Given that SQ¯¯¯¯¯ bisects ∠PSR and ∠SPQ≅∠SRQ, which of the following proves that PS¯¯¯¯¯≅SR¯¯¯¯¯? The figure shows two triangles P Q S and R Q S with a common side S Q.

Answers
A.
1. ∠SPQ≅ ∠SRQ (Given)2. SQ¯¯¯¯¯ bisects ∠PSR. (Given)3. ∠SQP≅∠SQR (Def. of ∠ bisect)4. SQ¯¯¯¯¯≅SQ¯¯¯¯¯ (Reflex. Prop. of ≅)5. △PQS≅△RQS (AAS Steps 1, 3, 4)6. PS¯¯¯¯¯≅SR¯¯¯¯¯ (CPCTC)

B.
1. ∠SPQ≅ ∠SRQ (Given)2. SQ¯¯¯¯¯ bisects ∠PSR. (Given)3. ∠SQP≅∠SQR (Def. of bisect)4. SQ¯¯¯¯¯≅SQ¯¯¯¯¯ (Reflex. Prop. of ≅)5. △PQS≅△RQS (SAS Steps 1, 3, 4)6. PS¯¯¯¯¯≅SR¯¯¯¯¯ (CPCTC)

C.
1. ∠SPQ≅ ∠SRQ (Given)2. SQ¯¯¯¯¯ bisects ∠PSR. (Given)3. SP¯¯¯¯¯≅SR¯¯¯¯¯ (Def. of bisect)4. SQ¯¯¯¯¯≅SQ¯¯¯¯¯ (Sym. Prop. of ≅)5. △PQS≅△RQS (SAS Steps 1, 3, 4)6. PS¯¯¯¯¯≅SR¯¯¯¯¯ (CPCTC)

D.
1. ∠SPQ≅ ∠SRQ (Given)2. SQ¯¯¯¯¯ bisects ∠PSR. (Given)3. ∠PSQ≅∠QSR (Def. of ∠ bisect)4. SQ¯¯¯¯¯≅SQ¯¯¯¯¯ (Reflex. Prop. of ≅)5. △PQS≅△RQS (AAS Steps 1, 3, 4)6. PS¯¯¯¯¯≅SR¯¯¯¯¯ (CPCTC)

Step-by-step explanation:

It is given that ∠SPQ≅∠SRQ

.

The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Angles S P Q and S R Q are highlighted in red.

It is also given that SQ⎯⎯⎯⎯⎯

bisects ∠PSR

.

By the definition of angle bisector, ∠PSQ≅∠QSR

.

The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Angles P S Q and R S Q are congruent and highlighted in red.

△PQS

and △RQS share a common side SQ⎯⎯⎯⎯⎯, and SQ⎯⎯⎯⎯⎯≅SQ⎯⎯⎯⎯⎯

by the Reflexive Property of Congruence.

The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Segment S Q is highlighted in red.

Two angles, ∠SPQ

and ∠PSQ, and a nonincluded side, SQ⎯⎯⎯⎯⎯, of △PQS are congruent to two angles, ∠SRQ and ∠QSR, and a nonincluded side, SQ⎯⎯⎯⎯⎯, of △RQS

.

The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Angles P S Q and R S Q are highlighted in red. Angles S P Q and S R Q are highlighted in red. Side S Q is highlighted in blue.

So, △PQS≅△RQS

by the Angle-Angle-Side (AAS) Congruence Theorem.

PS⎯⎯⎯⎯⎯

and SR⎯⎯⎯⎯⎯ are corresponding sides of congruent triangles, △PQS and △RQS. So, PS⎯⎯⎯⎯⎯≅SR⎯⎯⎯⎯⎯

by CPCTC.

The figure shows the same triangles P Q S and R Q S as in the beginning of the task. Sides S R and S P are congruent and highlighted in red.

Translate these six statements and reasons into a 2

-column proof,

1. ∠SPQ≅∠SRQ

(Given)

2. SQ⎯⎯⎯⎯⎯

bisects ∠PSR

. (Given)

3. ∠PSQ≅∠QSR

(Def. of ∠

bisect)

4. SQ⎯⎯⎯⎯⎯≅SQ⎯⎯⎯⎯⎯

(Reflex. Prop. of ≅

)

5. △PQS≅△RQS

(AAS Steps 1, 3, 4)

6. PS⎯⎯⎯⎯⎯≅SR⎯⎯⎯⎯⎯

(CPCTC)

There you go

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12