Draw PQ so that P is on YZ, Q is on YX, PQ ∥ XZ, and YP=ST
2.
Construction
3.
∠ YPQ ≅ ∠ Z
3.
Corresponding Angles Theorem
4.
∠ Y ≅ ∠ Y
4.
Reflexive Property of Congruence
5.
△ QYP ~ △ XYZ
5.
Angle-Angle (AA) Similarity Postulate
6.
QY/XY = YP/YZ = QP/XZ
6.
Corresponding sides of similar triangles are proportional
7.
QY/XY = ST/YZ
7.
Substitution
8.
QY/XY = RS/XY
8.
Substitution
9.
QY=RS
9.
Multiplying both sides by XY
10.
QY≅ RS and YP≅ ST
10.
Definition of congruent segments
11.
△ YPQ ≅ △ STR
11.
SAS Congruence Theorem
12.
∠ YPQ ≅ ∠ T
12.
Definition of congruent triangles
13.
∠ T ≅ ∠ Z
13.
Transitive Property of Congruence
14.
△ RST ~ △ XYZ
14.
AA Similarity Postulate
Practice makes perfect
We are asked to write a two-column proof of the following theorem.
Side-Angle-Side (SAS) Similarity
If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
Let's consider △ RST and △ XYZ such that ∠ S ≅ ∠ Y and RS XY = ST YZ.
Let P be a point on YZ such that YP = ST. Then, we will draw PQ such that PQ∥ RT.
The Side-Angle-Side (SAS) Congruence Theorem leads us to conclude that △ YPQ ≅ △ STR. This implies that ∠ YPQ ≅ ∠ T, since they are corresponding angles. Earlier we found that ∠ YPQ ≅ ∠ Z. By the Transitive Property of Congruence, this gives us the following.
∠ YPQ ≅ ∠ T and ∠ YPQ ≅ ∠ Z
⇓
∠ T ≅ ∠ Z
Here is what we found about angles in △ RST and △ XYZ.
ccc
∠ S ≅ ∠ Y & & Angle
∠ T ≅ ∠ Z & & Angle
One more time applying the AA Similarity Postulate, we conclude that △ RST ~ △ XYZ, which is what we wanted to prove.
Two-Column Proof
Let's summarize the proof we wrote in the following two-column table.
Given: & ∠ S ≅ ∠ Y and RSXY = STYZ
Prove: & △ RST ~ △ XYZ
Statements
Reasons
1.
∠ S ≅ ∠ Y and RS/XY = ST/YZ
1.
Given
2.
Draw PQ so that P is on YZ, Q is on YX, PQ ∥ XZ, and YP=ST
2.
Construction
3.
∠ YPQ ≅ ∠ Z
3.
Corresponding Angles Theorem
4.
∠ Y ≅ ∠ Y
4.
Reflexive Property of Congruence
5.
△ QYP ~ △ XYZ
5.
Angle-Angle (AA) Similarity Postulate
6.
QY/XY = YP/YZ = QP/XZ
6.
Corresponding sides of similar triangles are proportional
