Question

When proving ASTE - ANTE, the statement TE-TE can be used because...

TE bisects line segment SN.
Angles 1 and 3 are supplementary.
Congruent parts of congruent triangles are congruent (CPCTC).
Every segment is congruent to itself.

Answer 1

When proving congruence such as in ASTE - ANTE, TE-TE is used based on the reflexive property of congruence, which asserts that every segment is congruent to itself.

Step-by-step explanation:

When proving ASTE - ANTE, the statement TE-TE can be used because every segment is congruent to itself.

This principle, referred to as the reflexive property of congruence, stipulates that any geometric figure, including a line segment, is always congruent to itself.

It's a foundational concept in geometry that applies regardless of any additional details provided, such as TE bisecting line segment SN or angles 1 and 3 being supplementary.

Applying the reflexive property is particularly useful in the context of geometric proofs and is often used to establish the congruence of corresponding parts in congruent triangles, as per the CPCTC (Congruent Parts of Congruent Triangles are Congruent) postulate.

Answer 2

The statement TE-TE can be used when proving ASTE - ANTE because TE bisects line segment SN, and according to the Segment Bisector Theorem, it divides the segment into two congruent parts. Therefore, TE = TE, establishing an important equality in the proof.

Step-by-step explanation:

In the given scenario, the line segment TE is a bisector of SN. According to the Segment Bisector Theorem, when a segment is bisected, it divides the segment into two congruent parts. Mathematically, this can be expressed as TE = SE. This property is fundamental in our proof as it provides the basis for the subsequent statements.

Another key aspect is the supplementary nature of Angles 1 and 3. When two angles are supplementary, their measures add up to 180 degrees. In this case, if Angle 1 is congruent to Angle 3, we can express this relationship as m∠1 + m∠3 = 180°. This supplementary property is crucial for establishing relationships between angles in the triangles ASTE and ANTE.

Additionally, the statement "Congruent parts of congruent triangles are congruent (CPCTC)" plays a pivotal role. Once we establish the congruence of corresponding parts of triangles ASTE and ANTE, we can conclude that corresponding angles and sides are also congruent. This principle is a direct application of CPCTC and strengthens our overall proof.

Lastly, the assertion that every segment is congruent to itself is a basic property of equality. In mathematical terms, it is expressed as AB = AB, where AB is any segment. This property helps in reinforcing the equality relationships in the proof, providing a solid foundation for the logical progression of our argument.