Question

1. in \\( \triangle a b c \\), if \\( \overline{a c} \cong \overline{c b}, m \angle a=3 x+18, m \angle b=7 x-58 \\), and \\( m \angle c=2 x-8 \\), find \\( x \\) and the measure of each angle. \\( x= \\) \\( m \angle a= \\) \\( m \angle b= \\) \\( m \angle c= \\) 13. in \\( \triangle q r s \\), if \\( \overline{q r} \cong \overline{r s}, m \angle q=8 x-17, m \angle r=19 x+4 \\), and \\( m \angle s=5 x+1 \\), find \\( x \\) and the measure of each angle. \\( x= \\) \\( m \angle q= \\) \\( m \angle r= \\) \\( m \angle s= \\) 14. in \\( \triangle k l m \\), if \\( \angle k \cong \angle l, k l=9 x-40, l m=7 x-37 \\), and \\( k m=3 x+23 \\), find \\( x \\) and the measure of each side. \\( x= \\) \\( k l= \\) \\( l m= \\) \\( k m= \\) figure not drawn to scale. 15. use the figure above to find each measure. \\( a= \\) \\( b= \\) \\( c= \\) \\( d= \\) \\( e= \\) \\( f= \\) \\( g= \\) \\( h= \\) \\( k= \\) \\( n= \\) \\( p= \\) \\( r= \\)

Explanation:

Step1: Identify the isosceles triangle property

Since \(\overline{AC}\cong\overline{CB}\) in \(\triangle ABC\), then \(\angle A=\angle B\). But wait, no! Wait, in a triangle, the sum of interior angles is \(180^{\circ}\). And if \(\overline{AC}\cong\overline{CB}\), then \(\angle A = \angle B\). Wait, no: in a triangle, if two sides are congruent, the angles opposite those sides are congruent. So \(\angle A=\angle B\).
So \(m\angle A + m\angle B+m\angle C=180^{\circ}\). Substitute \(m\angle A = 3x + 18\), \(m\angle B = 7x-58\), \(m\angle C=2x - 8\) into the angle - sum formula:
\((3x + 18)+(7x-58)+(2x - 8)=180\)

Step2: Simplify the left - hand side of the equation

Combine like terms: \((3x+7x + 2x)+(18-58 - 8)=180\)
\(12x-48 = 180\)

Step3: Solve for \(x\)

Add \(48\) to both sides of the equation: \(12x-48 + 48=180 + 48\), so \(12x=228\)
Divide both sides by \(12\): \(x=\frac{228}{12}=19\)

Step4: Find \(m\angle A\)

Substitute \(x = 19\) into \(m\angle A=3x + 18\): \(m\angle A=3\times19+18=57 + 18=75^{\circ}\)

Step5: Find \(m\angle B\)

Substitute \(x = 19\) into \(m\angle B=7x-58\): \(m\angle B=7\times19-58=133 - 58=75^{\circ}\)

Step6: Find \(m\angle C\)

Substitute \(x = 19\) into \(m\angle C=2x - 8\): \(m\angle C=2\times19-8=38 - 8=30^{\circ}\)

Answer:

\(x = 19\), \(m\angle A=75^{\circ}\), \(m\angle B=75^{\circ}\), \(m\angle C=30^{\circ}\)