Line CB is perpendicular to line AD at B between A and D. Angle BCA is congruent to angle BCD and line AC is congruent to line DC. Which congruency statements (HL, AAS, ASA, SAS, and SSS) can you use to conclude that triangle ABC is congruent to DBC?
A)HL, AAS, ASA, SAS, and SSS
B) HL and ASA
C) HL, AAS, ASA, and SAS **
D) HL, AAS, and ASA
The congruency statements that can be used to conclude that triangle ABC is congruent to DBC are: HL, AAS, ASA, and SAS.
To determine which congruency statements can be used to conclude that triangle ABC is congruent to DBC, we need to analyze the given information in the problem.
First, we know that line CB is perpendicular to line AD at B. This means that angle ABD is a right angle, and angle CBD is also a right angle.
Next, we are given that angle BCA is congruent to angle BCD. This shows that angle ABC and angle DBC are congruent, since angles that are congruent to the same angle are congruent to each other.
We also know that line AC is congruent to line DC. This indicates that side AC is congruent to side DC.
Based on this information, we can conclude that the congruency statement SAS (Side-Angle-Side) can be used to prove that triangle ABC is congruent to DBC.
The SAS congruency statement 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.
In our case, we have side AC congruent to side DC (S), angle BCA congruent to angle BCD (A), and we can infer that side BC is congruent to side BC (S).
Hence, the correct answer is option C) HL, AAS, ASA, and SAS.