1. Properties of Triangles

2. Vocabulary Words Equidistant Locus Concurrent Point of concurrency
Circumcenter
Median
Centroid
Altitude
Orthocenter
Incenter

3. Perpendicular and Angle Bisectors
Equidistant – when a point is the same distance from two o r more objects. Theorems: Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

4. Example
Find each measure. PB
B. AB
C. AD

5. Distance and Angle Bisectors
Locus – a set of points that satisfies a given condition exp: The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. Angle Bisector Theorem – If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem – If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

6. Example
Find each measure ED
𝑚∠𝐹𝐴𝐷, given that 𝑚∠𝐴𝐹𝐸 = 112°
𝑚∠𝐹𝐴𝐷,

7. Bisectors of Triangles
Concurrent – when three or more lines intersect at one point. Point of concurrency – the point where three or more lines intersect. Circumcenter of the triangle – the point of concurrency of the three perpendicular bisectors of a triangle.

8. Cirmcumcenter Theorem
The circumcenter of a triangle is equidstant from the vertices of the triangle.

9. Example 𝐾𝑍 , 𝐿𝑍 , and 𝑀𝑍 are perpendicular bisectors of ∆𝐺𝐻𝐽. Find HZ.
Your turn GM GK JZ

10. Incenter Theorem
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle.
Incenter Theorem – The incenter of a triangle is equidistant from the sides to the triangle.
Unlike the circumcenter, the incenter is always inside the triangle.

11. Example JV and KV are angle bisectors of ∆𝐽𝐾𝐿. Find each measure.
The from V to KL.
𝑚∠𝑉𝐾𝐿

12. Your Turn QX and RX are angle bisectors of ∆𝑃𝑄𝑅. Find each measure.
The distance from X to PQ
19.2
2. m∠𝑃𝑄𝑋 52

13. Name Type Point of Concurrency
Perpendicular Bisector
A line segment with the midpoint of a side as an end point
Circumcenter
Angle Bisector
Bisects an angle on the interior of the triangle into two congruent angles
Incenter
Median
A line segment with endpoints from a vertex and the midpoint of the opposite side
Centroid
Altitude
Is a line segment from a vertex that is perpendicular to the side opposite the vertex
Orthocenter

14. Inequalities and Triangles
Page 247 Resource Book page 17 Review Exterior Angle Theorem Page 248 Exterior Angle Inequality Theorem

15. One Triangle Inequality
Page 261 Resource Book page 29

16. Determine whether the measures and
can be lengths of the sides of a triangle.
Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle.
Example 4-1a

17. Determine whether the measures 6. 8, 7. 2, and 5
Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle.
Check each inequality.
Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.
Example 4-1b

18. In and Find the range of the third side.
Example 4-2a

19. Inequalities Involving Two Triangles
Resource Book page 35