Q. 24 In the diagram, m鈭燰OZ=90聽.聽O... [FREE SOLUTION]

91影视

Geometry

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

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 5: Q. 24 (page 201)

In the diagram, $m 鈭� VOZ = 90$ .
$\overset{炉}{OW}$ is an altitude of $鈻� VOZ$ .
$\overset{炉}{OX}$ bisects $鈭� VOZ$ .
$\overset{炉}{OY}$ is a median of $鈻� VOZ$ .
Find the measures of the four numbered angles.
$m 鈭� Z = 30$

Short Answer

Expert verified

The measures of the four numbered angles $鈭� 1$ , $鈭� 2$ , $鈭� 3$ and $鈭� 4$ are $30 掳$ , $15 掳$ , 15 and 30 respectively.

Step by step solution

- Observe the given diagram.

The given diagram is:

- Description of step.

It is being given that $m 鈭� V O Z = 90$ .

Therefore it can be obtained that:

$鈭� 1 + 鈭� 2 + 鈭� 3 + 鈭� 4 = 90 掳$ .

As, $\overset{炉}{O X}$ bisects $鈭� V O Z$ .

Therefore, $鈭� V O Z = 2 鈭� V O X$ .

Therefore it can be obtained that:

$\begin{matrix} 鈭� V O Z = 2 鈭� V O X \\ 90 掳 = 2 鈭� V O X \\ \frac{90 掳}{2} = 鈭� V O X \\ 45 掳 = 鈭� V O X \end{matrix}$

Therefore, the measure of the angle $鈭� V O X$ is $45 掳$ .

As, $\overset{炉}{O X}$ bisects $鈭� V O Z$ .

Therefore, $鈭� V O X = 鈭� Z O X = 45 掳$

As, $鈭� V O X = 45 掳$ , therefore it can be obtained that:

$鈭� 1 + 鈭� 2 = 45 掳$

As, $鈭� Z O X = 45 掳$ , therefore it can be obtained that:

$鈭� 3 + 鈭� 4 = 45 掳$

- Description of step.

The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

As, $V Z$ is hypotenuse and $\overset{炉}{O Y}$ is a median of $鈻� V O Z$ .

Therefore, is the midpoint of the hypotenuse and therefore it is equidistant from the three vertices of the triangle.

That implies, $O Y = Y Z$ .

The angles opposite to the equal sides are also equal.

Therefore, it can be noticed that:

$鈭� Y O Z = 鈭� Y Z O$

Therefore, it can be obtained that:

$\begin{matrix} 鈭� Y O Z = 鈭� Y Z O \\ 鈭� 4 = 鈭� Z \\ 鈭� 4 = 30 掳 \end{matrix}$

Therefore, the measure of the angle $鈭� 4$ is $30 掳$ .

- Description of step.

As, $3 + 4 = 45 掳$ , therefore it can be obtained that:

$\begin{matrix} 鈭� 3 + 鈭� 4 = 45 掳 \\ 鈭� 3 + 30 掳 = 45 掳 \\ 鈭� 3 = 45 掳鈭� 30 掳 \\ 鈭� 3 = 15 掳 \end{matrix}$

Therefore, the measure of the angle $鈭� 3$ is $15 掳$ .

- Description of step.

As, the sum of all the angles of a triangle is $180 掳$ .

Therefore, the sum of the angles of triangle $V O Z$ is $180 掳$ .

Therefore, it can be obtained that:

$\begin{matrix} 鈭� V O Z + 鈭� V Z O + 鈭� Z V O = 180 掳 \\ 90 掳 + 30 掳 + 鈭� Z V O = 180 掳 \\ 鈭� Z V O + 120 掳 = 180 掳 \\ 鈭� Z V O = 180 掳鈭� 120 掳 \\ 鈭� Z V O = 60 掳 \end{matrix}$

As, $\overset{炉}{O W}$ is an altitude of $鈻� V O Z$ .

Therefore, $鈭� V W O = 90 掳$ .

- Description of step.

As, the sum of all the angles of a triangle is $180 掳$ .

Therefore, the sum of the angles of triangle $V W O$ is $180 掳$ .

Therefore, it can be obtained that:

$\begin{matrix} 鈭� V O W + 鈭� V W O + 鈭� W V O = 180 掳 \\ 鈭� 1 + 90 掳 + 鈭� Z V O = 180 掳 \\ 鈭� 1 + 90 掳 + 60 掳 = 180 掳 \\ 鈭� 1 + 150 掳 = 180 掳 \\ 鈭� 1 = 180 掳鈭� 150 掳 \\ 鈭� 1 = 30 掳 \end{matrix}$