Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the...
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Transcript of Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the...
Section 5-2Bisectors in Triangles
Vocabulary
• Distance from a point to a line: the length of the perpendicular segment from the point to the line.
Theorems
• Perpendicular Bisector Theorem- If a point lies on the perpendicular bisector of a line segment, then it is an equal distance away from both endpoints of the line segment.
• Angle Bisector Theorem- If a point lies on the angle bisector of an angle, then it is an equal distance away from both sides of the angle.
Converse of the Theorems
• Converse of the Perpendicular Bisector Theorem- If a point is an equal distance away from the endpoints of a line segment, then it lies on the perpendicular bisector of the line segment.
• Converse of the Angle Bisector Theorem- If a point in the interior of an angle is an equal distance away from both sides of the angle, then it lies on the angle bisector of the angle.
Perpendicular Bisector Theorem
Angle Bisector Theorem
Proof of Perpendicular Bisector Theorem
Statement Reason
BD is the bisector of AC┴ Given
∠ABD and CBD are right angles∠ Definition of perpendicular
∠ABD ∠ CBD ∠ All right angles are congruent
DB DB≅ Reflexive Property of Congruency
AB CB≅ Definition of bisector
∆ABD ∆CBD≅ SAS
AD CD≅ CPCTC
Proof of Angle Bisector TheoremStatement Reason
AD is the angle bisector of CAB∠ Given
CD is to AC┴ By construction
DB is to AB┴ By construction
∠ACB and ABD are right angles∠ Definition of perpendicular
∠ACB ∠ ABD ∠ All right angles are congruent
∠CAD BAD≅ ∠ Definition of angle bisector
AD AD≅ Reflexive Property pf Congruency
∆CAD ∆BAD≅ AAS
CD BD≅ CPCTC
Practice Problem
• Given: BE is the perpendicular bisector of AC, AED CEF, ∠ ≅ ∠DE FE.≅
• Prove: DAE FCE∠ ≅ ∠Answer on next slide⫸
Solution to Practice ProblemStatement Reason
BE is ┴ bisector of AC Given
AE CE≅ Perpendicular Bisector Theorem
DE FE≅ Given
∠AED CEF≅ ∠ Given
∆ADE ∆CFE≅ SAS
∠DAE FCE≅ ∠ CPCTC
Practice Problem 2
• Find the value of X and YAnswer on next slide⫸
Solution to Practice Problem 2
• Answer: X = -3, Y = 12
Extra Resources
• http://www.youtube.com/watch?v=oskp0T8aZJw (very weird jumpy guy explaining the perpendicular bisector theorem)
• http://www.youtube.com/watch?v=9k8QMHIFwOk&list=PL668AB35C6885A036&index=35 (same weird guy explaining the angle bisector theorem)