Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the...

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Section 5-2 Bisectors in Triangles

Transcript of Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the...

Page 1: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Section 5-2Bisectors in Triangles

Page 2: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Vocabulary

• Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Page 3: Section 5-2 Bisectors 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.

Page 4: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

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.

Page 5: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Perpendicular Bisector Theorem

Page 6: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Angle Bisector Theorem

Page 7: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

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

Page 8: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

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

Page 9: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Practice Problem

• Given: BE is the perpendicular bisector of AC, AED CEF, ∠ ≅ ∠DE FE.≅

• Prove: DAE FCE∠ ≅ ∠Answer on next slide⫸

Page 10: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

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

Page 11: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Practice Problem 2

• Find the value of X and YAnswer on next slide⫸

Page 12: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Solution to Practice Problem 2

• Answer: X = -3, Y = 12

Page 13: Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

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)