In a rhombus mpkn with an obtuse angle k the diagonals intersect each other at point e. the measure of one of the angles of a triangle pke is equal 16°. find the measures of the other angles of this triangle and the measures of the angles of triangle pmn. m∠kep = m∠ekp = m∠n pm= m∠m = m∠mnp =

m∠KEP=90°,  m∠NPM=16°, m∠KPE=16° , m∠EKP=74°, m∠PMN=148°, m∠MNP=16°

Step-by-step explanation:

Given In a rhombus MPKN with an obtuse angle K the diagonals intersect each other at point E.

The measure of one of the angles of a triangle PKE is equal 16°. we know diagonals of rhombus bisect at right angles i.e at 90°

Hence, m∠KEP=90°

As ∠K is given as obtuse angles so the only possible angle which measures 16° in triangle PKE is ∠KPE. Hence, m∠KPE=16°

In triangle KEP, by angle sum property

m∠EKP+m∠KEP+m∠KPE=180°

⇒ m∠EKP+90°+16°=180°

⇒ m∠EKP=74°

∵ The diagonals bisect the angles at vertices of rhombus.

⇒ ∠NPK=∠NPM=16°

and ∠PKN=2(∠PKE)=2(74°)=148°

∵ Opposite angles of rhombus are equal

⇒ ∠PKN=∠PMN=148°

In triangle PMN, By angle sum property

m∠NPM+m∠PMN+m∠MNP=180°

⇒ 16°+148°+m∠MNP=180°

⇒ m∠MNP=16°

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