Question

4.
given: abed is an isosceles trapezoid
prove: \\( \overline { a d } \cong \overline { b e } \\)
(hint: try to prove diagonals
are congruent with sas)
part ii: other diagonals

Explanation:

Step1: Properties of Isosceles Trapezoid

In isosceles trapezoid \(ABED\), \(AB\parallel ED\), \(AE = BD\) (legs of isosceles trapezoid are congruent), \(\angle AED=\angle BDE\) (base - angles of isosceles trapezoid are congruent)

Step2: SAS Congruence Criterion

In \(\triangle AED\) and \(\triangle BDE\):

• \(AE = BD\) (given as legs of isosceles trapezoid \(ABED\))
• \(\angle AED=\angle BDE\) (base - angles of isosceles trapezoid \(ABED\))
• \(ED = DE\) (common side)

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle AED\cong\triangle BDE\)

Step3: Corresponding Parts of Congruent Triangles

Since \(\triangle AED\cong\triangle BDE\), then \(AD = BE\) (corresponding parts of congruent triangles are congruent)

Answer:

Statement	Justification
In \(\triangle AED\) and \(\triangle BDE\): \(AE = BD\), \(\angle AED=\angle BDE\), \(ED = DE\)	Given and common side
\(\triangle AED\cong\triangle BDE\)	SAS congruence criterion
\(AD = BE\)	Corresponding parts of congruent triangles (CPCTC)