Q24. Copy everything shown and write ... [FREE SOLUTION]

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 2: Q24. (page 65)

Copy everything shown and write a two-column proof.
Make a diagram showing $鈭� P Q R$ bisected by role="math" localid="1646848681352" $\overset{鈫�}{Q X}$ . Choose a point Y on the ray opposite to $\overset{鈫�}{Q X}$ .
Prove: $鈭� P Q Y 鈮� 鈭� R Q Y$

Expert verified

The two-column proof is:

Statements	Reasons
Point Y is on the ray opposite to $\overset{鈫�}{Q X}$ , $鈭� P Q R$ is bisected by $\overset{鈫�}{Q X}$ .	Given
$鈭� 1 = 鈭� 2$	$鈭� P Q R$ is bisected by $\overset{鈫�}{Q X}$
$鈭� P Q Y + 鈭� 1 = 180$	Definition of supplementary angles
$鈭� R Q Y + 鈭� 2 = 180$	Definition of supplementary angles
$鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 2$	Substitution property
$鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 1$	As $鈭� 1 = 鈭� 2$
$鈭� P Q Y 鈮� 鈭� R Q Y$	By solving the equation $鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 1$ .

The diagram depicting the given situation is:

It is being given that point Y is on the ray opposite to $\overset{鈫�}{Q X}$ 补苍诲鈭�PQRis bisected by $\overset{鈫�}{Q X}$ .

础蝉,鈭�PQRis bisected by localid="1646849179080" $\overset{鈫�}{Q X}$ , therefore $鈭� 1 = 鈭� 2$ .

From the diagram, it can be noticed that the angles鈭�PQY 补苍诲鈭�1 forms the linear pair.

Therefore, the sum of the angles鈭�PQY 补苍诲鈭�1 is 180.

That implies, $鈭� P Q Y + 鈭� 1 = 180$ .

From the diagram, it can be noticed that the angles鈭�RQY 补苍诲鈭�2 forms the linear pair.

Therefore, the sum of the angles鈭�RQY 补苍诲鈭�2 is 180.

That implies, role="math" localid="1646849374087" $鈭� P Q Y + 鈭� 1 = 180$ .

Therefore, by using the substation property it can be obtained that:

$鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 2$

As, $鈭� 1 = 鈭� 2$ , therefore it can be obtained that:

$\begin{matrix} 鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 2 \\ 鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 1 \\ 鈭� P Q Y = 鈭� R Q Y \end{matrix}$

Therefore, $鈭� P Q Y 鈮� 鈭� R Q Y$ .

Hence proved.

Statements	Reasons
Point Y is on the ray opposite to $\overset{鈫�}{Q X}$ , $鈭� P Q R$ is bisected by $\overset{鈫�}{Q X}$ .	Given
$鈭� 1 = 鈭� 2$	$鈭� P Q R$ is bisected by
$鈭� P Q Y + 鈭� 1 = 180$	Definition of supplementary angles
$鈭� R Q Y + 鈭� 2 = 180$	Definition of supplementary angles
$鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 2$	Substitution property
$鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 1$	As $鈭� 1 = 鈭� 2$
$鈭� P Q Y 鈮� 鈭� R Q Y$	By solving the equation $鈭� P Q Y + 鈭� 1 = 鈭� R Q Y + 鈭� 1$ .