Question

Given parallelogram GRAM, prove ∠A and ∠R are supplementary.Fill in the blanks to complete the proof.(1 point)Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.Statements ReasonsABCD is a parallelogram.givenResponse areadefinition of a parallelogramResponse area are supplementaryConsecutive Interior Angles Theorem∠G and ∠AGR¯¯¯¯¯¯¯¯∥MA¯¯¯¯¯¯¯¯¯∠A and ∠RGR¯¯¯¯¯¯¯¯∥RA¯¯¯¯¯¯¯¯∠M and ∠RGM¯¯¯¯¯¯¯¯¯∥MA¯¯¯¯¯¯¯¯¯

Answer 1

To prove that angles ∠A and ∠R are supplementary in parallelogram GRAM, we can follow these statements and reasons:

Proof Structure:

1. Statement: GRAM is a parallelogram.
   Reason: Given.

2. Statement: ∠G and ∠A are consecutive interior angles.
   Reason: Definition of a parallelogram (opposite sides are parallel, and consecutive interior angles are supplementary).

3. Statement: GR || MA.
   Reason: Definition of a parallelogram (opposite sides are parallel).

4. Statement: ∠A + ∠G = 180°.
   Reason: Consecutive Interior Angles Theorem.

5. Statement: ∠R and ∠M are consecutive interior angles.
   Reason: Definition of a parallelogram.

6. Statement: GR || MA.
   Reason: Definition of a parallelogram (opposite sides are parallel).

7. Statement: ∠R + ∠M = 180°.
   Reason: Consecutive Interior Angles Theorem.

Since in parallelograms opposite angles are equal, we have ∠G = ∠M. Thus, substituting that into our earlier statement, we find:

8. Statement: ∠A + ∠R = 180°.
   Reason: Since ∠G = ∠M, substituting in yields the desired result.

Conclusion:

Thus, ∠A and ∠R are supplementary angles.

This completes the proof that in the parallelogram GRAM, the angles ∠A and ∠R are supplementary.