Question

Short Response Suppose \(A B=A D=4\), \(m \angle B=m \angle D=65\), and \(A C=3.5\). Is \(\triangle A B C \cong \triangle A D C ? B C D\) (Lesson 5-5)

Short answer

Yes, \(\triangle ABC \cong \triangle ADC\) by SAS criteria.

Step-by-step solution

Understand Congruence Criteria

To determine if two triangles are congruent, we can use congruence criteria such as ASA (Angle-Side-Angle), SAS (Side-Angle-Side), or SSS (Side-Side-Side). For this problem, given the angles and sides, we will explore the ASA and SAS criteria.

Identify Given Information

The given information is that \(AB = AD = 4\), \(m \angle B = m \angle D = 65^\circ\), and \(AC = 3.5\). This means that we have two equal sides and one equal angle for each triangle.

Consider Possible Congruence

For \( \triangle ABC \) and \( \triangle ADC \) to be congruent using ASA criteria, we need two angles and the non-included side to be equal, which is not available from the given information. However, with the given side and angle, verify if SAS applies with the unknown side \(BC\) and \(DC\).

Check the Unknown Side

Since \(BC\) and \(DC\) are both sides opposite to the equal angle \(65^\circ\) and within the congruent triangles, these must be equal for \( \triangle ABC \) to be congruent to \( \triangle ADC \) by SAS criteria.

Apply SAS Congruence Criteria

For \( \triangle ABC \) and \( \triangle ADC \), angles \( \angle B = \angle D = 65^\circ\) and the included side \(AB = AD = 4\), if it is given or can be shown that \(BC = DC\), then by SAS, \( \triangle ABC \cong \triangle ADC \). Since angle and the adjacent side are the same, by default, \(BC = DC\).

Conclusion: Congruence Determination

Since \( \angle B = \angle D\), \(AB = AD\), and \(BC = DC\), \ \(\triangle ABC \cong \triangle ADC\) using the SAS criteria. Therefore, the triangles are congruent.

Key concepts

ASA (Angle-Side-Angle)

The ASA, or Angle-Side-Angle, criteria for triangle congruence is a straightforward method.
It states that if two triangles have two angles and the included side equal, the triangles are congruent. To break it down:

• We need two angles in one triangle to be equal to two angles in another triangle.
• Additionally, the side that is in between these two angles (called the included side) must also be equal in length.

This criterion is useful when we have this specific pattern of known quantities.
However, in the case of the problem, unfortunately, ASA does not apply directly since we have only been given one angle and not two.
Therefore, ASA cannot be used as the congruence condition in this scenario.

SAS (Side-Angle-Side)

The SAS, or Side-Angle-Side, criterion is crucial for identifying the congruence of triangles.
This rule dictates that if two sides and the included angle of one triangle are equal to two sides and the included angle of another, the triangles are congruent. Here's a breakdown:

• Two sides, say \(AB\) and \(CA\), of one triangle should be equal to two sides of another triangle.
• The angle between these two sides, such as \(\angle BAC\), should also be equal to the corresponding angle in the other triangle.

In our problem, since we have \(AB = AD = 4\), \(\angle B = \angle D = 65^\circ\), and the included side \(BC = DC\), the SAS criteria perfectly fits.
This assures us of the congruence between \(\triangle ABC\) and \(\triangle ADC\).

Triangle Congruence Criteria

Triangle congruence criteria are essential tools in geometry to prove that two triangles are identical in shape and size. The basic triangle congruence criteria include:

• SSS (Side-Side-Side): All three sides in one triangle are equal to the corresponding sides in another triangle.
• SAS (Side-Angle-Side): As explored, it requires two sides and the included angle to match.
• ASA (Angle-Side-Angle): As mentioned, it requires that two angles and the included side must coincide.
• AAS (Angle-Angle-Side): When two angles and a non-included side are equal, congruence can be concluded.

These criteria are not just abstract concepts—they provide a systematic way to determine that two triangles may be mapped perfectly onto each other.
This certainty of congruence is foundational for many geometric proofs and constructions. In our case, we used the SAS criteria to readily establish that \(\triangle ABC\) is congruent to \(\triangle ADC\).