a) If $$ m \angle M O N=51^{\circ} $$, find $$ m \angle N O K $$. b) If $$ m \angle J O K=40^{\circ} $$ and $$ m \angle N O M=2 x+8 $$, find the value of $$ x $$. c) If $$ m \angle J O K=5 x^{\circ} $$ and $$ m \angle K O N=(17 x+26)^{\circ} $$, find the value of $$ x $$. 4. Find the distance between points $$ Q(2,4) $$ and $$ R(-3,9) $$.

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Let's tackle each part of your question step by step.

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### **a) Finding $$ m \angle N O K $$ Given $$ m \angle M O N = 51^{\circ} $$**

Assuming that angles $$ \angle M O N $$ and $$ \angle N O K $$ are adjacent angles that form a straight line at point $$ O $$, their measures should add up to $$ 180^{\circ} $$.

$$
m \angle M O N + m \angle N O K = 180^{\circ}
$$

$$
51^{\circ} + m \angle N O K = 180^{\circ}
$$

$$
m \angle N O K = 180^{\circ} - 51^{\circ} = 129^{\circ}
$$

**Answer:** $$ m \angle N O K = 129^{\circ} $$

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### **b) Finding the Value of $$ x $$ Given $$ m \angle J O K = 40^{\circ} $$ and $$ m \angle N O M = 2x + 8^{\circ} $$**

Assuming that angles $$ \angle J O K $$ and $$ \angle N O M $$ are also adjacent and form a straight line, their measures should add up to $$ 180^{\circ} $$.

$$
m \angle J O K + m \angle N O M = 180^{\circ}
$$

$$
40^{\circ} + (2x + 8^{\circ}) = 180^{\circ}
$$

$$
2x + 48^{\circ} = 180^{\circ}
$$

$$
2x = 180^{\circ} - 48^{\circ} = 132^{\circ}
$$

$$
x = \frac{132^{\circ}}{2} = 66
$$

**Answer:** $$ x = 66 $$

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### **c) Finding the Value of $$ x $$ Given $$ m \angle J O K = 5x^{\circ} $$ and $$ m \angle K O N = (17x + 26)^{\circ} $$**

Again, assuming that angles $$ \angle J O K $$ and $$ \angle K O N $$ form a straight line, their measures should add up to $$ 180^{\circ} $$.

$$
m \angle J O K + m \angle K O N = 180^{\circ}
$$

$$
5x^{\circ} + (17x + 26)^{\circ} = 180^{\circ}
$$

$$
22x + 26 = 180
$$

$$
22x = 180 - 26 = 154
$$

$$
x = \frac{154}{22} = 7
$$

**Answer:** $$ x = 7 $$

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### **4. Finding the Distance Between Points $$ Q(2,4) $$ and $$ R(-3,9) $$**

To find the distance between two points in a Cartesian plane, use the **distance formula**:

$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Plugging in the coordinates:

$$
\text{Distance} = \sqrt{(-3 - 2)^2 + (9 - 4)^2} = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
$$

**Answer:** The distance between points $$ Q $$ and $$ R $$ is $$ 5\sqrt{2} $$ units.

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I hope this clarifies all parts of your question!
a) $$ m \angle N O K = 129^{\circ} $$ b) $$ x = 66 $$ c) $$ x = 7 $$ 4) The distance between points $$ Q $$ and $$ R $$ is $$ 5\sqrt{2} $$ units.

a) If , find .
b) If and , find the value of .
c) If and , find the value of .
4. Find the distance between points and .

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