Properties of Triangles

Vocabulary Words

1. Equidistant2. Locus3. Concurrent4. Point of concurrency5. Circumcenter6. Median7. Centroid8. Altitude9. Orthocenter10. Incenter

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.

Example

Find each measure.

A. PB B. AB

C. AD

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.

Example

Find each measure

A. ED

B. given that = 112

C. ,

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.

Cirmcumcenter Theorem

The circumcenter of a triangle is equidstant from the vertices of the triangle.

Example

, , and are perpendicular bisectors of . Find HZ.

Your turn1. GM2. GK3. JZ

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.

Example

JV and KV are angle bisectors of

A. The from V to KL.

Your Turn

QX and RX are angle bisectors of Find each measure.

1. The distance from X to PQ 19.22. m 52

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

Inequalities and Triangles

Page 247

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Review Exterior Angle TheoremPage 248 Exterior Angle Inequality Theorem

One Triangle Inequality

Page 261

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Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle.

Determine whether the measures and

can be lengths of the sides of a triangle.

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.

In and Find the range of the third side.

Inequalities Involving Two Triangles

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