a Let's use the labeling suggested in the question to redraw the diamond.
Gathering Information
Since the sides of the diamond have the same length, these are congruent.
ST≅TR≅RF≅FS
There are also some perpendicular corners marked on the diagram. Since all right angles are congruent, these angles are also congruent.
∠ T≅∠ S≅∠ F
Let's use the labels to state the distances the question is asking about.
Description
Expression
Distance from first base to third base.
FT
Distance from home plate to second base.
RS
Formal Statement
We are now ready to use the labels to state the given information and the claim we are asked to prove.
Given: &ST≅TR≅RF≅FS
& ∠ T≅∠ S≅∠ F
Prove: &FT=RS
Explanation
Let's draw two copies of the diamond with the one-one diagonal added. Instead of the length of the sides, let's use markers to indicate congruence. On both copies we will focus on half of the diamond.
We can see congruent sides and angles in the two triangles.
TS&≅ SF
SF&≅ FR
∠ S&≅ ∠ F
Since two pairs of sides and the included angles are congruent, the Side-Angle-Side (SAS) Congruence Postulate guarantees that the triangles are congruent.
△ TSF≅ △ SFR
Since corresponding segments of congruent triangles are congruent, this means that the diagonals of the diamond are congruent.
FT≅ RS
Using the definition of congruent segments, this means that the diagonals have the same length.
FT= RS
Thus, we proved that the distance from first base to third base is the same as the distance from home plate to second base.
Two-column Proof
Given: &ST≅TR≅RF≅FS
& ∠ T≅∠ S≅∠ F
Prove: &FT=RS
Proof:
Statements
Reasons
1.
ST≅TR≅RF≅FS ∠ T≅∠ S≅∠ F
1.
Given
2.
△ TSF≅△ SFR
2.
Side-Angle-Side (SAS) Congruence Postulate
3.
FT≅RS
3.
Corresponding sides of congruent triangles are congruent
4.
FT=RS
4.
Definition of congruent segments
b We can use the labeling in Part A to express the angles in the question.
Description
Expression
Angle formed between second base, home plate, and third base.
∠ SRT
Angle formed between second base, home plate, and first base.
∠ SRF
Formal Statement
Like in Part A, we can formally state the claim.
Note that we are still working with the same given congruences.
Given: &ST≅TR≅RF≅FS
& ∠ T≅∠ S≅∠ F
Prove: &∠ SRT≅∠ SRF
Explanation
Let's include the diagonal SR on the diagram and mark the angles we are interested in.
Triangles △ SRT and △ SRF have two pairs of congruent sides and congruent included angles, so like in Part A they are congruent.
△ SRT≅ △ SRF
Since corresponding angles of congruent triangles are congruent, this means that the angles the question asks about are congruent.
∠ SRT≅ ∠ SRF
Thus, we proved that the angle formed between second base, home plate, and third base is congruent to the angle formed between second base, home plate, and first base.
