given: segment HI congruent to segment GJ, segment HI parallel to segment GJ
prove: triangle GJH congruent to triangle IHJ
Proof:
Since segment HI is congruent to segment GJ and parallel to segment GJ, then angles HGI and IHJ are congruent.
Since angles HGI and IHJ are congruent, then sides GH and IJ are congruent.
Since sides GH and IJ are congruent, then angles GHI and IJH are congruent.
Since angles GHI and IJH are congruent, then triangle GJH is congruent to triangle IHJ.
Therefore, triangle GJH is congruent to triangle IHJ.
To prove that triangle GJH is congruent to triangle IHJ, we can use the concept of the "Side-Angle-Side" (SAS) congruence criterion.
SAS Congruence Criterion states that if two triangles have two sides and the included angle of one triangle congruent to the corresponding two sides and included angle of the other triangle, then the triangles are congruent.
Given:
- Segment HI is congruent to segment GJ (HI ≅ GJ)
- Segment HI is parallel to segment GJ (HI || GJ)
To prove:
- Triangle GJH is congruent to triangle IHJ (GJH ≅ IHJ)
Proof:
1. Given: HI ≅ GJ, HI || GJ
2. The segment HI and GJ are parallel, so corresponding angles GHJ and IHJ are congruent by alternate interior angles theorem.
3. The segment HI and GJ are congruent, so corresponding sides GH and IH are congruent.
4. Triangle GJH and IHJ have side GH congruent to side IH (common side)
5. Triangle GJH and IHJ have angle GHJ congruent to angle IHJ (corresponding angles)
6. Triangle GJH and IHJ have side GJ congruent to side HI (given)
7. By SAS Congruence Criterion, triangle GJH is congruent to triangle IHJ.
Therefore, we have shown that triangle GJH is congruent to triangle IHJ based on the given information and the SAS congruence criterion.
To prove that triangle GJH is congruent to triangle IHJ, we can use the Side-Angle-Side (SAS) congruence criterion. Here is the step-by-step proof:
Given:
- Segment HI is congruent to segment GJ.
- Segment HI is parallel to segment GJ.
To prove:
- Triangle GJH is congruent to triangle IHJ.
Proof:
Step 1: Given information.
- Segment HI is congruent to segment GJ (Given).
- Segment HI is parallel to segment GJ (Given).
Step 2: Parallel lines cut by a transversal.
Since HI is parallel to GJ, it forms pairs of corresponding angles when intersected by a transversal. Therefore, we can conclude that angle JIH is congruent to angle GHJ and angle GJH is congruent to angle HIJ.
Step 3: Side-Angle-Side (SAS) congruence criterion.
- Side: Segment HI is congruent to segment GJ (Given).
- Angle: Angle GJH is congruent to angle HIJ (Step 2).
- Side: Segment GH is congruent to segment IJ (Given).
Step 4: Conclusion.
By the SAS congruence criterion, we have shown that triangle GJH is congruent to triangle IHJ.