   Chapter 4.3, Problem 45E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

In square ABCD (not shown), point E lies in the interior of ABCD in such a way that △ A B E is an equilateral triangle. Find m ∠ D E C .

To determine

To find:

mDEC

Explanation

Calculation:

Consider a square ABCD having E in the interior such a way that ABE is an equilateral triangle.

The 3 sides of the equilateral triangle are EB, AE, and AB.

Since AB is one of the sides of the triangle and is also one of the sides of the square, then each side of the equilateral triangle must be congruent to each side of the square.

Since ABE is an equilateral triangle, then each of its angles is equal to 60 degrees.

BEA=60EBA=60EAB=60

Since each angle of a square is equal to 90 degrees, and one part of the angle is 60 degrees, then the other part of the angle must be 30 degrees because 60 + 30 = 90.

We get:

BCE=30

2 isosceles triangle are formed from the congruent sides.

Those 2 isosceles triangle are:

Triangle ADE and triangle BCE

Since their vertex angles are 30 degrees, then each of their base angles must be equal to 75 degrees because base angles of an isosceles triangle are equal

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