Chapter 12.2, Problem 24E

a.

To determine

To express: the expression for $m\angle 1$ , $m\angle 2$ and $m\angle 3$ in terms of $x$ and $y$

Answer to Problem 24E

The expression for $m\angle 1$ , $m\angle 2$ and $m\angle 3$ in terms of $x$ and $y$ are $m\angle 1=x^{\circ }$ , $m\angle 2={180}^{\circ }-(x^{\circ }+y^{\circ })$ , $m\angle 3=y^{\circ }$ .

Explanation of Solution

Given information: $\angle 1$ , $\angle 2$ and $\angle 3$ are the internal angles of the triangle. And other two angles $x^{\circ }$ and $y^{\circ }$ are provided.

Concept used : Alternate interior angles are always equal and the straight angle measure ${180}^{\circ }$

  $m\angle 1=x^{\circ }$ (Alternate interior angle)

  $m\angle 2={180}^{\circ }-(x^{\circ }+y^{\circ })$ (Straight angle)

  $m\angle 3=y^{\circ }$ (Alternate interior angle)

The expression for $m\angle 1$ , $m\angle 2$ and $m\angle 3$ in terms of $x$ and $y$ are $m\angle 1=x^{\circ }$ , $m\angle 2={180}^{\circ }-(x^{\circ }+y^{\circ })$ , $m\angle 3=y^{\circ }$ .

b.

To determine

To proof: the sum of the measure of the angles in $△ABC$ is ${180}^{\circ }$

Answer to Problem 24E

Sum of the all three angle of the triangle is ${180}^{\circ }$

Explanation of Solution

Given information: The expression for $m\angle 1$ , $m\angle 2$ and $m\angle 3$ in terms of $x$ and $y$ are $m\angle 1=x^{\circ }$ , $m\angle 2={180}^{\circ }-(x^{\circ }+y^{\circ })$ , $m\angle 3=y^{\circ }$ .

Concept Used: Sum up the all three angles.

  $\begin{array}{l}{x}^{\circ }+y^{\circ }+{180}^{\circ }-\left(x^{\circ }+y^{\circ }\right)\\ =x^{\circ }+y^{\circ }+{180}^{\circ }-x^{\circ }-y^{\circ }\\ ={180}^{\circ }\end{array}$

Sum of the all three angle of the triangle is ${180}^{\circ }$