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.
A 2 column proof with the image of a quadrilateral is shown.
The image shows a quadrilateral R T U S with a horizontal diagonal drawn from S on the left to T on the right. Point R is the upper vertex and point U is the lower vertex.
Given
modifying above R S with bar is congruent to modifying above U T with bar and
modifying above R T with bar is congruent to modifying above U S with bar
Prove
triangle R S T is congruent to triangle U T S
Proof:
first row: first column: modifying above R S with bar is congruent to modifying above U T with bar, second column Given
second row first column: modifying above R T with bar is congruent to modifying above U S with bar, second column Given
third row first column: modifying above S T with bar is congruent to modifying above T S with bar, second column question mark
fourth row first column: triangle R S T is congruent to triangle U T S, second column question mark
(4 points)
To justify the last two steps of the proof, we need to use the property of congruent triangles called side-side-side (SSS) congruence.
For the third row, we can justify it using SSS congruence. We know that modifying above RS with bar is congruent to modifying above UT with bar, and modifying above RT with bar is congruent to modifying above US with bar. Additionally, we are given that modifying above ST with bar is congruent to modifying above TS with bar. These three pairs of congruent segments form the corresponding sides of both triangles. Therefore, we can conclude that triangle RST is congruent to triangle UTS by the SSS congruence.
For the fourth row, we can justify it using the Transitive Property of Congruence. Since we have proven that triangle RST is congruent to triangle UTS, and we are given that triangle RST is congruent to triangle UTS in the question, we can use the Transitive Property to state that triangle RST is congruent to triangle UTS.
Therefore, the last two steps are justified.