Advertisements
Advertisements
Question
Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see the given figure). Show that:
- ΔAPB ≅ ΔAQB
- BP = BQ or B is equidistant from the arms of ∠A.
Advertisements
Solution
We have, l is the bisector of ∠QAP.
∴ ∠QAB = ∠PAB
and ∠Q = ∠P ...[Each 90°]
⇒ ∠ABQ = ∠ABP ...[By angle sum property of △]
i. Now, in △APB and △AQB, we have
∠ABP = ∠ABQ ...[Proved above]
AB = BA ...[Common]
∠PAB ≅ ∠QAB ...[Given]
△APB ≅ △AQB ...[By ASA congruency]
ii. Since, △APB ≅ △AQB
⇒ BP = BQ ...[By Corresponding parts of congruent triangles]
i.e., [Perpendicular distance of B from AP] = [Perpendicular distance of B from AQ]
Thus, the point B is equidistant from the arms of ∠A.
APPEARS IN
RELATED QUESTIONS
You have to show that ΔAMP ≅ AMQ.
In the following proof, supply the missing reasons.
| Steps | Reasons | ||
| 1 | PM = QM | 1 | ... |
| 2 | ∠PMA = ∠QMA | 2 | ... |
| 3 | AM = AM | 3 | ... |
| 4 | ΔAMP ≅ ΔAMQ | 4 | ... |
In the figure, the two triangles are congruent.
The corresponding parts are marked. We can write ΔRAT ≅ ?
Which of the following statements are true (T) and which are false (F):
If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent.
The given figure shows a circle with center O. P is mid-point of chord AB.
Show that OP is perpendicular to AB.
If AP bisects angle BAC and M is any point on AP, prove that the perpendiculars drawn from M to AB and AC are equal.
ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively, such produced to E and F respectively, such that AB = BE and AD = DF.
Prove that: ΔBEC ≅ ΔDCF.
In the parallelogram ABCD, the angles A and C are obtuse. Points X and Y are taken on the diagonal BD such that the angles XAD and YCB are right angles.
Prove that: XA = YC.
In the following diagram, ABCD is a square and APB is an equilateral triangle.
- Prove that: ΔAPD ≅ ΔBPC
- Find the angles of ΔDPC.
In the following figure, ABC is an equilateral triangle in which QP is parallel to AC. Side AC is produced up to point R so that CR = BP.
Prove that QR bisects PC.
Hint: ( Show that ∆ QBP is equilateral
⇒ BP = PQ, but BP = CR
⇒ PQ = CR ⇒ ∆ QPM ≅ ∆ RCM ).
ABC is a right triangle with AB = AC. Bisector of ∠A meets BC at D. Prove that BC = 2AD.
