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Proofs using Triangle Congruence. Information. Perpendicular bisector theorem. Perpendicular bisector theorem : If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

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perpendicular bisector theorem
Perpendicular bisector theorem

Perpendicular bisector theorem:If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Given that AX is the perpendicular bisector of BC, prove that AB = AC.

by definition of perpendicular bisector:

BC⊥ AX

∠AXC = ∠AXB = 90°

∠AXC≅∠AXB

BX≅ XC

by the reflexive property of congruence:

AX≅ AX

by SAS congruence postulate:

△AXB ≅△AXC

by CPCTC:

AB = BC

AB≅ BC

converse of the perp bisector theorem
Converse of the perp. bisector theorem

Converse of the perpendicular bisector theorem:If a point is equidistant from endpoints of a segment, then it lies on the perpendicular bisector of the segment.

Set up the proof of the converse of the perpendicular bisector theorem.

given:

AB≅AC

need to show:

AX⊥ BC

BX≅ XC

angles in isosceles triangles
Angles in isosceles triangles

Which of the following choices could be the measures of three angles in an isosceles triangle?

C) 30°, 45°, 105°

A) 75°, 40°, 65°

D) 50°, 65°, 65°

B) 44°, 90°, 46°

What do you notice about the angles? Is this true in general of isosceles triangles?

base angle theorem
Base angle theorem

Base angle theorem: If two sides in a triangle are congruent, then the angles opposite these sides are also congruent.

Prove that the angles opposite the congruent sides in an isosceles triangle are congruent.

1. Since A is equidistant from the endpoints of BC, it lies on the perpendicular bisector of BC by the converse of the perpendicular bisector theorem.

2. It is given that AB and AC are congruent. X bisects BC, so BX and CX are congruent. With this, and since △ABX and △ACX share the side AX, △ABX and △ACX are congruent by SSS.

3. Corresponding parts of congruent triangles are congruent, therefore ∠B is congruent to ∠C.

angle bisector theorem and converse
Angle bisector theorem and converse

Angle bisector theorem and its converse:

A point lies on the bisector of an angle if and only if it is equidistant from the sides of the angle.

State the angle bisector theorem. As a challenge, prove it.

“If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle.” proof hint:Use AAS.

State the converse of the theorem. As a challenge, prove it.

“If a point is equidistant from the sides of the angle, then it lies on the bisector of the angle.” proof hint:Use SSS.

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