two circles

[logistic_guy]

\(\displaystyle \bold{Given}\): \(\displaystyle \Delta \text{ABC is isosceles} \ (\overline{\text{AB}} \ \cong \overline{\text{AC}}). \ Ⓢ \ \text{P and Q}\), \(\displaystyle \overline{\text{BC}} \parallel \overline{\text{PQ}}\).

\(\displaystyle \bold{Prove}\): \(\displaystyle \text{\large$\odot$}\text{P} \ \cong \text{\large$\odot$}\text{Q}\).

 

[khansaheb]

\(\displaystyle \bold{Given}\): \(\displaystyle \Delta \text{ABC is isosceles} \ (\overline{\text{AB}} \ \cong \overline{\text{AC}}). \ Ⓢ \ \text{P and Q}\), \(\displaystyle \overline{\text{BC}} \parallel \overline{\text{PQ}}\).

\(\displaystyle \bold{Prove}\): \(\displaystyle \text{\large$\odot$}\text{P} \ \cong \text{\large$\odot$}\text{Q}\).
View attachment 39750

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[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 1}}\)

\(\displaystyle \Delta \text{ABC is isosceles} \ (\overline{\text{AB}} \ \cong \overline{\text{AC}}) \longrightarrow \textcolor{grey}{\bold{Given}}\)

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 2}}\)

\(\displaystyle Ⓢ \ \text{P and Q}\), \(\displaystyle \overline{\text{BC}} \parallel \overline{\text{PQ}} \longrightarrow \textcolor{grey}{\bold{Given}}\)

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 3}}\)

\(\displaystyle \angle \text{ABC} \ \cong \angle \text{P}\) and \(\displaystyle \angle \text{ACB} \ \cong \angle \text{Q}\)

\(\displaystyle \textcolor{grey}{\bold{Since \ corresponding \ angles \ formed \ by \ parallel \ lines \ are \ congruent.}}\)

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 4}}\)

\(\displaystyle \angle \text{ABC} \ \cong \angle \text{ACB}\)

In \(\displaystyle \Delta\text{ABC}\), sides \(\displaystyle \text{AB}\) and \(\displaystyle \text{AC}\) are congruent. By the \(\displaystyle \text{Isosceles Triangle Theorem}\), the angles opposite these sides are also congruent.

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 5}}\)

\(\displaystyle \angle \text{P} \ \cong \angle \text{Q}\)

\(\displaystyle \textcolor{grey}{\bold{Transitive \ Property.}}\)

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 6}}\)

\(\displaystyle \overline{\text{AP}} \ \cong \overline{\text{AQ}}\)

In \(\displaystyle \Delta\text{APQ}\), angles \(\displaystyle \text{P}\) and \(\displaystyle \text{Q}\) are congruent. By the \(\displaystyle \text{Isosceles Triangle Theorem}\), the sides opposite those angles are also congruent.

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 7}}\)

\(\displaystyle \overline{\text{PB}} \ \cong \overline{\text{QC}}\)

\(\displaystyle \textcolor{grey}{\bold{Substract \ step \ 1 \ from \ step \ 6}}\)

 

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{step \ 8}}\)

\(\displaystyle \text{\large$\odot$}\text{P} \ \cong \text{\large$\odot$}\text{Q}\)

\(\displaystyle \textcolor{grey}{\bold{Circles \ that \ have \ congruent \ radii \ are \ congruent.}}\)