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
[tex]m\angle H=m\angle O[/tex]
The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
2. The diagram shows an isosceles triangle DGO because DG = GO. Angles adjacent to the base DO of isosceles triangle are congruent, so
[tex]m\angle D=m\angle O=63^{\circ}[/tex]
The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
3. The diagram shows an isosceles triangle SLO because LO = SO. Angles adjacent to the base SL of isosceles triangle are congruent, so
[tex]m\angle S=m\angle L[/tex]
The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
Angles OLE and L (SLO) are supplementary (add up to 180°), so
[tex]m\angle OLE=180^{\circ}-m\angle L\\ \\m\angle OLE=180^{\circ}-72.5^{\circ}\\ \\m\angle OLE=107.5^{\circ}[/tex]
4. The diagram shows an isosceles triangle AMR because [tex]m\angle A=m\angle M=68^{\circ}[/tex] (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
[tex]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}[/tex]
To legs in isosceles triangle are always congruent, so
[tex]RM=AR=35\ cm[/tex]
5. The diagram shows isosceles triangle RYD because YD = RD. Angles adjacent to the base RY are congruent, so
[tex]m\angle R=m\angle Y[/tex]
The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
To legs in isosceles triangle are always congruent, so
[tex]YD=RD=3.5\ cm[/tex]
6. The diagram shows an isosceles triangle UMD because UM = UD. Angles adjacent to the base MD of isosceles triangle are congruent, so
[tex]m\angle D=m\angle M=72^{\circ}[/tex]
The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
To legs in isosceles triangle are always congruent, so
[tex]UM=UD=14\ cm[/tex]
The perimeter of isosceles triangle MUD is 36.6 cm, so
[tex]UM+MD+UD=36.6\\ \\MD=36.6-14-14\\ \\MD=8.6\ cm[/tex]
7. The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
Triangle STB is isosceles triangle because [tex]m\angle S=m\angle T=78^{\circ}[/tex] (angles adjacent to the side ST are congruent, so triangle STB is isoseceles).
To legs in isosceles triangle are always congruent, so
[tex]SB=TB\\ \\y+22.5=38.5\\ \\y=38.5-22.5\\ \\y=16[/tex]
Hence,
[tex]ST=16\ cm\\ \\TB=SB=38.5\ cm[/tex]
and the perimeter of triangle STB is
[tex]P_{STB}=16+38.5+38.5=93\ cm[/tex]
8. The diagram shows isosceles triangle CNB because CN = CB. Angles adjacent to the base RY are congruent, so
[tex]m\angle N=m\angle B[/tex]
The sum of the measures of all interior angles is always 180°, thus
[tex]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}[/tex]
To legs in isosceles triangle are always congruent, so
[tex]CB=CN=2x+90\ m[/tex]
The perimeter of the triangle CNB is
[tex]2x+90+2x+90+x=555\\ \\5x+180=555\\ \\x+36=111\\ \\x=111-36\\ \\x=75\ m[/tex]
So, [tex]NB=75 \ m[/tex]
