = 13

    ∵ m∠KJL = (2x + 5) ° ⇒ substitute the value of x

    ∴ m∠KJL = 2 (13) + 5 = 26 + 5 = 31°

    ∵ m∠KJL = 1/2 m∠MJK

    ∴ m∠MJK = 2 m∠KJL

    ∴ m∠ MJK = 2 * 31° = 62°

    ∵ m∠MJK = m∠KLM ⇒ opposite angles in the rhombus

    ∴ m∠KLM = 62°
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