Here are the completed proofs for the geometry worksheet.
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Problem 1
Goal: Prove that triangle ABC is congruent to triangle DEC.
Reasoning:
1.
Given: We are told that C is the midpoint of two lines, BE and AD.
2.
Definition of Midpoint: A midpoint cuts a line into two equal parts. So, segment BC equals EC, and segment AC equals DC.
3.
Vertical Angles: Where the two lines cross at point C, they create "vertical angles" (the angles opposite each other). Vertical angles are always equal. So, angle BCA equals angle ECD.
4.
Congruence: We now have two sides and the angle in between them equal for both triangles (Side-Angle-Side).
Completed Table:
| Statement | Reason |
| :--- | :--- |
| 1. C is midpoint of BE and AD | 1. Given |
| 2. BC ≅ EC | 2. definition of a midpoint |
| 3.
AC ≅ DC | 3. definition of a midpoint |
| 4.
∠BCA ≅ ∠ECD | 4.
Vertical Angles Theorem |
| 5. ΔABC ≅ ΔDEC | 5.
SAS (Side-Angle-Side) |
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Problem 2
Goal: Prove that triangle ABC is congruent to triangle CDA.
Reasoning:
1.
Given: We know side BC is equal to side DA, and line AC cuts angle BCD in half.
2.
Angle Bisector: Since AC bisects angle BCD, it creates two equal angles: angle BCA and angle DCA.
3.
Reflexive Property: Both triangles share the side AC. A line segment is always equal to itself.
4.
Congruence: We have two sides and the angle in between them equal (Side-Angle-Side).
Completed Table:
| Statement | Reason |
| :--- | :--- |
| 1.
BC ≅ DA | 1.
Given |
| 2. AC bisects ∠BCD | 2. Given |
| 3.
∠BCA ≅ ∠DCA | 3.
Definition of Angle Bisector |
| 4. AC ≅ AC | 4.
Reflexive Property of Congruence |
| 5.
ΔABC ≅ ΔCDA | 5.
SAS (Side-Angle-Side) |
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Problem 3
Goal: Prove that triangle XGF is congruent to triangle HFG.
Reasoning:
1.
Given: We are given that angle X equals angle H, and line XG is parallel to line FH.
2.
Parallel Lines: When lines are parallel, "alternate interior angles" are equal. Looking at the "Z" shape formed by lines XG, FH, and transversal FG, the angles inside are equal. So, angle XGF equals angle HFG.
3.
Reflexive Property: Both triangles share the side FG.
4.
Congruence: We have two angles and a non-included side equal (Angle-Angle-Side).
Completed Table:
| Statement | Reason |
| :--- | :--- |
| 1.
∠X ≅ ∠H | 1. Given |
| 2.
XG || FH | 2. Given |
| 3. ∠FGX ≅ ∠GFH | 3.
Alternate Interior Angles Theorem |
| 4.
FG ≅ FG | 4.
Reflexive Property of Congruence |
| 5.
ΔXGF ≅ ΔHFG | 5.
AAS (Angle-Angle-Side) |
Parent Tip: Review the logic above to help your child master the concept of geometry triangle proof worksheet.
