Many angles

Geometry Level 2

Five lines: $l_1$ , $l_2$ , $1_3$ , $l_4$ and $l_5$ intersect at point O and they are intersected by five other lines $k_1$ , $k_2$ , $k_3$ , $k_4$ and $k_5$ (see the picture). What is the sum of the measures of the 10 shaded angles in degrees, shown in the picture below? Give your answer in degrees.

The answer is 720.

1 solution

Marta Reece
May 27, 2017

The five triangles containing the angles have a sum of internal angles equal to $5\times180^\circ$ .

The non-shaded angles included in this sum are all at point $O$ , and each is equal to an angle on the other side of $O$ betweem the same lines.

So the non-shaded angles included in the sum make up exactly one half of the $360^\circ$ around point $O$ , that is $180^\circ$ .

Subtracting that from the original sum will give us the answer:

$5\times180^\circ-180^\circ=4\times180^\circ=\boxed{720^\circ}$

Many angles

The answer is 720.

1 solution

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