Question

Proof of equal intercept theorem

Answer 1

$\mathfrak\color{palevioletred}{\underline{\underline{Equal\: Intercept\: Theorem}}}$

If a transversal makes equal intercepts on three or more parallel lines then any other line cutting them will also make equal intercepts

$\mathfrak\color{palevioletred}{\underline{\underline{Given:}}}$

Transversal AB makes equal intercepts on these parallel lines l, m and n

[ If l ∥ m ∥ n and PQ = QR ]

CD is another transversal with intercepts LM and MN

$\mathfrak\color{palevioletred}{\underline{\underline{To prove:}}}$

LM = MN

$\mathfrak\color{palevioletred}{\underline{\underline{Construction:}}}$

Draw PS and QT parallel to CD

$\mathfrak\color{palevioletred}{\underline{\underline{Proof:}}}$

In ∆PQS and ∆QRT

PQ = QR [given]

$<span>&#8736;</span>PQS =<span>&#8736;</span>QRT$

$<span>&#8736;</span>QPS = <span>&#8736; </span>RQT$

[ PS ∥ CD ∥ QT , corresponding angles]

∴ ∆PQR $<span>&#8773;</span>$ ∆QRT [ By ASA congruency ]

∴ PS = QT→① [ By C.P.C.T.C]

PSML is a parallelogram [ Both the pairs of opposite sides are parallel ]

PS = LM →② [ opposite sides of parallelogram are equal and parallel ]

QTNM is a parallelogram

From ① and ②

QT = LM → ③

QT = MN → ④ [opposite sides of parallelogram are equal ]

From ③ and ④

$\color{palevioletred}{\fbox{\fbox{ \color{black}{\rm\therefore LM = MN}}}}$

Answer 2

Answer:

Step-by-step explanation:

Hope it hps u❤