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This is a rectangle. Work out the angles that have a letter

Let's break down the diagram step by step to find the angles labeled with letters \(a\), \(b\), \(c\), \(d\), and \(e\).

### Step 1: Analyze the angles given
- We are working inside a rectangle, so all the internal angles at the corners of the rectangle are \(90^\circ\).
- The angles at the top left and top right are given as \(30^\circ\) and \(40^\circ\), respectively.
- The angles at the bottom left and bottom right are both given as \(20^\circ\).

### Step 2: Analyze the center angles
- The angle labeled \(140^\circ\) at the center is shared between angles \(a\), \(b\), \(c\), \(d\), and \(e\), which form a closed shape (circle around the point). So, the total sum of these angles must equal \(360^\circ\).

### Step 3: Finding angle \(a\)
The triangle formed between the top left corner, the center angle \(a\), and the \(30^\circ\) angle must sum to \(180^\circ\) because the sum of angles in a triangle is always \(180^\circ\).

- The top-left corner of the rectangle contributes \(90^\circ\), and we are given \(30^\circ\) for one of the angles, so:
  \[
  a = 180^\circ - 90^\circ - 30^\circ = 60^\circ
  \]

### Step 4: Finding angle \(b\)
For angle \(b\), we use the fact that the angle \(140^\circ\) is at the center, and angles around a point sum to \(360^\circ\). To find \(b\), we subtract the known angles.

- The angles around point \(b\) include angle \(a = 60^\circ\) and the central angle \(140^\circ\). So:
  \[
  b = 360^\circ - 140^\circ - 60^\circ = 160^\circ
  \]

### Step 5: Finding angles \(c\), \(d\), and \(e\)
- The lower part of the rectangle has two \(20^\circ\) angles, and similar logic applies to these angles being part of a triangle. Let's denote the sum of the angles at the center involving \(c\), \(d\), and \(e\).

Since we already know the values of \(a\) and \(b\),

This is a rectangle. Work out the angles that have a letter.

Solve for unknown angles labeled with letters in a rectangle. Apply the angle sum properties of triangles and the angles around a point theorem to find the values of angles a, b, c, d, and e. Suitable for Grades 8-10.

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