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Chapter 12: Chapter 12

Problem 194

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Prove that the diagonals of a parallelogram bisect each other

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Using the properties of a parallelogram and congruent triangles, we prove that triangles AEB and CED, formed by the diagonals of parallelogram ABCD, are congruent using the Side-Angle-Side (SAS) congruence postulate. Since triangle AEB is congruent to triangle CED, we have AE = EC and BE = DE, proving that diagonals AC and BD bisect each other at point E.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: 1. Draw a parallelogram with diagonals

Draw a parallelogram ABCD with diagonals AC and BD intersecting at point E. We will now show that the two triangles formed by the diagonals, triangle AEB and triangle CED, are congruent.

Step 2: 2. Identify corresponding sides of the parallelogram

In a parallelogram, opposite sides are parallel and equal in length. Therefore, we have the following relationships: - AB is parallel to CD and AB = CD - AD is parallel to BC and AD = BC

Step 3: 3. Identify corresponding angles of the parallelogram

In a parallelogram, consecutive angles are supplementary (sum up to 180 degrees), and opposite angles are equal. Therefore, we have the following relationships: - Angle A = Angle C (opposite angles) - Angle B = Angle D (opposite angles) - Angle A + Angle B = 180 degrees (consecutive angles) - Angle B + Angle D = 180 degrees (consecutive angles)

Step 4: 4. Prove triangles AEB and CED are congruent

We will use the Side-Angle-Side (SAS) congruence postulate to prove that triangles AEB and CED are congruent. According to the SAS postulate, if two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, then the triangles are congruent. In our case, we have the following equalities: - AB = CD (opposite sides of parallelogram) - AD = BC (opposite sides of parallelogram) - Angle AEB = Angle DEC (alternate angles corresponding to parallel lines AB and CD) Since we have two corresponding sides and an included angle equal, we can conclude that triangle AEB is congruent to triangle CED by the SAS postulate.

Step 5: 5. Prove that the diagonals bisect each other

Since triangle AEB is congruent to triangle CED, it follows that their corresponding parts are equal. Therefore, we have the following equalities: - AE = EC - BE = DE These equalities show that the diagonals AC and BD bisect each other at point E. This completes the proof that the diagonals of a parallelogram bisect each other.

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