Plane Geometry Question (1 Viewer)

[Zak Ambrose]

in the diagram ABCD is a quadrilateral.
the diagonals AC, BD intersect at right angles.
and

DAS =

BAS.

(i) draw a neat sketch of the diagram
(ii) giving reasons for each step, use congruent triangles to prove that DA = AB
(iii) hence prove that DC = BC

im having trouble with part (ii)

i can show that they are similiar by AAA
and that they share a common side, line AS.
which will give me angle, angle, side making them congruent.
im just unsure how to finish it off and prove DA = AB

any help would be greatly appreciated

 

[Timothy.Siu]

corresponding sides of congruent triangles

 

[studentcheese]

where did the line S come from?

 

[Zak Ambrose]

S in the centre. a.k.a the intersection of the lines AC and BD

 

[Drongoski]

Note: 'S' is the point of intersection; not the 'centre'

(ii) Proof:

In triangles ABS & ADS

1) angle ASB = angle ASD (both right angles)
2) angle BAS = angle DAS (given)
3) side AS = side AS (common side)

Therefore triangle ABS is congruent to triangle ADS (AAS)

Therefore corresponding sides AB = AD

Now in triangles ABC & ADC:

1) side AB = side AD (proven above)
2) side AC = side AC (common side)
3) included angle BAC = included angle DAC (given)

Therefore triangles ABC & ADC are congruent (SAS)
Therefore corresp. sides BC = DC


QED