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प्रश्न
In the following figures, the sides AB and BC and the median AD of triangle ABC are equal to the sides PQ and QR and median PS of the triangle PQR.
Prove that ΔABC and ΔPQR are congruent.
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उत्तर
Given: AB = PQ ; BC = QR ; AD = PS
To prove: ΔABC ≅ ΔPQR
Proof:
BC = QR
2 BD = 2 QS
BD = QS ...(i)
In Δ ABD and Δ PQS
AB = PQ ...[Given]
BD = QS ...[From equation (i)]
AD = PS ...[Given]
∴ ΔABD ≅ ΔPQS ...[by SSS rule]
Then, ∠B = ∠Q ...[by CPCTC] ...(ii)
In ΔABC and ΔPQR
AB = PQ ...[Given]
∠B = ∠Q ...[ii]
BC = QR ...[Given]
∴ ΔABC ≅ ΔPQR ...[by SAS rule]
Hence proved.
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संबंधित प्रश्न
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.
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 | ... |
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.
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In triangles ABC and CDE, if AC = CE, BC = CD, ∠A = 60°, ∠C = 30° and ∠D = 90°. Are two triangles congruent?
ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.
In a triangle ABC, D is mid-point of BC; AD is produced up to E so that DE = AD. Prove that:
AB is parallel to EC.
In the figure, given below, triangle ABC is right-angled at B. ABPQ and ACRS are squares. 
Prove that:
(i) ΔACQ and ΔASB are congruent.
(ii) CQ = BS.
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Show that LM and QS bisect each other.
In a triangle, ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB.
Prove that: AD = CE.
