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Presentation transcript:

5.1 and 5.2: Midsegments of Triangles Perpendicular and Angle bisectors Objectives: Students will be able to… Use properties of midsegments to solve problems Use properties of perpendicular and angle bisectors to find missing measurements

Midsegment of a Triangle Segment connecting the midpoints of 2 sides of a triangle B D E C A D is the midpoint of E is the midpoint of is the midsegment of

Triangle Midsegment Theorem If a segment joins the midpoints of 2 sides of a triangle, the segment is parallel to the 3rd side, and is ½ its length Do NOT assume it’s a midsegment unless they tell you or you prove it.

Triangle Midsegment Theorem is the midsegment of Therefore…. AND

EXAMPLES: Find the value of the variables. 1. 2. A B C x x+2 E D 18 20

Find the perimeter of D 5 3 E 7 A

Find the value of the variable. (6x)° 30°

In ∆XYZ, M, N, and P are midpoints. The perimeter of the ∆ MNP is 60 yd. Find NP and YZ. 22 M P 24 NAME ALL PARALLEL SEGMENTS: Y Z N

What is the measure of angle ANM? Angle A? Explain. 65° C B

Warm Up What is a perpendicular bisector of a segment? What is an angle bisector? What does equidistant mean?

is the perpendicular bisector of What do we know?

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 Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. IS THE PERPENDICULAR BISECTOR OF SEGMENT AB 6 6

EXAMPLES Find PB and AQ. 14 7

Find AD, x, and BC. 12 C D A 2x+6 3x+1 B

What do we know about P? 10 10

FYI When we refer to the distance from a point to a line, we are talking about the length of the perpendicular segment from point to a line.

Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. (perpendicular distance to sides is the same) 4 4

Converse of Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

You are designing a park, and you are in charge of building a walkway where every point on the walkway will be equidistant from 2 major monuments in the park. How would you figure out where to put the walkway?