If in a rhombus `LMNP, angleLNM=40^(@)` then what is the measure of `angleLPM`?

Answer

To find the measure of angle LPM in the rhombus LMNP where angle LNM = 40°, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus has the following properties:
- All sides are equal.
- The diagonals bisect each other at right angles (90°).
- Opposite angles are equal.

### Step 2: Identify the angles in the rhombus
Given that angle LNM = 40°, we can use the property of opposite angles. Therefore, angle LPM (which is opposite to angle LNM) will also be equal to angle LNM.

### Step 3: Use the property of diagonals
Let’s denote the intersection point of the diagonals as O. Since the diagonals bisect each other at right angles, we have:
- Angle LON = 90° (because diagonals bisect at 90°).

### Step 4: Analyze triangle LNM
In triangle LNM:
- Angle LNM = 40° (given)
- Angle LON = 90° (diagonal property)
- Therefore, angle LNM + angle LON + angle NLM = 180° (sum of angles in triangle)

### Step 5: Calculate angle NLM
Using the triangle angle sum property:
- 40° + 90° + angle NLM = 180°
- angle NLM = 180° - 130° = 50°

### Step 6: Relate angle LPM to angle NLM
Since angle LPM is equal to angle NLM (because they are opposite angles formed by the intersection of the diagonals), we have:
- angle LPM = angle NLM = 50°.

### Conclusion
Thus, the measure of angle LPM is **50°**.

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