Question

given:
\\( \angle r=\angle q \\)
\\( a s=a t \\)
prove:
\\( r s=q t \\).
helpful hint: start by identifying the given first.

1. \\( \angle r=\angle q, a s=a t \\)
2. \\( \angle 5=\angle 6 \\)
3. \\( \angle 4, \angle 5 \\) are supplementary and \\( \angle 6, \angle 7 \\) are supplementary
4. \\( \angle 4=\angle 7 \\)
5. triangle ras congruent to triangle qat
6. \\( r s=q t \\)

Explanation:

Step1: Identify the given information

Given \(\angle R=\angle Q\) and \(AS = AT\).

Step2: Use the property of isosceles triangle

Since \(AS = AT\), by the base - angles of an isosceles triangle theorem, \(\angle5=\angle6\).

Step3: Use the supplementary - angle property

\(\angle4\) and \(\angle5\) are supplementary (\(\angle4+\angle5 = 180^{\circ}\)), \(\angle6\) and \(\angle7\) are supplementary (\(\angle6+\angle7=180^{\circ}\)). Since \(\angle5=\angle6\), then \(\angle4=\angle7\) (two angles supplementary to equal angles are equal).

Step4: Prove triangle congruence

In \(\triangle RAS\) and \(\triangle QAT\), we have \(\angle R=\angle Q\) (given), \(\angle4=\angle7\) (proven above), and \(AS = AT\) (given). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle RAS\cong\triangle QAT\).

Step5: Use CPCTE (Corresponding Parts of Congruent Triangles are Equal)

Since \(\triangle RAS\cong\triangle QAT\), then \(RS = QT\) (CPCTE).

Answer:

1. Given
2. Base \(\angle\)'s of isosceles triangle are \(=\)
3. Exterior sides in opposite rays
4. Two \(\angle\)'s supplementary to equal \(\angle\)'s are \(=\)
5. AAS
6. CPCTE