1.5 What you should learn Why you should learn it Segment and Angle Bisectors What you should learn GOAL 1 Bisect a segment GOAL 2 Bisect an Angle. Why you should learn it To solve real-life problems, such as finding the angle measures of a kite.

1.5 Segment and Angle Bisectors 1 GOAL BISECTING A SEGMENT Vocabulary The point that divides a segment into two congruent segments is called the ________ of the segment. Another way to describe this point is that it _______ the segment. midpoint bisects Z X Y In the diagram, Y is the midpoint of if Y is on and This is shown by the red congruence marks.

A segment, ray, line, or plane that intersects a segment at its midpoint is called a _______________. segment bisector T U Z X Y Name all eight different bisectors of shown above. Click to check your answers. All eight figures named above are bisectors of

In geometry, a construction is a drawing that uses only two tools, a ________ and a ___________. compass straightedge Be sure to follow the steps on page 34 to perform the construction of a segment bisector and midpoint. You must be able to complete this construction on a quiz or test, and will use it again later in the course. Click on the camera below if you want to see a video of the construction. While watching the video, you may pause at any time by clicking the pause button.

You may need to find the coordinates of the midpoint of a segment in the coordinate plane. To do this, simply find the mean (average) of the x-coordinates and the mean of the y-coordinates. In algebraic terms,

THE MIDPOINT FORMULA If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of has coordinates y x A(x1, y1) B(x2, y2) y2 x2 y1 x1 EXAMPLE 1 Did you study Example 1???

Extra Example 1 Find the midpoint of with endpoints D(3, 5) and E(-4, 0). Click to see the solution. = = EXAMPLE 2

Extra Example 2 The midpoint of is M(3, -4). One endpoint is Y(-3, -1). Find the coordinates of the other endpoint. Click to see the solution. If the coordinates of X are (x, y), then by the Midpoint Formula and Solve: So the other endpoint is X(9, -7).

Checkpoint Find the midpoint of with endpoints P(0, -7) and Q(-2, -1). 2. The midpoint of is One endpoint is J(2, -2). Find the coordinates of the other endpoint. Click for the answers.

1.5 Angles and Their Measures 2 GOAL BISECTING AN ANGLE Vocabulary A ray that divides an angle into two adjacent angles that are congruent is called an ____________. angle bisector F H G K In the diagram, bisects since Note the red congruence marks.

Follow the steps on page 36 to perform the construction of an angle bisector. You must be able to complete this construction on a quiz or test, and will use it again later in the course. Click on the camera below if you want to see a video of the construction. While watching the video, you may pause at any time by clicking the pause button. EXAMPLE 3

Extra Example 3 bisects Given that what are the measures of Hint: Draw a sketch! Click to see a sample sketch. H J L K 42° Click for the answer. EXAMPLE 4

Extra Example 4 A cellular phone tower bisects the angle formed by the two wires that support it. Find the measure of the angle formed by the two wires. wire tower ? Click for the solution: EXAMPLE 5

Extra Example 5 In the diagram bisects The measures of the two congruent angles are Solve for x. M L O N Click for the solution:

Checkpoint In the diagram bisects Find x and use it to find D A C B Click to check your solution:

QUESTIONS?