Question

In the figure DE intersect the side of triangle ABC in point P and Q such that arc AD =arc DB and arc AE =arc EC show that angle PQC is congruent to angle BPQ

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Answer 1

ABC is a triangle, DE intersect it through point P and Q.

AD = DB {given}

AE = EC

Let's add B to Q, and P to C,here BQ and PC intersect in point O.

In triangle PQC and triangle BPQ:---

1). PQ is the base of both triangles,

2). angle POQ = angle BOC( opposite angles are equal)

3). angle POB = angle QOC ( opposite angles are equal).

therefore , triangle PQC and triangle BPQ both are congruent. hence proved.

Answer 2

Given: arc AC=arc DB

To prove: chord AB || chord CD

Proof :arc AC=arc DB---(GIVEN)

chord AC=chord BD---(Equal arcs makes equal chords)

AB=CD---(Distance between parallel lines are equal)

AB||CD

Hence proved.