Triangle RST has a median of RQ
and RS > RT. Explain why ∠RQT must be acute.

A.
∠RQS
is an exterior angle of
△RQT
. By the Corollary to the Triangle Exterior Angle Theorem, m
∠RQS
> m
∠RQT
. Also,
∠RQS
and
∠RQT
form a linear pair. So, since m
∠RQS
+ m
∠RQT
= 180 and m
∠RQS
> m
∠RQT
, m
∠RQS
must be greater than half of 180. Therefore, m
∠RQS
is obtuse and
∠RQT
must be acute.

B.
∠RQS
is an exterior angle of
△RQT
. By the Triangle Exterior Angle Theorem, m
∠RQS
> m
∠RQT
. Also,
∠RQS
and
∠RQT
form a linear pair, and

RQ
bisects
∠SRT
. So, since m
∠RSQ
+ m
∠RTQ
+
∠SRT
= 180 and m
∠RQS
> m
∠RQT
, m
∠RQT
must be greater than half of 180. Therefore, m
∠RQS
is obtuse and
∠RQT
must be acute.

C.
By the Converse of the Hinge Theorem, m
∠RQS
> m
∠RQT
. Also,
∠RQS
and
∠RQT
form a linear pair. So, since m
∠RQS
+ m
∠RQT
= 180 and m
∠RQS
> m
∠RQT
, m
∠RQS
must be greater than half of 180. Therefore, m
∠RQS
is obtuse and
∠RQT
must be acute.

D.
By the Hinge Theorem, m
∠RSQ
> m
∠RTQ
. Also,
∠RQS
and
∠RQT
form a linear pair, and

RQ
bisects
∠SRT
. So, since m
∠RSQ
+ m
∠RTQ
+
∠SRT
= 180 and m
∠RQS
> m
∠RQT
, m
∠RQT
must be greater than half of 180. Therefore, m
∠RQS
is obtuse and
∠RQT
must be acute.