a To bisect an angle means we want to cut it in half. An acute angle is between 0∘ and 90∘.
B
b Draw segments from E and C to the angle bisector to get two triangles. Can you prove that these are congruent?
A
aExample Solution:
B
b See solution.
Practice makes perfect
a An acute angle is between 0∘ and 90∘. Let's draw an arbitrary ∠CDE that is acute.
To bisect this angle, we need to use a compass. Start by placing the sharp end of the compass at the vertex of the angle. Draw an arc across both rays.
Next, keeping the compass set, place its sharp end at the point where one ray intersects the arc. Draw a new arc.
Do the same with the other ray, making sure this arc intersects the previous arc.
Use a ruler or a straightedge to draw a ray from the vertex through the intersection between the two arcs.
This ray is the angle bisector to ∠D, and divides the angle into two angles of equal measure.
b The Converse of the Angle Bisector Theorem says that 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. Therefore, we need to show that the distance from E and C to the angle bisector is the same.
We can prove that EF≅CF if we show that △DEF≅△DFC. From the diagram, we see that the triangles have two pairs of congruent angles. We also notice that the triangles share DF as a side, which means it is congruent by the Reflexive Property of Congruence.
Now we can claim that △DEF≅△DFC by the AAS (Angle-Angle-Side) Congruence Theorem. Since EF and CF are corresponding sides we know that they are congruent, which means F is equidistant from the sides of ∠CDE. Therefore, F has to be on the angle bisector according to the Converse of the Angle Bisector Theorem.
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