Verify Euler’s formula for these solids.

Explanation

To verify Euler’s formula for the given solids, we will follow these steps:

### Step 1: Understand Euler's Formula
Euler's formula states that for any convex polyhedron, the relationship between the number of faces (F), vertices (V), and edges (E) is given by:
\[ F + V - E = 2 \]

### Step 2: Identify the First Solid
Let’s analyze the first solid.

#### Step 2.1: Count the Faces (F)
- The first solid has **7 faces**.

#### Step 2.2: Count the Vertices (V)
- The solid has **10 vertices** (5 on the top face and 5 on the base).

#### Step 2.3: Count the Edges (E)
- The solid has **15 edges** (5 edges on the top face, 5 edges on the base, and 5 edges connecting the top to the base).

### Step 3: Apply Euler's Formula for the First Solid
Now, we will substitute the values into Euler's formula:
\[ F + V - E = 7 + 10 - 15 \]
Calculating this gives:
\[ 17 - 15 = 2 \]

### Step 4: Verify for the First Solid
Since the result is equal to 2, we have verified Euler's formula for the first solid.

### Step 5: Identify the Second Solid
Now, let’s analyze the second solid.

#### Step 5.1: Count the Faces (F)
- The second solid has **9 faces** (4 triangular faces and 1 base face).

#### Step 5.2: Count the Vertices (V)
- The second solid has **9 vertices**.

#### Step 5.3: Count the Edges (E)
- The second solid has **16 edges**.

### Step 6: Apply Euler's Formula for the Second Solid
Now, we will substitute the values into Euler's formula:
\[ F + V - E = 9 + 9 - 16 \]
Calculating this gives:
\[ 18 - 16 = 2 \]

### Step 7: Verify for the Second Solid
Since the result is equal to 2, we have verified Euler's formula for the second solid as well.

### Conclusion
Both solids satisfy Euler's formula, confirming that \( F + V - E = 2 \) holds true for both cases.

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