For questions 13 – 15, find x so that q || r. State the converse used.

13. x = 7 ; Converse : Alternate interior angles are equal .

14.  x=12; Converse used : Linear pair

15.  x = 23; Converse: Corresponding angles are equal

Step-by-step explanation:

13.

Refer the attached figure

Given: q || r

(Alternate interior angles )

So,  15x+3=108

15x=108-3

15x=105

x=7

So, x = 7 ; Converse : Alternate interior angles are equal .

14.

Refer the attached figure

Given q||r

(Alternate interior angles

So,

Now(Linear pair)

So,7x-8+11x-28=180

18x-36= 180

18x=216

x=12

So, x=12; Converse used : Linear pair

15.

Refer the attached figure

Given q||r

(corresponding angles)

90+2x=5x+21

90-21=5x-2x

69=3x

23=x

So, x = 23; Converse: Corresponding angles are equal

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< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)