Short Answer
Note: Your teacher will grade your responses to questions 7–10 to ensure you receive proper credit for your answers.
Justify the last two steps of the proof.
Given: RS~=UT and RT ~= US
Prove: angle RST ~= angle UTS
1. RS~= UT 1. Given
2. RT~= US 2. Given
3. ST ~= TS 3. ?
4. angle RST ~= angle UTS 4. ?
3. The third step, ST ~= TS, is not given in the original statement. This step requires additional information or a reasoning process to justify it. Without more information or reasoning, we cannot determine the equality of ST and TS.
4. The fourth step, angle RST ~= angle UTS, follows from the previous steps because corresponding sides of congruent angles are congruent. This is a property of congruent triangles. Therefore, we can conclude that angle RST is congruent to angle UTS.
To justify the last two steps of the proof:
3. ST ~= TS: This step can be justified by the segment symmetry property. The segment symmetry property states that if two segments have the same length, then they are congruent. Since RS ~= UT in step 1 and RT ~= US in step 2, the transitive property can be applied to conclude that RS + ST = UT + TS, which implies ST = TS.
4. angle RST ~= angle UTS: This step can be justified by the angle side-angle (ASA) congruence theorem. The ASA congruence theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. In this case, since RS ~= UT, RT ~= US, and ST ~= TS, we have two corresponding sides and the included angle of one triangle congruent to two corresponding sides and the included angle of another triangle. Therefore, by the ASA congruence theorem, it can be concluded that angle RST is congruent to angle UTS.
To justify the last two steps of the proof, we need to use the previously given information and apply known properties of congruent triangles.
Step 3: ST ~= TS (Justification ?)
To prove that ST is congruent to TS, we can use the fact that the two pairs of sides RS and UT, and RT and US, are given to be congruent. According to the Side-Side-Side (SSS) congruence criterion, if three corresponding sides of two triangles are congruent, then the triangles themselves are congruent. So, by using the given information of RS ~= UT and RT ~= US, we can conclude that triangle RST is congruent to triangle UTS (by SSS).
Step 4: angle RST ~= angle UTS (Justification ?)
Once we have established that triangle RST is congruent to triangle UTS (by SSS), we can conclude that the corresponding angles of the congruent triangles are also congruent. This is known as the Angle-Angle-Side (AAS) congruence criterion. The AAS criterion states that if two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the two triangles are congruent. So, by the virtue of RST being congruent to UTS (by SSS), we can conclude that angle RST is congruent to angle UTS (by AAS).
Therefore, both Step 3 and Step 4 can be justified by applying congruence criteria and using the given information in the proof.