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Authoritative PPT Summary
"Angle Bisector" PPT Courseware 2
1. Do it with your hands
Draw a ∠BAC randomly on the paper, fold it in half so that the two sides of the corner coincide, then unfold the paper and flatten it to get a crease. What do you find?
An angle is an axially symmetrical figure, and the straight line where the bisector of the angle is located is its axis of symmetry.
2. Drawing with ruler and compass
Observe and understand practices, explore and think about proof methods:
practice:
1. With O as the center of the circle and any length as the radius, draw an arc and intersect OA and OB at points M and N respectively.
2. Take points M and N as the center of the circle respectively. Greater than 1/2 MN. The length of MN is the radius. Draw an arc inside the angle and intersect at point C.
3. Make ray OC．
Ray OC is the desired graph.
3. Theoretical basis
Think about it: Why is OC the bisector of ∠AOB?
Proof: Connect CM, CN.
In △OMC and △ONC,
OM=ON，
MC=NC，
OC=OC，
∴ △OMC ≌ △ONC.
∴∠MOC =∠NOC.
That is, OC bisects ∠AOB.
5. Determination theorem of angle bisectors
Determination theorem: A point inside an angle that is equidistant from both sides of the angle is on the bisector of the angle.
Expressed in symbolic language as:
∵ PD ⊥OA, PE ⊥OB, PD=PE,
∴ Point P is on the bisector of ∠AOB.
6. Give it a try
Known: As shown in the figure, in △ABC, AB=AC, AD is the bisector of ∠BAC, DE⊥AB, DF⊥AC, and the vertical feet are E and F respectively. Determine whether the following conclusions are correct:
(1)DE=DF. ( )
(2)BD=CD. ( )
(3) The distance from any point on AD to AB and AC is equal. ( )
(4) The distance from any point on AD to points B and C is equal. ( )
10. Summary and evaluation
What did we learn in this lesson?
Together, summarize the main knowledge learned in this lesson:
(1) Use a ruler and compass to draw the bisector of the angle.
(2) The property theorem of angle bisectors:
A point on the bisector of an angle is equidistant from both sides of the angle.
(3) Determination theorem of angle bisector:
A point equidistant from both sides of an angle is on the bisector of the angle.
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