abcd is a parallelogram.if the line joining the mid-point of side bc to vertex a bisects angle a,then prove that bisector of angle b also bisects ad
let M and N be where these angle bisectors meet BC and AD,
Let E be the point where they intersect each other.
Since consecutive angles of a parallelogram are supplementary, their bisectors are complementary. Thus, triangles ABE and ANE are right triangles, and by AAA they are similar. Since they share side AE, they are congruent. Thus, AN ≅ BM and BN bisects AD.
To prove that the bisector of angle b also bisects ad in the parallelogram abcd, we can use the concept of midpoints and the properties of parallelograms.
1. Given that abcd is a parallelogram, we have the following properties:
- Opposite sides are parallel (ab || cd, ad || bc).
- Opposite angles are congruent (angle a = angle c, angle b = angle d).
- Diagonals bisect each other (the line segment joining the midpoints of ac and bd bisect each other).
2. Let M be the midpoint of side bc. Since the line joining the midpoint of side bc (M) to vertex a bisects angle a, we can conclude that angle EAM (where E is the intersection point of M and ad) is congruent to angle BAM (angle b).
3. To prove that the bisector of angle b also bisects ad, we need to show that the angles EAD and BAD are congruent.
4. Since angle BAM and angle EAM are congruent (as mentioned in step 2), and angle b = angle d (property of parallelograms), we can conclude that angle BAD and angle EAD are congruent.
Therefore, the bisector of angle b also bisects ad in parallelogram abcd.
Note: Make sure to draw the diagram of the parallelogram abcd to visualize the given information and the relationships between angles and lines.