In the accompanying diagram of triangle ABC, AC=BC, D is point on `overline(AC), overline(AB)` is extended to E, and `overline(DEF)` is drawn so that `triangleADE-triangleABC`. If `mangleC=30`, what is the value of x?

Answer

To solve the problem step by step, let's follow the reasoning based on the information provided in the question and the video transcript.

### Step 1: Understand the properties of triangle ABC
Since AC = BC, triangle ABC is an isosceles triangle. This means that the base angles are equal. Given that angle C = 30°, we can find the other two angles.

**Hint:** Recall that the sum of angles in a triangle is 180°.

### Step 2: Calculate the base angles of triangle ABC
Let the base angles be denoted as angle A and angle B. Since AC = BC, we have:
\[ \text{angle A} = \text{angle B} \]
Using the triangle angle sum property:
\[ \text{angle A} + \text{angle B} + \text{angle C} = 180° \]
Substituting the known value:
\[ \text{angle A} + \text{angle A} + 30° = 180° \]
This simplifies to:
\[ 2 \times \text{angle A} + 30° = 180° \]

**Hint:** Isolate the variable to find the value of angle A.

### Step 3: Solve for angle A
Subtract 30° from both sides:
\[ 2 \times \text{angle A} = 180° - 30° \]
\[ 2 \times \text{angle A} = 150° \]
Now divide both sides by 2:
\[ \text{angle A} = 75° \]

**Hint:** Remember that since angle A and angle B are equal, they both have the same measure.

### Step 4: Establish the relationship between triangles ADE and ABC
Since triangle ADE is similar to triangle ABC (as stated in the problem), the corresponding angles are equal. Therefore:
\[ \text{angle C} = \text{angle E} \]
Thus:
\[ \text{angle E} = 30° \]

**Hint:** Use the property of similar triangles that states corresponding angles are equal.

### Step 5: Relate angle X to the angles in triangle ADE
In triangle ADE, we know that:
- angle A = 75°
- angle E = 30°
- angle D = angle X (since they are vertically opposite angles)

Using the exterior angle theorem, we know:
\[ \text{angle A} = \text{angle D} + \text{angle E} \]
Substituting the known values:
\[ 75° = X + 30° \]

**Hint:** Rearranging the equation will help you isolate X.

### Step 6: Solve for X
Subtract 30° from both sides:
\[ X = 75° - 30° \]
\[ X = 45° \]

### Final Answer
The value of \( x \) is \( 45° \).

---