In the diagram AB is parallel to DC.
Angle ABC = 70°
Angle ACD = 55°
Not drawn accurately
Show that triangle ABC is isosceles.
You must give reasons in your working.

Triangle ABC is an isosceles triangle.

Solution:

Given data:

AB is parallel to CD and AC is the transversal.

∠ABC = 70° and ∠ACD = 55°

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

∠BAC ≅ ∠ACD

m∠BAC = m∠ACD

m∠BAC = 55°

Sum of the angles in a straight line = 180°

m∠ACD + m∠ACB + m∠ABC = 180°

55° + m∠ACB + 70° = 180°

m∠ACB + 125° = 180°

Subtract 125° from both sides, we get

m∠ACB = 55°

In triangle ABC,

∠BAC = 55° and ∠ACB = 55°

∠BAC = ∠ACB

Two angles in the triangle are equal.

Therefor triangle ABC is an isosceles triangle.