The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30°. The measure of the obtuse angle is

Options

A. 100°
B. 150°
C. 105°
D. 120°

Answer

To find the measure of the obtuse angle in the parallelogram given that the angle between the two altitudes through the same vertex of an obtuse angle is 30°, we can follow these steps:

### Step 1: Understand the Geometry of the Parallelogram
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (they add up to 180°). The two altitudes from the same vertex of an obtuse angle create two right angles with the sides of the parallelogram.

### Step 2: Set Up the Problem
Let’s denote the parallelogram as ABCD, where angle A is the obtuse angle. The two altitudes from vertex A meet the opposite sides at points E and F, forming two right angles (90°) with sides AB and AD respectively.

### Step 3: Identify the Angles
We know that:
- The angle between the two altitudes AE and AF is given as 30°.
- The angles at vertex A are angle A and its adjacent angles (B and D) are each 90°.

### Step 4: Use the Sum of Angles in a Quadrilateral
The sum of all angles in a quadrilateral is 360°. Therefore, we can express the angles in terms of angle A:
- Angle A + Angle B + Angle C + Angle D = 360°
- Since angle B and angle D are both 90°, we have:
\[ \text{Angle A} + 90° + 90° + \text{Angle C} = 360° \]
\[ \text{Angle A} + \text{Angle C} = 180° \]

### Step 5: Relate Angle C to Angle A
Since angle C is the angle opposite to angle A, we can express angle C as:
\[ \text{Angle C} = 180° - \text{Angle A} \]

### Step 6: Calculate the Measure of Angle A
From the information given, the angle between the two altitudes (30°) can be related to the angles around point A. The angles at point A are:
- 90° (from altitude AE)
- 30° (the angle between the two altitudes)
- 90° (from altitude AF)

Thus, we can write:
\[ \text{Angle A} + 30° = 90° \]
This implies:
\[ \text{Angle A} = 90° - 30° = 60° \]

However, since angle A is obtuse, we need to find the obtuse angle that corresponds to this configuration.

### Step 7: Find the Obtuse Angle
Since angle A is obtuse, we can find it as follows:
\[ \text{Obtuse Angle A} = 180° - 60° = 120° \]

### Step 8: Final Calculation
Now, we can find the obtuse angle:
\[ \text{Obtuse Angle A} = 180° - \text{Angle A} = 180° - 30° - 90° = 150° \]

Thus, the measure of the obtuse angle is **150°**.

### Conclusion
The measure of the obtuse angle in the parallelogram is **150°**.

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