Answer with Step-by-step explanation:
1.In triangle ABC
AB=BC
Let AB=BC=x and AC=y
Perimeter of triangle ABC=25
[tex]x+x+y=25[/tex]
[tex]2x+y=25[/tex]...(1)
[tex]x-y=4[/tex]...(2)
Adding equation (1) and (2)
[tex]3x=29[/tex]
[tex]x=\frac{29}{3}=9.67[/tex]
Substitute x=9.67 in equation (2)
[tex]9.67-y=4[/tex]
[tex]y=9.67-4=5.67[/tex]
[tex]AB=BC=9.67[/tex]
[tex]AC=5.67[/tex]
2.[tex]m\angle R=2x+11[/tex]
[tex]m\angle S=3x+23[/tex]
[tex]m\angle T=x+42[/tex]
[tex]m\angle R+m\angle S+m\angle T=180^{\circ}[/tex]
By using triangle angle sum property
Substitute the values then we get
[tex]3x+42+4x-11+x+13=180[/tex]
[tex]8x+44=180[/tex]
[tex]8x=180-44=136[/tex]
[tex]x=\frac{136}{8}=17[/tex]
Substitute the value
[tex]m\angle R=3(17)+42=93^{\circ}[/tex]
[tex]m\angle S=4(17)-11=57^{\circ}[/tex]
[tex]m\angle T=17+13=30^{\circ}[/tex]
[tex]m\angle R>m\angle S[/tex]
[tex]m\angle S>m\angle T[/tex]
[tex]ST>RT[/tex] (Side ST is opposite to angle R, Side RT is opposite to angle S
[tex]RT>RS[/tex] (side RS is opposite to angle T)
When a>b
Then , opposite side of a> opposite side of b
RS<RT<ST
3.[tex]m\angle R=2x+11[/tex]
[tex]m\angle S=3x+23[/tex]
[tex]\angle T=x+42[/tex]
[tex]m\angle R+m\angle S+m\angle T=180^{\circ}[/tex]
By using triangle angle sum property
Substitute the values then we get
[tex]2x+11+3x+23+x+42=180[/tex]
[tex]6x+76=180[/tex]
[tex]6x=180-76[/tex]
[tex]6x=104[/tex]
[tex]x=\frac{104}{6}=17.3[/tex]
Substitute the value
[tex]m\angle R=2(17.3)+11=45.6[/tex]
[tex]m\angle S=3(17.3)+23=74.9[/tex]
[tex]m\angle T=17.3=42=59.3[/tex]
[tex]m\angle S>m\angle T[/tex]
[tex]m\angle T>m\angle R[/tex]
RT>RS
RS>ST
ST<RS<RT
