In rhombus MPKN with an obtuse angle K the diagonals intersect each other at point E. The measure of one of the angles of a ∆PKE is equal 16°. Find the measures of all angles of ∆PKE and ΔPMN.
Since the diagonals bisect the vertex angles, if ∠K is obtuse, ∠PKE cannot be just 16°.
Anyway, to solve this just remember that two consecutive angles of a rhombus are supplementary, so if half of one of them is 16°, half of the next one is 90-16=74°
The diagonals bisect each other, forming four congruent triangles.
To find the measures of all angles in triangles ∆PKE and ∆PMN, we need to use the properties of rhombus MPKN and the given information.
In a rhombus, opposite angles are congruent. Therefore, angle K of rhombus MPKN is also 16°.
Now, let's analyze triangle ∆PKE. We know that angle PKM is 90° because the diagonals of a rhombus are perpendicular bisectors of each other. Since angle K is 16° and angle PKM is 90°, we can find the measure of angle PMK using the fact that the sum of the angles in a triangle is 180°.
Angle PMK = 180° - (angle PKM + angle K)
= 180° - (90° + 16°)
= 180° - 106°
= 74°
So, in triangle ∆PKE, the measures of angles are:
Angle P = 90°
Angle K = 16°
Angle E = 16°
Angle PKM = 90°
Angle PMK = 74°
Since opposite angles in a rhombus are congruent, angle NKM is also 90°. And since the diagonals of the rhombus bisect each other, angle NKM and angle PKM are vertical angles and therefore congruent.
Now, let's analyze triangle ∆PMN. We can find the measure of angle PMN using the fact that the sum of the angles in a triangle is 180°.
Angle PMN = 180° - (angle PMK + angle NKM)
= 180° - (74° + 90°)
= 180° - 164°
= 16°
So, in triangle ∆PMN, the measures of angles are:
Angle P = 90°
Angle M = 16°
Angle N = 16°
Angle PKM = 90°
Angle PMK = 74°
Angle PMN = 16°
Therefore, the measures of all angles in triangles ∆PKE and ∆PMN are as follows:
In ∆PKE:
Angle P = 90°
Angle K = 16°
Angle E = 16°
Angle PKM = 90°
Angle PMK = 74°
In ∆PMN:
Angle P = 90°
Angle M = 16°
Angle N = 16°
Angle PKM = 90°
Angle PMK = 74°
Angle PMN = 16°