https://file.aiquickdraw.com/mathbot/file/17255726822668nrzdt2j.jpg

In this image, we are tasked with finding the measure of angle \( e \). The diagram shows a triangle with three internal angles given: 34°, 115°, and 30°. To justify the solution and label the angles, let's proceed step-by-step:

### Step 1: Analyze the Triangle
1. The triangle is split into two smaller triangles, which share a common interior line.
2. The sum of the internal angles of any triangle is always 180°.

### Step 2: Label the Angles
Let's label the triangle vertices as follows:
- Call the topmost vertex \( A \), where the angle is 34°.
- Call the vertex where the 115° angle is located as \( B \).
- The vertex adjacent to angle \( e \) and the 30° angle will be called \( C \).
- The vertex at the bottom of the 30° angle will be \( D \), and the point adjacent to angle \( e \) will be point \( E \).

### Step 3: Solve for the Angle

#### Interior Angles at \( B \)
- The angle at vertex \( B \) is 115° (an exterior angle), which corresponds to two interior angles. 
- The interior angle \( \angle ABC \) is supplementary to the 115° exterior angle. Thus, the interior angle \( \angle ABC = 180° - 115° = 65° \).

#### Sum of Angles in Triangle \( ABC \)
- Now, in triangle \( ABC \), we already know two angles:
  - \( \angle A = 34° \)
  - \( \angle ABC = 65° \)
- We can find the third angle \( \angle ACB \) by subtracting the sum of the known angles from 180°:
  \[
  \angle ACB = 180° - 34° - 65° = 81°
  \]

#### Angle \( e \) in Triangle \( ACD \)
- Now, let’s look at triangle \( ACD \). The sum of angles in any triangle is 180°. We already know two of the angles:
  - \( \angle ACB = 81° \)
  - \( \angle D = 30° \)
- To find angle \( e \), we subtract the known angles from 180°:
  \[
  e = 180° - 81° - 30° = 69°
  \]

Thus, the measure of angle \( e \) is **69°**.

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Would you like any additional details or explanations on specific steps?

Here are five related questions to further enhance your understanding:
1. What is the sum of interior angles in a quadrilateral?
2. How do supplementary and complementary angles differ?
3. How can exterior angles of a triangle help in determining interior angles?
4. What is the angle sum property of polygons with more than 3 sides?
5. How do you classify triangles based on their angles?

**Tip:** In any geometric shape, always remember that adjacent angles along a straight line are supplementary (i.e., they sum to 180°).

Find the measure of angle e (add in labels/letters so you can justify your angles).

Learn how to find the measure of angle e in a triangle where three angles are given as 34°, 115°, and 30°. This geometry problem uses concepts like the sum of angles in a triangle and supplementary angles. Suitable for students in Grade 8-10.

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