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...
![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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/1.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/2.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/3.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/4.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/5.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/6.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/7.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/8.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/9.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/10.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/11.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/12.jpg)
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.](https://reader036.fdocuments.us/reader036/viewer/2022072016/56649ef15503460f94c018a2/html5/thumbnails/13.jpg)
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)