Question

determine whether each statement about the triangles is true or false. a graphing calculator may be used for all questions on this test. a. triangles abc and dcb are congruent by the angle - angle triangle congruence theorem. b. triangles abc and bcd are congruent by the angle - side - angle triangle congruence theorem. c. triangles abc and bcd are congruent by the side - side - side triangle congruence theorem. d. triangles abc and dcb are congruent by the side - angle - side triangle congruence theorem. e. triangles abc and dcb are congruent by the side - side - side triangle congruence theorem. f. there is not enough information to determine if the triangles are congruent. graphing calculator

Explanation:

Step1: Identify congruent parts

In \(\triangle ABC\) and \(\triangle DCB\), we have \(AC = DB\) (marked with one - hash), \(\angle A=\angle D\) (marked with one - arc), and \(AB = DC\) (marked with two - hashes).

Step2: Recall congruence theorem

The Side - Angle - Side (SAS) congruence theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, in \(\triangle ABC\) and \(\triangle DCB\), \(AB\) and \(DC\) are two sides, \(\angle A\) and \(\angle D\) are the included angles, and \(AC\) and \(DB\) are the other two sides.

Step3: Evaluate each statement

• A. Not by Angle - Angle (AA) as we have side information too. False.
• B. Not \(\triangle ABC\) and \(\triangle BCD\), wrong triangles considered. False.
• C. Not \(\triangle ABC\) and \(\triangle BCD\), wrong triangles considered. False.
• D. Not by Side - Side - Side (SSS) as we have an angle involved. False.
• E. By SAS, \(\triangle ABC\cong\triangle DCB\). True.
• F. There is enough information. False.

Answer:

A. False
B. False
C. False
D. False
E. True
F. False