Certainly, let's examine the problem statement step-by-step to determine what Noah knows about the top layer of his cake.
### Step-by-Step Solution:
#### Step 1: Understanding the Given Information
Noah has a two-layer cake:
- The bottom layer is a circle.
- The top layer is a triangle.
For the top layer, the triangle, we are given:
- Segment [tex]\(\overline{AB} = 6\)[/tex] inches.
- [tex]\(\widehat{AB} \cong \widehat{AC}\)[/tex].
#### Step 2: Interpreting [tex]\(\widehat{AB} \cong \widehat{AC}\)[/tex]
- [tex]\(\widehat{AB} \cong \widehat{AC}\)[/tex] tells us that the segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{AC}\)[/tex] are congruent.
If two sides of a triangle are congruent, then the corresponding angles opposite those sides are equal. But here, each segment [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{AC}\)[/tex] being equal specifies more about the shape formed.
#### Step 3: Determining the Type of Triangle
Given [tex]\(\overline{AB} = \overline{AC}\)[/tex]:
- We see that we have two congruent sides.
- To check if [tex]\(\triangle ABC\)[/tex] is an equilateral triangle, we note that the statement implies [tex]\(\overline{AB}\)[/tex] is also equal to [tex]\(\overline{BC}\)[/tex] since all sides of an equilateral triangle are the same length.
This all aligns with the defining properties of an equilateral triangle where:
- All three sides are of equal length.
- All three internal angles are equal (each measuring 60 degrees).
#### Step 4: Concluding Properties of [tex]\(\triangle ABC\)[/tex]
Since [tex]\(\triangle ABC\)[/tex] is an equilateral triangle:
- Each side, including [tex]\(\overline{AC}\)[/tex] and [tex]\(\overline{BC}\)[/tex], is also 6 inches.
- Each internal angle [tex]\(\angle A\)[/tex], [tex]\(\angle B\)[/tex], and [tex]\(\angle C\)[/tex] measures 60 degrees.
### Final Summary:
Noah knows the top layer of his cake, which is [tex]\(\triangle ABC\)[/tex], has the following properties:
- All three sides ([tex]\(\overline{AB}\)[/tex], [tex]\(\overline{AC}\)[/tex], and [tex]\(\overline{BC}\)[/tex]) are of equal length, each measuring 6 inches.
- All three internal angles ([tex]\(\angle A\)[/tex], [tex]\(\angle B\)[/tex], and [tex]\(\angle C\)[/tex]) are equal, each measuring 60 degrees.
Therefore, [tex]\(\triangle ABC\)[/tex] is an equilateral triangle with sides being 6 inches each and angles being 60 degrees each.
