Given: bisects rts r = s prove: 1. segment tq bisects ∠rts; ∠r = ∠s definition of perpendicular. 2. ∠1 = ∠2 definition of angle bisector. 3. tq = tq aas 4. triangle rtq congruent to triangle stq given 5. ∠3 = ∠4 cpcte 6. segment tq ⊥ segment rs reflexive

Solution:

Given : T Q bisects ∠RT S. And , ∠R=∠S

To Prove : T Q ⊥ R S

Proof : In Δ R T Q and Δ ST Q

T Q bisects ∠ R T S.

So, ∠1=∠2  →→→→    [Definition of angle bisector.]

T Q is Common.

∠R =∠S→→[Given]

Δ R T Q ≅ ST Q→→[AS A]

∵ ∠3 = ∠4  →→→[C P C T E]

But ∠3 and  ∠4 forms Linear pair.

∠3 + ∠4 = 180°

2 ∠3 = 180°

∠3 = 90°

∠3 = ∠4 =90°

So, Segment T Q ⊥ Segment RS.

Hence proved.