Geometric Proofs: SAS & SSS Guided Notes and Practice - Free Printable

Looking at problem 4, I can see we need to prove two triangles are congruent using the SAS (Side-Angle-Side) postulate. The given information is:
- M is a midpoint of DF (which means DM ≅ MF by definition of midpoint)
- DT ≅ FT (side)

Since M is the midpoint of DF, we know DM ≅ MF. We also have DT ≅ FT as given. To use SAS, we need the included angle between these sides to be congruent. The included angles would be ∠DMT and ∠FMT.

I notice that these two angles share the side MT and are formed by the segments DM, MF (which are congruent since M is the midpoint) and DT, FT (which are given as congruent). Since we have two sides and the included angle, this should be a SAS proof.

Let me now solve each problem systematically.

Problem 1 Solution:
Given: AB ≅ DE, ∠C ≅ ∠F, BC ≅ EF
We need to prove: △ABC ≅ △DEF

Proof:
1. AB ≅ DE (Given)
2. ∠C ≅ ∠F (Given)
3. BC ≅ EF (Given)
4. Therefore, △ABC ≅ △DEF (SAS Postulate - Side-Angle-Side)

Problem 2 Solution:
Given: AB ≅ DE, BC ≅ EF, ∠A ≅ ∠D
Wait, this appears to be an error. For SSS, we need three sides, not two sides and an angle. This seems to be intended for SAS congruence, but the worksheet labels it as SSS.

If we proceed with the given information, we could use SAS if ∠A and ∠D were the included angles between the given sides. However, ∠A is between AB and AC, not between AB and BC.

This problem may contain an error in the worksheet. If we assume it's meant to be SAS, then we'd need ∠B ≅ ∠E (the included angle between AB and BC, and DE and EF respectively).

Problem 3 Solution:
Given: AB ≅ DE, BC ≅ EF, AC ≅ DF
We need to prove: △ABC ≅ △DEF

Proof:
1. AB ≅ DE (Given)
2. BC ≅ EF (Given)
3. AC ≅ DF (Given)
4. Therefore, △ABC ≅ △DEF (SSS Postulate - Side-Side-Side)

Problem 4 Solution:
Given: M is a midpoint of DF, DT ≅ FT
We need to prove: △DTM ≅ △FTM

Proof:
1. M is a midpoint of DF (Given)
2. Therefore, DM ≅ MF (Definition of Midpoint)
3. DT ≅ FT (Given)
4. TM ≅ TM (Reflexive Property - same segment)
5. Therefore, △DTM ≅ △FTM (SSS Postulate - Side-Side-Side)

Actually, looking at problem 4 again, if we're to use SAS, we need to identify the included angle. Since we have DM ≅ MF and DT ≅ FT, and the included angle would be ∠DMT and ∠FMT. But we don't have information about these angles being congruent.

However, if we consider the reflexive property for side TM, then we actually have SSS congruence, not SAS.

The worksheet appears to have some inconsistencies. Let me provide the most logical solutions based on the given information:

Final Solutions:

Problem 1:
Statement 1: AB ≅ DE
Reason 1: Given
Statement 2: ∠C ≅ ∠F
Reason 2: Given
Statement 3: BC ≅ EF
Reason 3: Given
Statement 4: △ABC ≅ △DEF
Reason 4: SAS Postulate

Problem 2:
There appears to be an error in this problem. For SSS, we need three sides, but only two sides and an angle are given. If we assume it's meant to be SAS, we would need the included angle between the given sides. As written, this problem cannot be solved with SSS.

Problem 3:
Statement 1: AB ≅ DE
Reason 1: Given
Statement 2: BC ≅ EF
Reason 2: Given
Statement 3: AC ≅ DF
Reason 3: Given
Statement 4: △ABC ≅ △DEF
Reason 4: SSS Postulate

Problem 4:
Statement 1: M is a midpoint of DF
Reason 1: Given
Statement 2: DM ≅ MF
Reason 2: Definition of Midpoint
Statement 3: DT ≅ FT
Reason 3: Given
Statement 4: TM ≅ TM
Reason 4: Reflexive Property
Statement 5: △DTM ≅ △FTM
Reason 5: SSS Postulate

Note: Problem 4 is actually proven with SSS, not SAS as indicated in the worksheet title. The worksheet may contain errors in labeling the congruence postulates for problems 2 and 4.