Isosceles trapezoid problem

**mjoshua** · December 14th 2010, 01:38 PM

The attached image shows an isosceles trapezoid ABCD and AB is parallel to DC with angle BDC measuring 25 and BCA measuring 35. What is the measure of angle DBC?

I know the answer is 95 but I don't know how to justify (what do I need to show or prove??)

**Plato** · December 14th 2010, 01:54 PM

It is clear that $m\left( {\angle ACD} \right) = 25^ \circ \;\& \,m\left( {\angle ACB} \right) = 35^ \circ$
That should do it.

**mjoshua** · December 14th 2010, 01:57 PM

But how do we justify that ACD = 25 ? Why is it clear?? The other angle was given anyway..

**Plato** · December 14th 2010, 02:06 PM

It is an isosceles trapezoid.

**Archie Meade** · December 14th 2010, 02:09 PM

Originally Posted by mjoshua

The attached image shows an isosceles trapezoid ABCD and AB is parallel to DC with angle BDC measuring 25 and BCA measuring 35. What is the measure of angle DBC?

I know the answer is 95 but I don't know how to justify (what do I need to show or prove??)

As the trapezoid is isosceles...

$BDC=DBA=25^o$

$BCA=BDA=35^o$

$\Rightarrow\ ADC=BCD=60^o$

$\Rightarrow\ ABC=180^o-60^o=120^o$

$\Rightarrow\ DBC=ABC-25^o=95^o$

**mjoshua** · December 14th 2010, 02:10 PM

Right, the BASE angles are congruent big angle D and big angle C, but how do you justify that the others of the triangle are? How do we know that that triangle is isosceles?

How do you justify that ADC = BCD = 60?? I just dont see it

**Archie Meade** · December 14th 2010, 02:23 PM

Originally Posted by mjoshua

Right, the BASE angles are congruent big angle D and big angle C, but how do you justify that the others of the triangle are? How do we know that that triangle is isosceles?

There is no isosceles triangle, except the ones touching at the centre with bases DC and AB.

If the trapezoid is isosceles, then side AD equals side BC in length.

Sorry about cutting across you Plato, my silly portable modem is running very slowly.

**Archie Meade** · December 14th 2010, 02:28 PM

modem error .... please delete

**mjoshua** · December 14th 2010, 02:28 PM

I read all that archie, I'm struggling actually proving it tho.. How do you know that BDA is 35? If anything it should be the answer of 95 so I'm still stuck..

**Archie Meade** · December 14th 2010, 02:32 PM

Originally Posted by mjoshua

I read all that archie, I'm struggling actually proving it tho.. How do you know that BDA is 35? If anything it should be the answer of 95 so I'm still stuck..

Triangles ABC and ABD are congruent (identical).

**mjoshua** · December 14th 2010, 02:35 PM

Check again, they arent congruent that way, how do you know that BDA is 35 and not 95, how do you find it?? I know all the properties of the isosc trap, I'm struggling proving that it is 95 guys.. The vertices dont match the angles that should be congruent either..

**Plato** · December 14th 2010, 02:39 PM

Do you not see the congruence $\Delta ADC \doteq \Delta BCD~?$

**Archie Meade** · December 14th 2010, 02:40 PM

Originally Posted by mjoshua

Check again, they arent congruent that way, how do you know that BDA is 35 and not 95, how do you find it?? I know all the properties of the isosc trap, I'm struggling proving that it is 95 guys.. The vertices dont match the angles that should be congruent either..

If you label the point at the centre "O",
then do you see that triangles BOC and AOD are identical ?

**mjoshua** · December 14th 2010, 02:41 PM

Yes Plato by SAS..

No archie those do not match up!! Angle CBO is NOT congruent to angle DAO

**Plato** · December 14th 2010, 02:45 PM

Originally Posted by mjoshua

Yes Plato by SAS..

No archie those do not match up!! Angle CBO is NOT congruent to angle DAO

Well that is the whole point! Go with it.

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