A figure shows two parallel lines, JG and HK. A line connects JK and HG. These two lines intersect at M.
Complete the proof.
It is given that M is the midpoint of segment
. Therefore
by the definition of a midpoint. It is also given that Angles
and
are right angles. Thus,
because all right angles are congruent. Since vertical angles are congruent by the vertical angles theorem,
. Then, it follows that
by the
criteria. In congruent triangles, corresponding sides are congruent, thus
. In conclusion
by definition of congruent angles.
By the definition of a midpoint, we can conclude that JM = MK.
Since angles JHG and JKJ are right angles, they are congruent.
By the vertical angles theorem, we know that angles JHG and HKM are congruent.
Therefore, triangle JHK is congruent to triangle JKM by the angle-side-angle (ASA) criteria.
In congruent triangles, corresponding sides are congruent. Therefore, JH = JM and HK = MK.
By the definition of congruent angles, we can conclude that angle JHM is congruent to angle JKM.