"Properties of Angle Bisectors" PPT Courseware 2

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"Properties of Angle Bisectors" PPT Courseware 2

As shown in the figure, it is an instrument that bisects an angle. AB=AD, CB=CD. Place point A at the vertex of the angle. Place AB and AD along both sides of the angle. Draw a ray AE along AC. AE is the bisector of the angle. Wire. Can you make sense of it?

According to SSS, we know that two triangles are congruent

∴∠1=∠2

From the above exploration, can you figure out how to construct the angle bisector of an angle?

Known: ∠AOB

Find: bisector of ∠AOB

Method: (1) Take O as the center of the circle, draw an arc with an appropriate length as the radius, intersect OA at M, and intersect OB at N;

(2) Take M and N as the center of the circle respectively, and draw an arc with a length greater than 1/2MN as the radius. The two arcs intersect at point C inside ∠AOB.

(3) Make ray OC. Ray OC is what you want.

Can you explain why?

Do the exercises on page P108 and answer the questions.

Do it

1. Can you fold the bisector of an angle?

Fold ∠AOB in half

2. Then fold a right triangle (make the angle bisector the hypotenuse, OA and OB the right-angled sides)

What conclusion can you draw by observing the three creases made by two folds?

The two creases made the second time are equal in length.

Can you explain why?

What conclusion can you draw from the above experiment?

Example 1 Known: As shown in the figure, E is a point on the bisector of ∠BAC, EB⊥AB,

EC⊥AC, B, and C are vertical feet respectively. Prove: ∠EBC=∠ECB

Prove: ∵ E is a point on the bisector of ∠BAC, EB⊥AB, EC⊥AC

∴EB=EC

(A point on the bisector of an angle is equidistant from both sides of the angle)

∴∠EBC=∠ECB (in a triangle, equal sides correspond to equal angles)

Think about it: Is BC in the question bisected perpendicularly by AE?

∵∠ABE=∠ACE=Rt∠

∠1＝∠ 2∴∠3＝∠4

Also ∵EB=EC

∴ AE bisects BC perpendicularly

Summary:

1 A point on the bisector of an angle is equidistant from both sides of the angle.

2 To a point equidistant from both sides of an angle, on the bisector of the angle.

3 Properties of bisectors of angles Theorem 1 and Theorem 2 are new ways to prove that angles are equal and line segments are equal. Theorem 1 is mostly used to prove that line segments are equal, and Theorem 2 is mostly used to prove that angles are equal or points are on angle bisectors.

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Update Time: 2024-11-18

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