Use the diagram to prove the given theorem. In the diagram, EC is drawn parallel to AB. Isoscele

step 1 Okay, we can say that if B, C, D, is a trapezoid with angle A equivalent to angle D and angle on

Use the diagram to prove the given theorem. In the diagram,...

Use the diagram to prove the given theorem. In the diagram, $\overline{\mathrm{EC}}$ is drawn parallel to $\overline{\mathrm{AB}}$. Isosceles Trapezoid Base Angles Theorem (Theorem 7.15) Given ABCDis a trapezoid. $\angle \mathrm{A} \cong \angle \mathrm{D}, \overline{\mathrm{BC}} \square \overline{\mathrm{AD}}$ Prove $\quad \mathrm{ABCD}$ is an isosceles trapezoid.

Use the diagram to prove the given theorem. In the diagram, $\overline{\mathrm{EC}}$ is drawn parallel to $\overline{\mathrm{AB}}$.
Isosceles Trapezoid Base Angles Theorem (Theorem 7.15)
Given
ABCDis a trapezoid.
$\angle \mathrm{A} \cong \angle \mathrm{D}, \overline{\mathrm{BC}} \square \overline{\mathrm{AD}}$
Prove $\quad \mathrm{ABCD}$ is an isosceles trapezoid.

Key Concepts

Auxiliary Constructions

Drawing additional lines, such as a segment parallel to a base, is an effective strategy in geometric proofs. These auxiliary constructions aid in creating familiar geometric scenarios that reveal angle and side congruencies, facilitating a clearer pathway to the proof of the theorem.

Congruent Angles and Triangle Congruence

Identifying congruent angles, especially those that result from properties of parallel lines, often sets the stage for applying triangle congruence postulates such as ASA or AAS. Proving triangles congruent within a geometric figure can subsequently prove the congruence of corresponding sides or angles, thereby establishing the properties of isosceles trapezoids.

Trapezoid Properties

A trapezoid is a quadrilateral with one pair of parallel sides, known as the bases. This concept is fundamental when discussing the properties of trapezoids, as it underpins many of the relationships between angles and sides in the figure.

Isosceles Trapezoid

An isosceles trapezoid is a special type of trapezoid where the non-parallel sides (legs) are congruent, which in turn makes the base angles congruent. Understanding this definition is key to proving that a given trapezoid is isosceles when the appropriate angle or side congruencies are established.

Parallel Lines and Transversals

When two lines are parallel and intersected by a transversal, corresponding and alternate interior angles are congruent. This principle is critical in geometric proofs involving parallel lines, as it establishes the necessary angle relationships that help relate different parts of the figure.

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