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. ?
On side square with line in middle in corners R, S, T, U.
3. The line ST is congruent to itself. This is the Reflexive Property of Congruence in Geometry.
4. angle RST is congruent to angle UTS. This is the Angle-Side-Angle (ASA) Congruence Postulate in Geometry. Since ST is congruent to itself (by the Reflexive Property) and RS and UT are congruent (given), the two triangles RST and UTS are congruent by the ASA postulate. Therefore, angle RST is congruent to angle UTS.
To justify the last two steps of the proof:
Step 3: ST ~= TS
This step can be justified by the fact that the line segment ST is equal in length to the line segment TS. This can be deduced from the nature of line segments where endpoints are interchangeable.
Step 4: angle RST ~= angle UTS
This step can be justified by the fact that the angles RST and UTS are corresponding angles formed by the intersection of the same line (TU) with a transversal (RS). Corresponding angles formed by two parallel lines and a transversal are congruent. Since RS and UT are not parallel, we can conclude that the angles RST and UTS are congruent.
To justify the last two steps of the proof, we need to use the given information and any relevant properties of the figure.
Step 3: ST ~= TS
To justify this step, we can use the fact that ST and TS are opposite sides of the same square. In a square, all sides are congruent, so ST and TS are equal in length. Thus, we can conclude that ST ~= TS.
Step 4: angle RST ~= angle UTS
To justify this step, we can use the fact that the diagonals of a square bisect each other and form congruent angles. In the given figure, RS and UT are the diagonals of the square. Since RS ~= UT (given) and RT ~= US (given), we can conclude that the diagonals RS and UT bisect each other at point T.
By the Vertical Angle Theorem, when two lines intersect, the opposite angles formed are congruent. Therefore, angle RST is congruent to angle UTS. Hence, we can justify the statement angle RST ~= angle UTS.