Answer:
1. 79°
2. 54°
3. 107.5°
4. 44°, 35 cm
5. 76°, 3.5 cm
6. m∠U=36°, m∠M=m∠D=72°, MD=8.6 cm
7. 78°, 93 cm
8. 81°, 75 cm
Step-by-step explanation:
1. The diagram shows an isosceles triangle because TH = OT. Angles adjacent to the base OH of isosceles triangle are congruent, so
![m\angle H=m\angle O](https://tex.z-dn.net/?f=m%5Cangle%20H%3Dm%5Cangle%20O)
The sum of the measures of all interior angles is always 180°, thus
![m\angle H+m\angle O+m\angle T=180^{\circ}\\ \\2m\angle H+22^{\circ}=180^{\circ}\\ \\2m\angle H=180^{\circ}-22^{\circ}\\ \\2m\angle H=158^{\circ}\\ \\m\angle H=79^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20H%2Bm%5Cangle%20O%2Bm%5Cangle%20T%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20H%2B22%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20H%3D180%5E%7B%5Ccirc%7D-22%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20H%3D158%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20H%3D79%5E%7B%5Ccirc%7D)
2. The diagram shows an isosceles triangle DGO because DG = GO. Angles adjacent to the base DO of isosceles triangle are congruent, so
![m\angle D=m\angle O=63^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20D%3Dm%5Cangle%20O%3D63%5E%7B%5Ccirc%7D)
The sum of the measures of all interior angles is always 180°, thus
![m\angle D+m\angle O+m\angle G=180^{\circ}\\ \\63^{\circ}+63^{\circ}+m\angle G=180^{\circ}\\ \\m\angle G=180^{\circ}-63^{\circ}-63^{\circ}\\ \\m\angle G=54^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20D%2Bm%5Cangle%20O%2Bm%5Cangle%20G%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C63%5E%7B%5Ccirc%7D%2B63%5E%7B%5Ccirc%7D%2Bm%5Cangle%20G%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20G%3D180%5E%7B%5Ccirc%7D-63%5E%7B%5Ccirc%7D-63%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20G%3D54%5E%7B%5Ccirc%7D)
3. The diagram shows an isosceles triangle SLO because LO = SO. Angles adjacent to the base SL of isosceles triangle are congruent, so
![m\angle S=m\angle L](https://tex.z-dn.net/?f=m%5Cangle%20S%3Dm%5Cangle%20L)
The sum of the measures of all interior angles is always 180°, thus
![m\angle S+m\angle L+m\angle O=180^{\circ}\\ \\2m\angle L+35^{\circ}=180^{\circ}\\ \\2m\angle L=180^{\circ}-35^{\circ}\\ \\2m\angle L=145^{\circ}\\ \\m\angle L=72.5^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20S%2Bm%5Cangle%20L%2Bm%5Cangle%20O%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20L%2B35%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20L%3D180%5E%7B%5Ccirc%7D-35%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20L%3D145%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20L%3D72.5%5E%7B%5Ccirc%7D)
Angles OLE and L (SLO) are supplementary (add up to 180°), so
![m\angle OLE=180^{\circ}-m\angle L\\ \\m\angle OLE=180^{\circ}-72.5^{\circ}\\ \\m\angle OLE=107.5^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20OLE%3D180%5E%7B%5Ccirc%7D-m%5Cangle%20L%5C%5C%20%5C%5Cm%5Cangle%20OLE%3D180%5E%7B%5Ccirc%7D-72.5%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20OLE%3D107.5%5E%7B%5Ccirc%7D)
4. The diagram shows an isosceles triangle AMR because
(angles adjacent to the side AM are congruent, so triangle AMR is isoseceles).
The sum of the measures of all interior angles is always 180°, thus
![m\angle A+m\angle M+m\angle R=180^{\circ}\\ \\m\angle R+2\cdot 68^{\circ}=180^{\circ}\\ \\m\angle R=180^{\circ}-2\cdot 68^{\circ}\\ \\m\angle R=44^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20A%2Bm%5Cangle%20M%2Bm%5Cangle%20R%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20R%2B2%5Ccdot%2068%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20R%3D180%5E%7B%5Ccirc%7D-2%5Ccdot%2068%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20R%3D44%5E%7B%5Ccirc%7D)
To legs in isosceles triangle are always congruent, so
![RM=AR=35\ cm](https://tex.z-dn.net/?f=RM%3DAR%3D35%5C%20cm)
5. The diagram shows isosceles triangle RYD because YD = RD. Angles adjacent to the base RY are congruent, so
![m\angle R=m\angle Y](https://tex.z-dn.net/?f=m%5Cangle%20R%3Dm%5Cangle%20Y)
The sum of the measures of all interior angles is always 180°, thus
![m\angle R+m\angle Y+m\angle D=180^{\circ}\\ \\2m\angle Y+28^{\circ}=180^{\circ}\\ \\2m\angle Y=180^{\circ}-28^{\circ}\\ \\2m\angle Y=152^{\circ}\\ \\m\angle Y=76^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20R%2Bm%5Cangle%20Y%2Bm%5Cangle%20D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20Y%2B28%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20Y%3D180%5E%7B%5Ccirc%7D-28%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20Y%3D152%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20Y%3D76%5E%7B%5Ccirc%7D)
To legs in isosceles triangle are always congruent, so
![YD=RD=3.5\ cm](https://tex.z-dn.net/?f=YD%3DRD%3D3.5%5C%20cm)
6. The diagram shows an isosceles triangle UMD because UM = UD. Angles adjacent to the base MD of isosceles triangle are congruent, so
![m\angle D=m\angle M=72^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20D%3Dm%5Cangle%20M%3D72%5E%7B%5Ccirc%7D)
The sum of the measures of all interior angles is always 180°, thus
![m\angle D+m\angle M+m\angle U=180^{\circ}\\ \\72^{\circ}+72^{\circ}+m\angle U=180^{\circ}\\ \\m\angle U=180^{\circ}-72^{\circ}-72^{\circ}\\ \\m\angle U=36^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20D%2Bm%5Cangle%20M%2Bm%5Cangle%20U%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C72%5E%7B%5Ccirc%7D%2B72%5E%7B%5Ccirc%7D%2Bm%5Cangle%20U%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20U%3D180%5E%7B%5Ccirc%7D-72%5E%7B%5Ccirc%7D-72%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20U%3D36%5E%7B%5Ccirc%7D)
To legs in isosceles triangle are always congruent, so
![UM=UD=14\ cm](https://tex.z-dn.net/?f=UM%3DUD%3D14%5C%20cm)
The perimeter of isosceles triangle MUD is 36.6 cm, so
![UM+MD+UD=36.6\\ \\MD=36.6-14-14\\ \\MD=8.6\ cm](https://tex.z-dn.net/?f=UM%2BMD%2BUD%3D36.6%5C%5C%20%5C%5CMD%3D36.6-14-14%5C%5C%20%5C%5CMD%3D8.6%5C%20cm)
7. The sum of the measures of all interior angles is always 180°, thus
![m\angle T+m\angle S+m\angle B=180^{\circ}\\ \\m\angle T+78^{\circ}+24^{\circ}=180^{\circ}\\ \\m\angle T=180^{\circ}-78^{\circ}-24^{\circ}\\ \\m\angle T=78^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20T%2Bm%5Cangle%20S%2Bm%5Cangle%20B%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20T%2B78%5E%7B%5Ccirc%7D%2B24%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20T%3D180%5E%7B%5Ccirc%7D-78%5E%7B%5Ccirc%7D-24%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20T%3D78%5E%7B%5Ccirc%7D)
Triangle STB is isosceles triangle because
(angles adjacent to the side ST are congruent, so triangle STB is isoseceles).
To legs in isosceles triangle are always congruent, so
![SB=TB\\ \\y+22.5=38.5\\ \\y=38.5-22.5\\ \\y=16](https://tex.z-dn.net/?f=SB%3DTB%5C%5C%20%5C%5Cy%2B22.5%3D38.5%5C%5C%20%5C%5Cy%3D38.5-22.5%5C%5C%20%5C%5Cy%3D16)
Hence,
![ST=16\ cm\\ \\TB=SB=38.5\ cm](https://tex.z-dn.net/?f=ST%3D16%5C%20cm%5C%5C%20%5C%5CTB%3DSB%3D38.5%5C%20cm)
and the perimeter of triangle STB is
![P_{STB}=16+38.5+38.5=93\ cm](https://tex.z-dn.net/?f=P_%7BSTB%7D%3D16%2B38.5%2B38.5%3D93%5C%20cm)
8. The diagram shows isosceles triangle CNB because CN = CB. Angles adjacent to the base RY are congruent, so
![m\angle N=m\angle B](https://tex.z-dn.net/?f=m%5Cangle%20N%3Dm%5Cangle%20B)
The sum of the measures of all interior angles is always 180°, thus
![m\angle N+m\angle B+m\angle C=180^{\circ}\\ \\2m\angle N+18^{\circ}=180^{\circ}\\ \\2m\angle N=180^{\circ}-18^{\circ}\\ \\2m\angle N=162^{\circ}\\ \\m\angle N=81^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20N%2Bm%5Cangle%20B%2Bm%5Cangle%20C%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20N%2B18%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20N%3D180%5E%7B%5Ccirc%7D-18%5E%7B%5Ccirc%7D%5C%5C%20%5C%5C2m%5Cangle%20N%3D162%5E%7B%5Ccirc%7D%5C%5C%20%5C%5Cm%5Cangle%20N%3D81%5E%7B%5Ccirc%7D)
To legs in isosceles triangle are always congruent, so
![CB=CN=2x+90\ m](https://tex.z-dn.net/?f=CB%3DCN%3D2x%2B90%5C%20m)
The perimeter of the triangle CNB is
![2x+90+2x+90+x=555\\ \\5x+180=555\\ \\x+36=111\\ \\x=111-36\\ \\x=75\ m](https://tex.z-dn.net/?f=2x%2B90%2B2x%2B90%2Bx%3D555%5C%5C%20%5C%5C5x%2B180%3D555%5C%5C%20%5C%5Cx%2B36%3D111%5C%5C%20%5C%5Cx%3D111-36%5C%5C%20%5C%5Cx%3D75%5C%20m)
So, ![NB=75 \ m](https://tex.z-dn.net/?f=NB%3D75%20%5C%20m)