Advertisements
Advertisements
Question
In the following figure, OA = OC and AB = BC.
Prove that:
(i) ∠AOB = 90o
(ii) ΔAOD ≅ ΔCOD
(iii) AD = CD
Advertisements
Solution
Given:
In the figure, OA=OC, AB =BC
We need to prove that,
AOB = 90°
(i) In ΔABO and ΔCBO,
AB = BC ...[given ]
AO = CO ...[ given ]
OB = OB ...[ common ]
∴By Side-Side-Side criterion of congruence, we have
ΔABO ≅ ΔCBO
The corresponding parts of the congruent triangles are congruent.
∴∠ABO = ∠CBO ...[c. p.c.t. ]
⇒ ∠ABD = ∠CBD
and ∠AOB = ∠COB ...[c. p.c t ]
We have
∠AOB + ∠COB = 180° .....[ linear pair ]
⇒ ∠AOB = ∠ COB= 90° and AC ⊥ BD
(ii) In ΔAOD and ΔCOD,
OD = OD ...[ common ]
∠AOD = ∠COD ...[ each=90° ]
AO = CO ...[ given]
∴By Side-Angel-Side criterion of congruence, we have
ΔAOD ≅ ΔCOD
(iii) The corresponding parts of the congruent
triangles are congruent.
∴AD = CD ...[c. p.c t ]
Hence proved.
APPEARS IN
RELATED QUESTIONS
In the given figure, ∠P ≅ ∠R seg, PQ ≅ seg RQ. Prove that, ΔPQT ≅ ΔRQS.
In the following diagram, ABCD is a square and APB is an equilateral triangle.
(i) Prove that: ΔAPD≅ ΔBPC
(ii) Find the angles of ΔDPC.
Prove that:
- ∆ ABD ≅ ∆ ACD
- ∠B = ∠C
- ∠ADB = ∠ADC
- ∠ADB = 90°
Which of the following pairs of triangles are congruent? Give reasons
ΔABC;(∠B = 90°,BC = 6cm,AB = 8cm);
ΔPQR;(∠Q = 90°,PQ = 6cm,PR = 10cm).
In the figure, AB = EF, BC = DE, AB and FE are perpendiculars on BE. Prove that ΔABD ≅ ΔFEC
In the figure, BM and DN are both perpendiculars on AC and BM = DN. Prove that AC bisects BD.
In ΔABC, X and Y are two points on AB and AC such that AX = AY. If AB = AC, prove that CX = BY.
PQRS is a quadrilateral and T and U are points on PS and RS respectively such that PQ = RQ, ∠PQT = ∠RQU and ∠TQS = ∠UQS. Prove that QT = QU.
∆ABC and ∆PQR are congruent under the correspondence:
ABC ↔ RQP
Write the parts of ∆ABC that correspond to
(i) `bar"PQ"`
(ii)∠Q
(iii) `bar"RP"`
Two figures are congruent, if they have the same shape.
