Using the Angle-Side-Angle Method to Prove Triangles Congruent
How to Do an Indirect Proof
Using the Hypotenuse-Leg-Right Angle Method to Prove Triangles Congruent

Using a Point on a Perpendicular Bisector to Prove two Segments Congruent

You can use a point on a perpendicular bisector to prove that two segments are congruent. If the point is on the perpendicular bisector of a segment, then it’s equidistant from the endpoints of the segment. (Here’s an abbreviated version: If you have a perpendicular bisector, then there’s one pair of congruent segments.)

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The above figure shows you how this equidistance theorem works.

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Note that you can see the reasoning behind the short form of the theorem in the above diagram:

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Here’s a proof that uses this equidistance theorem:

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Statement 1:

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Reason for statement 1: Given.

Statement 2:

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Reason for statement 2: If sides, then angles.

Statement 3:

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Reason for statement 3: Given.

Statement 4:

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Reason for statement 4: If two congruent angles (angle 2 and angle 3) are added to two other congruent angles (angle 1 and angle 4), then the sums are congruent.

Statement 5:

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Reason for statement 5: If angles, then sides.

Statement 6:

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Reason for statement 6: If two points (M and Q) are equidistant from the endpoints of a segment (segment LN; see statements 1 and 5), then they determine the perpendicular bisector of that segment.

Statement 7:

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Reason for statement 7: If a point (point P) is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.

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Using the Side-Angle-Side Method to Prove Triangles Congruent
Using Two Equidistant Points to Determine a Perpendicular Bisector
Using the Angle-Angle-Side Method to Prove Triangles Congruent
Using the Side-Side-Side Method to Prove Triangles Congruent
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