Question

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Answer 1

Answer:

Given :

GH ║ NO , and ∠MNO is bisected by NR , OR bisects ∠MON

To prove

GH = NG + OH

Construction

Draw line segment RS perpendicular to ON

Proof

Consider Δ NRS and Δ PRS

RS = RS  (common)

∠ RNS = ∠ RPS ( given )

NR =  PR ( angles opposite to equal sides are equal ) _____  (1)

∴ Δ NRS ≅ Δ PRS ( S.A.S ) congruence rule

Since point R is the mid - point on GH

So GR = RH

Now consider Δ GNR and Δ RHP

NR = PR ( From 1 ) ( CPCT ) *

GR = RH ( given )

∠ GNR = ∠ RHP ( given )

∴ Δ GNR ≅ Δ HDR ( S.A.S )

So GN = HP ( CPCT )*

So ∠ GNR + ∠ RHP = 180° ( co - interior ) _______ (2)

Line GH = 180° ( since it is a line ) _______ (3)

Substitute (1) and (2)

We get GH = ∠  GNR + ∠ RHP

or GH = NG + OH

Hence Proved

* CPCT - ( congruent parts of congruent triangles are equal )