Question

B In the given figure, by how much a is bigger than b? (a) 50° (b) 45° D (c) 5° (d) 10° 275°(B 245°

Answer 1

In order to solve this question it's useful to remember that the sum of the internal angles of a triangle is equal to 180°.

Firstly, let's take a look at the angles which have D as their vertex. These two angles are BDC and BDA and they both share a side whereas their remaining sides are in the same line. This means that these two are supplementary which means that their sum is equal to 180°. Knowing that the measure of BDC is equal to 90° we get:

`\begin{gathered} BDC+BDA=90^(\circ)+BDA=180^(\circ) \\ 90^(\circ)+BDA=180^(\circ) \end{gathered}`

If we substract 90° from both sides:

`\begin{gathered} 90^(\circ)+BDA-90^(\circ)=180^(\circ)-90^(\circ) \\ BDA=90^(\circ) \end{gathered}`

So both angles are right angles.

Having found the measure of angle BDA let's take a look at triangle BDA. We have a 45° angle, a 90° angle and angle b. Since the sum of these 3 must be equal to 180° so we get:

`\begin{gathered} b+45^(\circ)+90^(\circ)=180^(\circ) \\ b+135^(\circ)=180^(\circ) \end{gathered}`

Then we substract 135° from both sides:

`\begin{gathered} b+135^(\circ)-135^(\circ)=180^(\circ)-135^(\circ) \\ b=45^(\circ) \end{gathered}`

So we have found b. We still need to find a. Let's take a look at vertex B. The external angle is equal to 275° whereas there are two internal angles: a 45° angle and angle DBC. The sum of the external and internal angles in a vertex must be equal to 360° then we have:

`\begin{gathered} 360^(\circ)=275^(\circ)+45^(\circ)+DBC=320^(\circ)+DBC \\ 360^(\circ)=320^(\circ)+DBC \end{gathered}`

If we substract 320° from both sides we get:

`\begin{gathered} 360^(\circ)-320^(\circ)=320^(\circ)+DBC-320^(\circ) \\ 40^(\circ)=DBC \end{gathered}`

Now that we have angle DBC we can find angle a by looking at triangle BDC. The sum of its internal angles is 180° and we already know two of them: DBC and BDC. Then we get:

`\begin{gathered} 180^(\circ)=DBC+BDC=40^(\circ)+90^(\circ)+a=130^(\circ)+a \\ 180^(\circ)=130^(\circ)+a \end{gathered}`

Then if we substract 130° from both sides we get:

`\begin{gathered} 180^(\circ)-130^(\circ)=130^(\circ)+a-130^(\circ) \\ a=50^(\circ) \end{gathered}`

So now we know that a=50° and b=45°. Then a is bigger than b by 5° and the answer is option c.