Question

Given: 6SQ = 10TS; 6PS = 10MSProve: △△QPS ~ △△TMS

Answer 1

The proof establishes triangle similarity using Division and Transitive Properties, the definition of vertical angles, and the SAS Similarity Theorem, affirming proportional sides and a congruent included angle.

The given proof establishes the similarity of two triangles, SQ/TS and PS/MS. Starting with the given ratios SQ/TS = 10/6 and PS/MS = 10/6, the Division Property of Equality is applied to express these ratios as 6SQ = 10TS and 6PS = 10MS. The Transitive Property of Equality is then employed to equate these expressions, leading to PS/MS = SQ/TS.

Furthermore, Statement 4 invokes the definition of vertical angles, asserting that vertical angles are congruent. This congruence is crucial in Statement 5, where the SAS (Side-Angle-Side) Similarity Theorem is applied. The theorem relies on two triangles having proportional sides and a congruent included angle, which, in this case, are satisfied by the given conditions.

In summary, the proof progresses logically by applying mathematical properties such as the Division Property of Equality, the Transitive Property of Equality, and the definition of vertical angles. Ultimately, the triangles SQ/TS and PS/MS are deemed similar using the SAS Similarity Theorem, as sides are proportional and one angle is congruent.

Answer 2

PROOF:

1. Statement 1 is already given in the question.

2. Statement 2 is true by Division Property of Equality.

SQ/TS = 10/6 when cross-multiplied is just equal to 6SQ = 10TS.

PS/MS = 10/6 when cross-multiplied is just equal to 6PS = 10MS.

3. Statement 3 is true by Transitive Property of Equality.

The transitive property of equality states that if a = b and b = c, then a = c.

Since SQ/TS and PS/MS are both equal to 10/6, then we can say that PS/MS = SQ/TS.

4. Statement 4 is true by the definition of vertical angles. Vertical Angles are congruent.

5. Lastly, Statement 5 is true by SAS Similarity Theorem.

The corresponding sides of the two triangles are proportional to each other and have a ratio of 10/6 as stated in statements 2 and 3. Therefore, we now have two corresponding sides of each triangle that are proportional.

In addition, one included angle of these two sides as stated by Statement 4 is congruent to each other.

Since we have two sides that are proportional and an included angle that is congruent, the two triangles are similar by SAS or Side-Angle-Side Similarity Theorem.