Question

1. modeling mathematics in the photo, the road is perpendicular to the support beam and \\( \overline{ab} \cong \overline{cb} \\). which theorem allows you to conclude that \\( \overline{ad} \cong \overline{cd} \\)?

Explanation:

Step1: Identify Given Information

We know that \( DB \perp AC \) (the road is perpendicular to the support beam), so \( \angle ABD \) and \( \angle CBD \) are right angles (\( 90^\circ \)). Also, \( AB \cong CB \) (given), and \( DB \) is a common side to both \( \triangle ABD \) and \( \triangle CBD \).

Step2: Recall Triangle Congruence Theorems

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. In \( \triangle ABD \) and \( \triangle CBD \):

• \( AB \cong CB \) (given)
• \( \angle ABD \cong \angle CBD = 90^\circ \) (since \( DB\perp AC \))
• \( DB \cong DB \) (common side, reflexive property)

By the SAS congruence theorem, \( \triangle ABD\cong\triangle CBD \).

Step3: Use Congruent Triangles to Prove \( AD\cong CD \)

Corresponding parts of congruent triangles are congruent (CPCTC). Since \( \triangle ABD\cong\triangle CBD \), the corresponding sides \( AD \) and \( CD \) are congruent. So the theorem that allows us to conclude \( AD\cong CD \) after proving the triangles congruent by SAS is the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, and the triangle congruence is established by SAS. If we are talking about the theorem for the congruence of the segments \( AD \) and \( CD \) given the perpendicularity and \( AB\cong CB \), the key is the SAS congruence of the triangles followed by CPCTC. But the main triangle congruence theorem used to set up the congruence of the triangles (which leads to \( AD\cong CD \)) is the SAS (Side - Angle - Side) congruence theorem.

Answer:

The Side - Angle - Side (SAS) Congruence Theorem (and then CPCTC) allows us to conclude \( \overline{AD}\cong\overline{CD} \). The primary triangle congruence theorem here is the SAS Congruence Theorem.