Question

question 10
r, s, and t are the vertices of one triangle. e, f, and d are the vertices of another triangle. ( mangle r = 60 ), ( mangle s = 80 ), ( mangle f = 60 ), ( mangle d = 40 ), ( rs = 4 ), and ( ef = 4 ). are the two triangles congruent? if yes, explain and tell which segment is congruent to ( overline{rt} ).
no, the two triangles are not congruent.
yes, by sas; ( overline{ed} )
yes, by aas; ( overline{ed} )
yes, by asa; ( overline{fd} )

Explanation:

Step1: Find \(m\angle T\)

In \(\triangle RST\), using the angle - sum property of a triangle (\(m\angle R + m\angle S+m\angle T=180^{\circ}\)).
\(m\angle T=180-(60 + 80)=40^{\circ}\)

Step2: Analyze congruence

In \(\triangle RST\) and \(\triangle EFD\), \(m\angle R=m\angle F = 60^{\circ}\), \(RS = EF=4\), \(m\angle T=m\angle D = 40^{\circ}\).
By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle RST\cong\triangle EFD\).

Step3: Find the congruent segment

Since \(\triangle RST\cong\triangle EFD\), the segment congruent to \(\overline{RT}\) is \(\overline{FD}\) (corresponding parts of congruent triangles).

Answer:

Yes, by ASA; \(\overline{FD}\)