Chapter 3: Q34 (page 99)

A pair of same-side interior angles are trisected (divided into three congruent angles) by the red lines in the diagram. Find out what you can about the angles of ABCD.

Expert verified

The angles of ABCD is given by $∠ A = 120 ° - ∠ Y, ∠ B = 60 °, ∠ C = 120 ° - ∠ y$ and $∠ D = 120 °$ .

Step by step solution

If two parallel lines are cut by transversal then same side interior angles are supplementary. Then,

$\begin{matrix} 3 ∠ x + 3 ∠ y = 180 \\ 3 (∠ x + ∠ y) = 180 \\ ∠ x + ∠ y = 60 ° \end{matrix}$

The sum of the measure of all angles of a triangle is 180. Therefore,3

$\begin{matrix} ∠ x + ∠ y + ∠ D = 180 \\ 60 + ∠ D = 180 \\ ∠ D = 120 ° \end{matrix}$

The sum of the measure of all angles of a triangle is 180. Therefore,

$\begin{matrix} 2 ∠ x + 2 ∠ y + ∠ B = 180 \\ 2 (∠ x + ∠ y) + ∠ B = 180 \\ 120 + ∠ B = 180 \\ ∠ B = 60 ° \end{matrix}$

The sum of the measure of all angles of a triangle is 180. Therefore,

$\begin{matrix} ∠ x + 2 ∠ y + ∠ A = 180 \\ ∠ x + ∠ y + ∠ y + ∠ A = 180 \\ 60 + ∠ y + ∠ A = 180 \\ ∠ A = 120 ° - ∠ y \end{matrix}$

The sum of the measure of all angles of a triangle is 180. Therefore,

$\begin{matrix} 2 ∠ x + ∠ y + ∠ C = 180 \\ ∠ x + ∠ x + ∠ y + ∠ C = 180 \\ 60 + ∠ x + ∠ C = 180 \end{matrix}$

$∠ C = 120 ° - ∠ x$

The angles of ABCD is given by $∠ A = 120 ° - ∠ y, ∠ B = 60 °, ∠ C = 120 ° - ∠ y$ and $∠ D = 120 °$

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