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Given: Isosceles trapezoid EFGH with EF¯¯¯¯¯¯¯¯≅GH¯¯¯¯¯¯¯¯ Prove: Opposite angles ∠EFG and ∠EHG are supplementary. When naming angles, put the letters in alphabetical order as much as possible. Statements Reasons 1. Isosceles trapezoid EFGH with EF¯¯¯¯¯¯¯¯≅GH¯¯¯¯¯¯¯¯ . 1. __________ 2. ∠EHG≅∠FEH 2. __________ 3. ___________ 3. Same side interior angles are supplementary. 4. m∠EHG+∠FGH=180° 4. ___________ 5. ∠EFG+∠EHG=180° 5. Substitution Property 6. __________ 6. definition of supplementary angles

1 answer

To complete the proof, we'll fill in the blanks with appropriate statements and reasons. Here’s how the proof looks:

| Statement | Reason | |---------------------------------------------------|-----------------------------------------------| | 1. Isosceles trapezoid EFGH with EF ≅ GH | 1. Given | | 2. ∠EHG ≅ ∠FEH | 2. Base angles of isosceles trapezoids are congruent | | 3. m∠EHG + m∠FGH = 180° | 3. Same side interior angles are supplementary | | 4. m∠EHG + m∠FGH = 180° | 4. Substitution (from 2 into 3) | | 5. ∠EFG + ∠EHG = 180° | 5. Substitution Property | | 6. ∠EFG and ∠EHG are supplementary | 6. Definition of supplementary angles |

Expanded steps for clarity:

  1. The first statement is just the given information about the trapezoid.
  2. From the properties of isosceles trapezoids, we know that the base angles (∠EHG and ∠FEH) are congruent.
  3. Since we are considering angles formed by parallel lines (EF and GH) crossed by a transversal (EH), the angle pair ∠EHG and ∠FGH must be supplementary (they are same-side interior angles).
  4. We substitute our earlier result that ∠EHG is congruent to ∠FEH into the equation regarding supplementary angles to relate them correctly.
  5. Lastly, we finish the proof by asserting that ∠EFG and ∠EHG sum to 180°, which coincides with the definition of supplementary angles.

Thus, we have proved that angles ∠EFG and ∠EHG are indeed supplementary.