what conclusion states the equality of the measures of two angles for CD bisects angle ABC?
The conclusion that states the equality of the measures of two angles when CD bisects angle ABC is known as the Angle Bisector Theorem. According to this theorem, if a line segment CD bisects an angle ABC, then the measure of angle ACD will be equal to the measure of angle BCD.
The conclusion that states the equality of the measures of two angles when CD bisects angle ABC is called the Angle Bisector Theorem.
To understand and prove this theorem, we can follow these steps:
Step 1: Draw the diagram.
Draw a straight line segment AB and a point C on it. Then, draw another line segment CD that intersects AB.
Step 2: State the given information.
The given information is that CD bisects angle ABC, which means it divides angle ABC into two equal angles.
Step 3: Express the information algebraically.
Let the measures of angle ABC and angle CBD be ∠ABC and ∠CBD, respectively.
Step 4: Write the theorem.
The Angle Bisector Theorem states that if CD is an angle bisector of angle ABC, then the measures of ∠CBD and ∠ABC are equal.
Step 5: Prove the theorem.
To prove the theorem, we can divide the problem into two cases:
- Case 1: CD lies entirely inside ∠ABC.
- Case 2: CD lies on the extension of segment AB.
For both cases, using geometric properties and the definition of angle bisector, we can demonstrate that the measures of ∠CBD and ∠ABC are equal.
Step 6: State the conclusion.
The conclusion of the Angle Bisector Theorem is that when CD bisects angle ABC, the measures of ∠CBD and ∠ABC are equal.
By following these steps, you can understand and prove the Angle Bisector Theorem, which states the equality of the measures of two angles when CD bisects angle ABC.