Explanation:
(h)
we need to prove that both triangles are congruent (that means that each pair of corresponding sides is equal, and also the angles are the same between both triangles.
DC = CB
gives us already 1 side in each triangle. these sides are congruent.
because of the same angles of an intersecting line with parallel lines we know that
angle A = angle E
angle D = angle B
and because of the law that the sum of all 3 angles in a triangle is always 180°, we know then that also the inner angles at C are equal.
so, we can apply the ASA proof (angle - side - angle).
when 2 angles and the side length between them are the same between 2 triangles, then both triangles are congruent.
as this is the case here (D - DC - C and B - BC - C), both triangles are congruent, and then all the other sides, heights,..., and angles are also the same between the 2 triangles.
therefore,
AC = CE
AB = DE
(i)
we know that ABD is an isoceles triangle (both legs have the same length). that means automatically that
angle B = angle D
we also know that
angle 1 (at C) = angle 2 (at C) = 90°
and as the height (AC) in an isoceles triangle must cut the top angle at A in half, we know that
angle 1 (at A) = angle 2 (at A) = 180 - 90 - B = 90 - B.
and the height (AC) also cuts the baseline in half. so,
BC = CD.
and the height AC is the same for both ABC and ADC.
so we know that all angles and all sides are the same between ABC and ADC.
so,
SSS (side- side- side) applies (all 3 sides are equal).
SAS (side- angle- side) applies (2 sides and the angle between them are equal).
ASA (angle- side- angle) applies (2 angles and the side between them are equal).
so, we can prove it even in 3 ways ...