Question

Geometry?

Answer 1

hint: XYZ is the shared overlapping angle between the two triangles

Answer 2

this is what I put in: The congruence theorem would be Side-Angle-Side proving triangle XBY is congruent to triangle ZAY.

Answer 3

for 2b.

Answer 4

We're already given
\( \angle X \cong \angle Z\)
and
\( \overline{XY} \cong \overline{ZY}\)
That takes care of the "A" and "S" respectively

As mentioned with the overlapping angles, if we knew that \(\angle XYB \cong \angle ZYA\), then we would have enough information to use ASA.

Answer 5

Once we've proven \(\triangle XBY \cong \triangle ZAY\), we use CPCTC to lead to \(\overline{AZ} \cong \overline{BX}\)

CPCTC = corresponding parts of congruent triangles are congruent

Answer 6

oh its ASA and not SAS?

Answer 7

correct

Answer 8

We know angle X is congruent to angle Z

So that's one "A" of ASA

Answer 9

and we know that XY and YZ are the same length
That's "S" of ASA

Answer 10

If we knew that
angle XYB = angle ZYA
then we have enough for ASA

Answer 11

note how the marked sides (the sides with the double tickmarks) are between the marked angles

So that tells us we aren't using AAS but instead we use ASA

Answer 12

ok thank you @jimthompson5910

Answer 13

No problem

Answer 14

do you have time for 2 more? lol if not that ok.

Answer 15

sure

Answer 16

how far did you get with problem 1?

Answer 17

I have only done the given's, I haven't got farther than that I am stuck

Answer 18

What do you notice about ZX?

Answer 19

its in the middle but also a bisector. Is that what you mean?

Answer 20

it might help to break up the triangles like so

Answer 21

Would you agree that ZX = ZX?

Answer 22

yes by reflexive prpperty

Answer 23

property*

Answer 24

yes

Answer 25

Do we have enough to prove these triangles congruent?

Answer 26

yes by SAS

Answer 27

yep

Answer 28

But that was in the given already that those two trangles are congruent

Answer 29

Then by CPCTC, we know that
angle WXZ = angle YXZ

Answer 30

but those angles I just marked also add up to 180 since angle WXY of the original drawing is a straight angle

let p = measure of angle WXZ = measure of angle YXZ

You need to solve p+p = 180

Answer 31

by triangle sum theorem

Answer 32

what do you get when you solve for p in the equation p+p = 180

Answer 33

ill be honest idk im lost...

Answer 34

p+p turns into 2p, correct?

Answer 35

oh.. yes sorry

Answer 36

so if 2p = 180, then p = ??

Answer 37

i am this far so far

Answer 38

p = 90

Answer 39

oh so right angle

Answer 40

that takes care of the "perpendicular" part

Answer 41

to prove the "bisector" part, you need to show that WX = XY

Answer 42

since "bisect" means "cut in half" or "cut into two smaller equal pieces"

Answer 43

yes, sorry I went afk but im back

Answer 44

its ok

Answer 45

so how can you show that WX = XY ?

Answer 46

transitive?

Answer 47

go back to this drawing

Answer 48

We've proven the triangles to be congruent, so we know that the corresponding pieces WX and XY are the same length

Answer 49

Because WX = XY, this means segment ZX bisects (cuts in half) segment WY

Answer 50

oh by the HL theorem

Answer 51

no we proved the triangles congruent by SAS

Answer 52

yes but isnt the last proof HL

Answer 53

the bisector one

Answer 54

you only use HL if you know the triangles are right triangles

But we already proved them to be congruent using SAS. So using HL is unnecessary

Answer 55

ok thank you

Answer 56

that is all I needed, I went ahead and did the other one, thank you so much for your help

Answer 57

You're welcome