6. in the diagram shown below, point o is located on $ \overline{mp} $ such that $ \overline{mn} \perp \overline{on} $, $ \overline{mn} \cong \overline{on} $, and $ \overline{on} \cong \overline{op} $.
(a) name two different isosceles triangles in this picture.
(b) what is the measure of $ \angle m $? show how you found your answer.
(c) what is the measure of $ \angle nop $? show how you found your answer. (d) what is the measure of $ \angle p $? show how you found your answer.
(e) what is the measure of $ \angle mnp $. show how you found your answer.

Question

Accepted Answer

### (a)
# Brief Explanations:
An isosceles triangle has two equal sides. Since $ \overline{MN}\cong\overline{ON}$, $\triangle MNO$ is isosceles. Also, because $ \overline{ON}\cong\overline{OP}$, $\triangle NOP$ is isosceles.
# Answer:
$\triangle MNO$ and $\triangle NOP$

### (b)
# Explanation:
## Step1: Use the property of right - isosceles triangle
In $\triangle MNO$, $ \overline{MN}\perp\overline{ON}$ (so $\angle MNO = 90^{\circ}$) and $ \overline{MN}\cong\overline{ON}$. For a right - isosceles triangle $\triangle MNO$, the sum of angles in a triangle is $180^{\circ}$. Let $\angle M=\angle NOM = x$. Then $x + x+90^{\circ}=180^{\circ}$.
## Step2: Solve the equation
$$
\begin{align*}
2x&=180^{\circ}- 90^{\circ}\
2x&=90^{\circ}\
x& = 45^{\circ}
\end{align*}
$$
# Answer:
$45^{\circ}$

### (c)
# Explanation:
## Step1: Use the angle - supplementary relationship
$\angle NOM$ and $\angle NOP$ are supplementary angles (they form a linear pair). From part (b), $\angle NOM = 45^{\circ}$.
## Step2: Calculate $\angle NOP$
Since $\angle NOM+\angle NOP = 180^{\circ}$, then $\angle NOP=180^{\circ}-\angle NOM$. Substituting $\angle NOM = 45^{\circ}$, we get $\angle NOP = 135^{\circ}$
# Answer:
$135^{\circ}$

### (d)
# Explanation:
## Step1: Use the property of isosceles triangle $\triangle NOP$
In $\triangle NOP$, $ \overline{ON}\cong\overline{OP}$, so $\angle P=\angle ONP$. Let $\angle P = y$. The sum of angles in $\triangle NOP$ is $180^{\circ}$, and $\angle NOP = 135^{\circ}$. Then $y + y+135^{\circ}=180^{\circ}$
## Step2: Solve the equation
$$
\begin{align*}
2y&=180^{\circ}-135^{\circ}\
2y&=45^{\circ}\
y& = 22.5^{\circ}
\end{align*}
$$
# Answer:
$22.5^{\circ}$

### (e)
# Explanation:
## Step1: Express $\angle MNP$ as a sum of angles
$\angle MNP=\angle MNO+\angle ONP$. We know $\angle MNO = 90^{\circ}$ (from $ \overline{MN}\perp\overline{ON}$) and $\angle ONP=\angle P = 22.5^{\circ}$ (from part (d), since $ \overline{ON}\cong\overline{OP}$ in $\triangle NOP$)
## Step2: Calculate $\angle MNP$
$\angle MNP=90^{\circ}+22.5^{\circ}=112.5^{\circ}$
# Answer:
$112.5^{\circ}$

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QUESTION IMAGE

Question

Explanation:

(a)

Step1: Use the property of right - isosceles triangle

Step2: Solve the equation

Step1: Use the angle - supplementary relationship

Step2: Calculate \(\angle NOP\)

Answer:

(b)