This problem will prepare you for the Concept Connection on...

This problem will prepare you for the Concept Connection on page 238 . Marc folded a rectangular sheet of paper, $A B C D,$ in half along $\overline{E F}$. He folded the resulting square diagonally and then unfolded the paper to create the creases shown. a. Use the Pythagorean Theorem to find $D E$ and $C E$ b. What is the $\mathrm{m} \angle D E C ?$ c. Classify $\triangle D E C$ by its side lengths and by its angle measures.

This problem will prepare you for the Concept Connection on page 238 . Marc folded a rectangular sheet of paper, $A B C D,$ in half along $\overline{E F}$. He folded the resulting square diagonally and then unfolded the paper to create the creases shown.
a. Use the Pythagorean Theorem to find $D E$ and $C E$
b. What is the $\mathrm{m} \angle D E C ?$
c. Classify $\triangle D E C$ by its side lengths and by its angle measures.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in Euclidean geometry that relates the lengths of the sides in a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, and it is used to compute unknown lengths when two sides of a right triangle are known.

Square Properties

A square is a regular quadrilateral with equal side lengths and right angles. Key properties include congruent diagonals that bisect each other and the fact that the diagonal creates two congruent isosceles right triangles. Understanding these properties helps in analyzing relationships between segments formed when a square is folded or divided.

Triangle Classification by Side Lengths

Triangles can be classified by the lengths of their sides. An equilateral triangle has all three sides equal, an isosceles triangle has two equal sides, and a scalene triangle has all sides of different lengths. This classification aids in determining the symmetry and potential congruencies within the triangle.

Triangle Classification by Angle Measures

Triangles are also classified by their angles. A triangle can be acute (all angles less than 90°), right (one angle equal to 90°), or obtuse (one angle greater than 90°). This aspect of classification provides insight into the triangle’s shape and is crucial when using geometric rules and theorems related to angles.

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