0
$\begingroup$

I need help understanding how the three angles are equal, it definitely has to do with geometry. enter image description here

  • 5
    Without more info, they are equal...because that's how they were constructed. If you don't say *what is given*, what is a construction out of the given and etc., it is hard to know. – 2017-01-02

4 Answers 4

1

Figure 1

As can be seen in the picture,

$\theta_1$ = A.

Also, B + A = $90^\unicode{xb0}$.

Therefore, B + $\theta_1$ = $90^\unicode{xb0}$.

But B + $\theta_2$ = $90^\unicode{xb0}$.

Therefore, $\theta_1$ = $\theta_2$.

Similarly, $\theta_2$ + C = $90^\unicode{xb0}$.

But C + $\theta_3$ = $90^\unicode{xb0}$.

Therefore $\theta_3$ = $\theta_2$.

  • 0
    I am sorry but i lost you in the first line, how is theta1 =A?, and could you start deriving knowing only theta3 is given. Thanks for the edit – 2017-01-02
  • 0
    Come on, use the same logic I used to show how $\theta_1$ = $\theta_2$ – 2017-01-02
  • 0
    Ok surely but how is theta1=A?? thats also the question – 2017-01-02
0

Because the three angles have mutually perpendicular sides (it is a a quite known elementary fact in geometry).

EDITION.- As promised in comment here a proof of equality. With the two angles $\angle ABC$ and $\angle DEF$ if they have their sides mutually perpendicular, when translating $\angle DEF$ in such a way that the vertices coincide it will be clear that both angles have equal complement $x$ as shown in the figure below.enter image description here

  • 0
    Can you give more information like theorem or source for this elementary fact? I encounter this problem a lot! – 2017-01-02
  • 0
    Actually the angles can be either equal or suplementary (i.e. sum=180º) but this second possibility is discarded here. Later I'll give you a proof (very easy). – 2017-01-02
0

Generally, if you rotate a square Euclidean grid (dashed/gray/red) through some acute angle $\theta$ to obtain a new grid (blue), the lines of the original grid meet the lines of the rotated grid at angles of $\theta$ or $\frac{\pi}{2} - \theta$ (or, if you work in degrees, $90^{\circ} - \theta$).

Particularly, for $\theta < \frac{\pi}{4}$ ($= 45^{\circ}$), the "small angles" are all equal to $\theta$. In your diagram, all three angles are "small".

0

Rotn by$-\theta$

Red lines as a rigid right angle corner with extended legs are rotated to Blue lines position. So the angle between straight lines before and after is same.

By Thales theorem right angles are subtended in a semicircle, the other angles $\theta$ in the segment you marked are same. The last one is also same by considering parallels.