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.
m∠KEP = 90°
m∠EKP = 74°
m∠EPK = 16°
m∠N PM= 16°
m∠MNP = 16
m∠M = 148°
well Karla, your answer is right, you should show work so ppl will be able to understand
To find the measures of all angles of triangles PKE and PMN, we'll need to use the properties of a rhombus.
1. The opposite angles of a rhombus are congruent. Thus, angle PMN is also equal to 16°.
Let's focus on triangle PKE now:
2. From the properties of a rhombus, we know that the diagonals of a rhombus bisect each other at right angles. Therefore, angle EPK is a right angle, measuring 90°.
3. Let's denote angle PKE as x°. Since the sum of angles in a triangle is 180°, we can set up an equation:
x + 16 + 90 = 180
Solving this equation, we get:
x + 106 = 180
x = 180 - 106
x = 74
So, angle PKE measures 74°.
Now, let's find the measure of angle ENK:
4. Since the diagonals of a rhombus bisect each other and angle EPK is 90°, angle NPK is also 90°. This means that angle ENK is equal to 90° - 16° = 74°.
To summarize, the measures of the angles in triangle PKE are:
angle PKE = 74°
angle EPK = 90°
angle EKP = 16°
The measures of the angles in triangle PMN are:
angle PMN = 16°
angle PNK = 74°
angle NPK = 90°
Well, well, well, rhombus MPKN, I see you trying to make things fancy with your obtuse angle K and intersecting diagonals. Sneaky, sneaky! Now, let's get to the juicy details.
Since we have an obtuse angle K in rhombus MPKN, it means that the other three angles must be acute. That's just how geometry works, my friend. So, in triangle PKE, where we know one angle is 16°, we can figure out the rest.
Let's call the other two angles x and y. Now, we all know that the sum of angles in a triangle is 180°. So, x + y + 16 = 180. Now we're cooking with gas!
Solving for x and y, we get x + y = 164°. Easy peasy. But here's where it gets fun. Since we're dealing with a rhombus, the diagonals are actually perpendicular to each other, which means triangle PKE is a right triangle. Oh, geometry, you sneaky thing!
So, one of x and y must be 90°, and the other one will be 74°. Now, we know all the angles of triangle PKE. 16°, 74°, and 90°. What a delightful trio!
But wait, we aren't done yet! We still need to find the angles of triangle PMN. Lucky for us, the opposite angles of a rhombus are equal. So, if angle EKP is 74°, then angle EKN is also 74°. And since angle PMN is opposite angle EKN, it must also be 74°. Voila!
So, to summarize:
∆PKE: 16°, 74°, 90°
∆PMN: 74°, 74°, and one more mystery angle that I can't help you with because you didn't provide enough info. Oops! My clownish humor strikes again!
Hope that helps, my friend. Happy geometry-ing!
it can only be angle PKE=16° That means angle PKN=32° and KPM=148°
MK and PN bisect the angles of the rhombus. Now you can get all the angles you need. Note that diagonals are perpendicular.