1. You have to show that m<ADC = m < ACD ( ADC is an isosceles triangle)
hi, i have 2 hard geometry questions that i just cannot do. any help would be great
1) In a pentagon ABCDE, AB=AE, BC=ED, and BD=EC Prove that AD=AC
2)In a quadrilateral ABCD, the diagonal AC bisects <BAD and AB=BC=CD. Prove that the other diagonal BD, bisects <ADC
thanx
Nice problems. Good for practice.
Let me show it not in statements-reasons form.
1) In a pentagon ABCDE, AB=AE, BC=ED, and BD=EC Prove that AD=AC
You have to sketch the figure on paper. I do not know how to sketch figures on the board.
The idea is to show that triangle ABC is congruent to riangle AED so that AD=AC.
First, triangle BCD is congruent to triangle EDC by SSS, because
BC=ED, BD=EC as givens, and CD=DC for the one and the same side.
So, angle CBD = angle DEC. -------------(i)
Draw line segment BE.
Triangles CBE and DEB are conguent, by SSS again.
BC=ED, BD=EC as givens, and BE=BE for the one and the same side.
So, angle DBE = angle CEB. -------------(ii)
In triangle ABE, since AB=AE, then the triangle is isosceles, and so,
angle EBA = angle BEA ------------------(iii)
So, using (i),(ii) and (iii), anlge ABC = angle AED.
Hence, triangles ABC and AED are congruent, by SAS, because
AB=AE, angle ABC = angle AED, and BC=ED.
Therefore, AD = AC.
==================================
2)In a quadrilateral ABCD, the diagonal AC bisects <BAD and AB=BC=CD. Prove that the other diagonal BD, bisects <ADC
This one is tougher. Took me many tries. The SSA (or ASS) was a block.
Draw the figure on paper. Draw the diagonal AC.
AB=BC, so triangle ABCis isosceles, and so angle BAC = angle BCA.
angle BAC = angle DAC -------given, or they are half each of angle BAD.
So angle DAC = angle BAC = angle BCA.
Hence, side AD is parallel to side BC. -----reverse of alternate interior angles of parallel lines (and a transversal line, diagonal AC here) are equal.
So, using diagonal BD as the transversal,
angle ADB = angle CBD ----alternate interior angles are equal. ----(a)
But in isosceles triangle BCD, since BC=CD,
angle CBD = angle CDB -----------------------(b)
So, using (a) and (b),
angle ADB = angle CDB.
Therefore, diagonal BD bisects angle ADC.
Construct the perpendicular bisector of this passes through
the point of intersection of and it also passes through .
(sorry this does not quite look right in the attachment, but the extra line
is the perpendicular bisector that I require, can you see why it passes through
and the point of intersection of and ?)
The triangles formed by the bisector the two halves of and
and are right triangles with two equal sides and so congruent,
and as and are corresponding sides in these triangles they
are congrent.
RonL
consider the bisector of CD that passes through the point of intersection
and label the mid point of CD M and the point of intersection I for
convenience.
Then CI is congruent to DI as triangles DBC and CED are congruent and CD
is a common side (angles BDC and ECD are corresponding angles in these
triangles and so equal which makes triangle DIC isosceles). So triangles CIM
and DIM are congruent by SSA. But angles CMI and DMI are corresponding
angles and so congruent, but they are also supplementary angles and so sum
to 180 degrees, so each or them is 90 degrees.
Hence the line through I and M is a perp bisector of CD.
(alternatively it is obvious by symmetry that the bisector of CD passing
through M is a perp bisector)
RonL