# In Fig. 7.7, l || m and M is the mid-point of a line segment AB. Show that M is also the mid-point of any line segment CD, having its end points on l and m, respectively.

Through the mid point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting
Through the mid point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting

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In Fig. 7.7, l || m and M is the mid-point of a line segment AB. Show that M is also the mid-point of any line segment CD, having its end points on l and m, respectively.

Solution:

Given, l is parallel to m

M is the midpoint of a line segment AB

We have to show that M is also the mid-point of any line segment CD, having its end points on l and m, respectively.

Since M is the midpoint of AB

AB = AM + BM

AM = BM

We know that the alternate interior angles are equal

∠CAB = ∠ABD

We know that vertically opposite angles are equal.

∠AMC = ∠DMB

Considering triangles AMC and BMD,

∠CAB = ∠ABD

Given, AM = BM

∠AMC = ∠DMB

ASA criterion states that two triangles are congruent, if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle.

By ASA criterion, ∆ AMC ≅ ∆ BMD

By CPCTC, MC = MD
Therefore, M is the point of the line segment CD.

✦ Try This: In the given figure, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meets at point T, then prove that ∠QTR= 1/2 ∠QPR.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7

NCERT Exemplar Class 9 Maths Exercise 7.3 Problem 8

In Fig. 7.7, l || m and M is the mid-point of a line segment AB. Show that M is also the mid-point of any line segment CD, having its end points on l and m, respectively

Summary:

In Fig. 7.7, l || m and M is the mid-point of a line segment AB. It is shown that M is also the mid-point of any line segment CD, having its end points on l and m, respectively by CPCTC which states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements

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