Question

Geometry help with lines.The floor pattern correspond to two overlapping squares.

Answer 1

1. We have a shaded shape made of 8 equal kites.

We can take one kite and name the angles as:

The angle ∠1 is also one of the angles of one of the squares. As all the angles of the square are right angles, then ∠1 is also a right angle.

Then, m∠1 = 90°.

If we look at ∠3, we can see that it is 1/8 of the full circle, as it shares the center of the pattern with 8 equal angles that complete a full circle.

Then, we can write:

`m\angle3=(360)/(8)=45\degree`

Now we are left with ∠2 and ∠4.

As the kite has a axis of symmetry at the long diagonal, the two angles will have the same measure.

We also know that all quadrilateral, including the kites, have measures of the interior angles that add 360°.

Then, we can calculate ∠2 and ∠4 as:

`\begin{gathered} m\angle1=90\degree \\ m\angle3=45\degree \\ m\angle2=m\angle4 \\ \Rightarrow m\angle1+m\angle2+m\angle3+m\angle4=360\degree \\ 90+m\angle2+45+m\angle2=360 \\ 2\cdot m\angle2=360-90-45 \\ 2\cdot m\angle2=225 \\ m\angle2=(225)/(2) \\ m\angle2=m\angle4=112.5\degree \end{gathered}`

The measures are then:

m∠1 = 90°

m∠2 = 112.5°

m∠3 = 45°

m∠4 = 112.5°

2. We can add the measures of the angles to the two kites as:

We start by proving that AB and CD are parallel sides.

If we add all the angles from vertex C to vertex B, we can check that the result is 360°:

`112.5\degree+90\degree+112.5\degree+45\degree=360\degree`

This means that the direction of CD and BA are the same, as 360° is equal to a net rotation of 0°.

Then, if CD and AB are parallel, the segments BC and AD will also be parallel as they join vertex of parallel sides.

Then, if we have a quadrilateral of two pair of parallel sides, we have a parallelogram.

ABCD is a parallelogram.