Question

11/14 - triangle congruence schoology assignment
possible points: 2.02
given the figure shown to the right, write
a two - column proof to prove \\( \angle cae \cong \angle cea \\).

Explanation:

Step1: Identify congruent angles and sides

From the figure, we can see that \(\angle BAC\cong\angle EDC\), \(\angle ABC\cong\angle EDC\), \(\overline{BC}\cong\overline{DC}\) (Given).

Step2: Prove triangle congruence

Since we have two angles and the included side congruent (\(\angle BAC\cong\angle EDC\), \(\angle ABC\cong\angle EDC\), \(\overline{BC}\cong\overline{DC}\)), by the Angle - Side - Angle (ASA) congruence criterion, \(\triangle ABC\cong\triangle EDC\).

Step3: Use congruent triangles property

If \(\triangle ABC\cong\triangle EDC\), then by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, \(\overline{AC}\cong\overline{EC}\).

Step4: Identify isosceles triangle

Since \(\overline{AC}\cong\overline{EC}\), \(\triangle ACE\) is isosceles (by the definition of isosceles triangle: a triangle with at least two congruent sides).

Step5: Use isosceles triangle property

In an isosceles triangle \(\triangle ACE\) with \(\overline{AC}\cong\overline{EC}\), the base angles are congruent. So, \(\angle CAE\cong\angle CEA\) (by the Isosceles Triangle Theorem: if two sides of a triangle are congruent, then the angles opposite those sides are congruent).

Answer:

Statements	Reasons
2. \(\triangle ABC\cong\triangle EDC\)	2. ASA (Angle - Side - Angle)
3. \(\overline{AC}\cong\overline{EC}\)	3. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
4. \(\triangle ACE\) is isosceles	4. Definition of isosceles
5. \(\angle CAE\cong\angle CEA\)	5. Isosceles Triangle Theorem