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Noah is baking a two-layer cake, with the bottom layer being a circle and the top layer being a triangle. If [tex]$\overline{AB} = 6$[/tex] inches and [tex]$\widehat{AB} \cong \widehat{AC}$[/tex], what does Noah know about the top layer of his cake?

[tex]$\overline{AB}$[/tex] is twice the length of [tex]$\overline{AC}$[/tex] because their arcs are congruent; therefore, [tex]$\triangle ABC$[/tex] is an equilateral triangle.

Sagot :

GinnyAnsweridn
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.
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