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Proofs using Triangle Congruence

Proofs using Triangle Congruence. Information. Exploring perpendicular bisectors. 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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Proofs using Triangle Congruence

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  1. Proofs using Triangle Congruence

  2. Information

  3. Exploring perpendicular bisectors

  4. 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

  5. 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

  6. Proof of the converse

  7. 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?

  8. 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. 

  9. 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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