IN PARALLELOGRAM ABCD, BISECTOR OF ANGLE A BISECTS BC,WILL BISECTOR OF ANGLE B BISECT AD,EXPLAIN HOW.
I encourage you to sketch parallelogram ABCD and mark it up accordingly as you read through this.
Let point E be the midpoint of BC, and hence also the point where angle A's bisector meets BC.
Let point F be the point where angle B's bisector meets AD.
Draw a line from E to F.
We now have quadrilateral ABEF, with diagonals AE and BF. Because these diagonals bisect angles A and B, respectively (given), then ABEF is a rhombus; i.e., AB = BE = EF = FA.
Now, BE = EC (given), so AB = EC.
We also know that AB = DC ( original parallelogram).
It follows that parallelogram ECDF is also a rhombus, of the same dimensions as rhombus ABEF. Thus F is the midpoint of AD or, in other words, angle B's bisector bisects AD.
In parallelogram ABCD, we are given that the bisector of angle A bisects side BC. We need to show that the bisector of angle B bisects side AD.
To prove this, let's consider the following steps:
Step 1: Draw parallelogram ABCD and mark the given points and angles.
Step 2: Let the bisector of angle A intersect side BC at point E.
Step 3: Draw segment AE.
Step 4: Since the bisector of angle A bisects side BC, we can conclude that BE = EC.
Step 5: Now, let's prove that the bisector of angle B bisects side AD.
Step 6: Connect points D and E.
Step 7: Since ABCD is a parallelogram, DE is parallel to AB.
Step 8: Also, since BE = EC (as we proved earlier), triangle ABE is an isosceles triangle.
Step 9: In an isosceles triangle, the angle bisectors also bisect the opposite side.
Step 10: Hence, in triangle ABE, the bisector of angle A would also bisect side AB.
Step 11: Since AB is parallel to DE, and the bisector of angle A bisects AB, it also bisects DE.
Step 12: Therefore, the bisector of angle B also bisects side AD.
By following these steps, we have shown that in parallelogram ABCD, the bisector of angle A bisects side BC, and the bisector of angle B bisects side AD.
To explain how the bisector of angle A in parallelogram ABCD bisects side BC, and how the bisector of angle B bisects side AD, we can start with the properties of a parallelogram.
In a parallelogram, opposite sides are parallel and congruent, and opposite angles are also congruent. Let's use this information to understand how the mentioned bisectors are achieved.
1. Bisector of Angle A bisects BC:
- Step 1: Draw the parallelogram ABCD with given angle measures.
- Step 2: Construct the bisector of angle A (let's call it line AE), which divides angle A into two equal angles.
- Step 3: Extend line AE until it intersects side BC (let's call this point F).
- Step 4: Now, using the properties of a parallelogram, we know that opposite sides in a parallelogram are congruent. Therefore, since AB is parallel to DC and AD is parallel to BC, we have AF = FC.
- Step 5: Hence, the bisector of angle A bisects side BC at point F.
2. Bisector of Angle B bisects AD:
- Step 1: Draw the parallelogram ABCD with the given angle measures.
- Step 2: Construct the bisector of angle B (let's call it line BG), which divides angle B into two equal angles.
- Step 3: Extend line BG until it intersects side AD (let's call this point H).
- Step 4: Similar to the previous case, using the properties of a parallelogram, we can conclude that opposite sides in the parallelogram are congruent. Therefore, since AB is parallel to DC and AD is parallel to BC, we have AH = HD.
- Step 5: Thus, the bisector of angle B bisects side AD at point H.
In summary, the bisector of angle A in parallelogram ABCD bisects side BC, and the bisector of angle B bisects side AD. These bisectors are possible due to the properties of parallelograms, specifically the congruence of opposite sides and angles.