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प्रश्न
Measures of some angles in the figure are given. Prove that `"AP"/"PB" = "AQ"/"QC"`.
उत्तर
Given: ∠APQ = 60∘, ∠ABC = 60∘
To Prove: `"AP"/"PB" = "AQ"/"QC"`.
Proof:
∠APQ = ∠ABC = 60∘ ...(Given)
∴ ∠APQ ≅ ∠ABC
∴ Seg PQ || Seg BC ...(Corresponding angles test for parallel lines )(I)
In ΔABC,
Seg PQ || Seg BC ...[From I]
By Basic proportionality theorem,
∴ `"AP"/"PB" = "AQ"/"QC"`
Hence proved.
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In ∆PQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY || QR.
Complete the proof by filling in the boxes.
In △PMQ, ray MX is bisector of ∠PMQ.
∴ `square/square = square/square` .......... (I) theorem of angle bisector.
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∴ `square/square = square/square` .......... (II) theorem of angle bisector.
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∴ `(PX)/(XQ) = (PY)/(YR)`
∴ XY || QR .......... converse of basic proportionality theorem.
In the given fig, bisectors of ∠B and ∠C of ∆ABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find `"AX"/"XY"`.
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In Δ ABC and Δ PQR,
∠ ABC ≅ ∠ PQR, seg BD and
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