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Download 02 - Parallelogram Proof Unscramble ANSWERS
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Names: __________________________________________ 2 PARALLELOGRAMS PROOFS UNSCRAMBLE Directions: Cut out the steps of the proof and arrange them (and glue them) logically to prove that the shape is parallelogram. Mark the quadrilateral with the information in the proof as you go. Proof #1: If opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Μ Μ Μ Μ and πΆπ· Μ Μ Μ Μ are congruent and that π΄π· Μ Μ Μ Μ and π΅πΆ Μ Μ Μ Μ We are given that π΄π΅ are congruent. Μ Μ Μ Μ and π·π΅ Μ Μ Μ Μ are congruent using the reflexive property because π΅π· they are the same line. So, triangle ABC is congruent to triangle CDB by the Side-Side-Side congruence. This means that angle ABC and angle CDB are congruent because corresponding parts of congruent triangles are congruent. Then we can see that β π΄π΅πΆ and β πΆπ·π΅ are alternate interior angles formed by the lines Μ Μ Μ Μ π΄π΅ and Μ Μ Μ Μ πΆπ· with the transversal Μ Μ Μ Μ π΅π·. Because they are congruent alternate interior angles, we know Μ Μ Μ Μ and πΆπ· Μ Μ Μ Μ are parallel. that π΄π΅ Similarly, angle BDA is congruent to angle DBC since corresponding parts of congruent triangles are congruent. Then we can see that β π΅π·π΄ and β π·π΅πΆ are alternate interior angles formed by the lines Μ Μ Μ Μ π΄πΆ and Μ Μ Μ Μ π΅π· with the transversal Μ Μ Μ Μ π΅πΆ . Because they are congruent alternate interior angles, we know that Μ Μ Μ Μ π΄π· and Μ Μ Μ Μ π΅πΆ are parallel. Since we have two sets of parallel sides that form quadrilateral ABCD, we know that ABCD is a parallelogram. 1 β Attempt at proof unscramble is incomplete. 2 β There are large logical errors in the proof or diagrams. 3 β There are small errors in the proof or diagrams. 4 β Proof is unscrambled correctly and most diagrams are marked with correct information. 5 β Proof is unscrambled correctly and all diagrams are marked with correct information. Names: __________________________________________ 2 PARALLELOGRAMS PROOFS UNSCRAMBLE Directions: Cut out the steps of the proof and arrange them (and glue them) logically to prove that the shape is parallelogram. Mark the quadrilateral with the information in the proof as you go. Proof #2: If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram. We are given that Μ Μ Μ Μ π΄π΅ and Μ Μ Μ Μ π·πΆ are parallel. We are also given that Μ Μ Μ Μ is congruent to π·πΆ Μ Μ Μ Μ . π΄π΅ Μ Μ Μ Μ and π·πΆ Μ Μ Μ Μ are parallel, we Μ Μ Μ Μ , since π΄π΅ If we draw in the diagonal π΅π· form congruent alternate interior angles β π΄π΅π· and β πΆπ·π΅. Μ Μ Μ Μ π΅π· is congruent to Μ Μ Μ Μ π·π΅ since they are the same line using the reflexive property. Then, we can use the Side-Angle-Side triangle congruence theorem to show that Ξπ΄π΅π· is congruent to ΞπΆπ·π΅. Then we also know that β πΆπ΅π· is congruent to β π΄π·π΅, since corresponding parts of congruent triangles are congruent. But β πΆπ΅π· and β π΄π·π΅ are also alternate interior angles formed by the lines Μ Μ Μ Μ π΄π· and Μ Μ Μ Μ π΅πΆ with the transversal Μ Μ Μ Μ π΅π·. Since they are congruent alternate interior angles, we know that Μ Μ Μ Μ π΄π· and Μ Μ Μ Μ π΅πΆ are parallel. Since we have two sets of parallel sides that form quadrilateral ABCD, we know that ABCD is a parallelogram. 1 β Attempt at proof unscramble is incomplete. 2 β There are large logical errors in the proof or diagrams. 3 β There are small errors in the proof or diagrams. 4 β Proof is unscrambled correctly and most diagrams are marked with correct information. 5 β Proof is unscrambled correctly and all diagrams are marked with correct information. Names: __________________________________________ 2 PARALLELOGRAMS PROOFS UNSCRAMBLE Directions: Cut out the steps of the proof and arrange them (and glue them) logically to prove that the shape is parallelogram. Mark the quadrilateral with the information in the proof as you go. Proof #3: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Μ Μ Μ Μ and ππΆ Μ Μ Μ Μ are congruent and that π·π Μ Μ Μ Μ and ππ΅ Μ Μ Μ Μ We are given that π΄π Μ Μ Μ Μ Μ Μ Μ Μ are congruent. First, we will prove that π΅πΆ and π΄π· are parallel. Because β π΄ππ· and β πΆππ΅ are vertical angles, we know that they are congruent. So Ξπ΄ππ· and ΞπΆππ΅ are congruent by the Side-Angle-Side triangle congruence theorem. Because of this, we know that β ππ·π΄ and β ππ΅πΆ are congruent because corresponding parts of congruent triangles are congruent. But β ππ·π΄ and β ππ΅πΆ are also alternate interior angles formed by Μ Μ Μ Μ and π΄π· Μ Μ Μ Μ and the transversal π΅π· Μ Μ Μ Μ . π΅πΆ Since they are congruent alternate interior angles, we know that Μ Μ Μ Μ and π΄π· Μ Μ Μ Μ are parallel. π΅πΆ We can apply the same logic to prove that Μ Μ Μ Μ π΄π΅ and Μ Μ Μ Μ πΆπ· are parallel. β π΄ππ΅ and β πΆππ· are congruent because they are vertical angles. Ξπ΄ππ΅ and Ξπ΄ππ΅ are congruent triangles because of the SideAngle-Side congruence theorem. β π΅π΄π and β π·πΆπ are congruent because corresponding parts of congruent triangles are congruent. β π΅π΄π and β π·πΆπ are also alternate interior angles formed by Μ Μ Μ Μ π΄π΅ Μ Μ Μ Μ and the transversal π΄πΆ Μ Μ Μ Μ . and πΆπ· Since they are congruent alternate interior angles, we know that Μ Μ Μ Μ π΄π΅ and Μ Μ Μ Μ πΆπ· are parallel. Since we have two sets of parallel sides that form quadrilateral ABCD, we know that ABCD is a parallelogram. 1 β Attempt at proof unscramble is incomplete. 2 β There are large logical errors in the proof or diagrams. 3 β There are small errors in the proof or diagrams. 4 β Proof is unscrambled correctly and most diagrams are marked with correct information. 5 β Proof is unscrambled correctly and all diagrams are marked with correct information. Names: __________________________________________ 2 PARALLELOGRAMS PROOFS UNSCRAMBLE Directions: Cut out the steps of the proof and arrange them (and glue them) logically to prove that the shape is parallelogram. Mark the quadrilateral with the information in the proof as you go. Proof #4: If one angle is supplementary to both consecutive angles in a quadrilateral, the quadrilateral is a parallelogram. We are given that angle A and angle B are supplementary. We are also given that angle A and angle D are supplementary. Because angle A and angle B are same side interior angles formed by Μ Μ Μ Μ π΄π· and Μ Μ Μ Μ π΅πΆ with the transversal Μ Μ Μ Μ π΄π΅ that are supplementary, we Μ Μ Μ Μ and π΅πΆ Μ Μ Μ Μ are parallel. know that π΄π· Because angle A and angle D are same side interior angles formed by Μ Μ Μ Μ π΄π΅ and Μ Μ Μ Μ π·πΆ and the transversal Μ Μ Μ Μ π΄π· that are supplementary, we Μ Μ Μ Μ and π΅πΆ Μ Μ Μ Μ are parallel. know that π΄π΅ Μ Μ Μ Μ and π΅πΆ Μ Μ Μ Μ are parallel and π΄π΅ Μ Μ Μ Μ and π΅πΆ Μ Μ Μ Μ are parallel, we Because π΄π· know that ABCD is a parallelogram, since it is formed by two pairs of parallel sides. 1 β Attempt at proof unscramble is incomplete. 2 β There are large logical errors in the proof or diagrams. 3 β There are small errors in the proof or diagrams. 4 β Proof is unscrambled correctly and most diagrams are marked with correct information. 5 β Proof is unscrambled correctly and all diagrams are marked with correct information.
