Answer :
Answer:
All of the angles along the center of the parallelograms in the arm are congruent because they are either vertical angles or opposite angles of a parallelogram. By the Same-Side Interior Angles Postulate ∠1 is supplementary to these angles, so m∠1 = 64.
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Step-by-step explanation:
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We are given that three parallelograms are connected at each vertex to create an arm that can extend and collapse for an exploratory spaceship robot.
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The angles at the center of the connected parallelograms are congruent because they either fall under the property of opposite angles of a parallelogram or are congruent vertical angles formed by the intersection of lines at the points of connection.
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Same-side interior angles are pairs of angles that lie on the same side of a transversal and between two parallel lines, formed when the transversal intersects those lines. In the case of a parallelogram, the adjacent sides act as transversals to each pair of parallel sides, making adjacent angles same-side interior angles. Since same-side interior angles are supplementary (meaning they sum to 180°), each pair of adjacent angles in a parallelogram is also supplementary. Therefore, the measure of ∠1 can be found by subtracting the measure of the given angle 116° from 180°:
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[tex]m\angle 1 + 116^{\circ}=180^{\circ}[/tex]
[tex]m\angle 1=180^{\circ}-116^{\circ}[/tex]
[tex]m\angle 1=64^{\circ}[/tex]
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Therefore, the completed statement is:
- All of the angles along the center of the parallelograms in the arm are congruent because they are either vertical angles or opposite angles of a parallelogram. By the Same-Side Interior Angles Postulate ∠1 is supplementary to these angles, so m∠1 = 64.
