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.
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.
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
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!
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°