Math  /  Geometry

Question3. 4.
5. In Triangle MNO, N\angle N is 5 times the measure of M\angle M, and O\angle O is 4 times the measure of M\angle M. \qquad M=\angle M= \qquad J=\angle \mathrm{J}= \qquad G=H=I=\begin{array}{l} \angle G= \\ \angle H= \\ \angle I= \end{array} K=\angle K= \qquad L=\angle L= \qquad \qquad

4 \qquad N=\angle N= \qquad O=\angle O=

Studdy Solution

STEP 1

What is this asking? We need to find the measure of each angle in triangle MNO, given that angle N is 5 times the measure of angle M, and angle O is 4 times the measure of angle M. Watch out! Remember that the angles in a triangle always add up to 180 degrees.
Don't forget your units (degrees)!

STEP 2

1. Set up the equation
2. Solve for angle M
3. Solve for angles N and O

STEP 3

We're told that N \angle N is **5** times M \angle M , so we can write that as N=5M \angle N = 5 \cdot \angle M .
Similarly, O \angle O is **4** times M \angle M , so O=4M \angle O = 4 \cdot \angle M .
This helps us relate all the angles to a single unknown, which is super helpful!

STEP 4

We know that the sum of the angles in any triangle is **180** degrees.
So, for triangle MNO, we have M+N+O=180 \angle M + \angle N + \angle O = 180^\circ .
This is a fundamental fact about triangles, and it's the key to unlocking this problem!

STEP 5

Now, let's **substitute** the expressions we found for N \angle N and O \angle O into the triangle angle sum equation: M+(5M)+(4M)=180 \angle M + (5 \cdot \angle M) + (4 \cdot \angle M) = 180^\circ .
See how we're bringing everything together?

STEP 6

We can **combine** the terms with M \angle M : 1M+5M+4M=10M 1 \cdot \angle M + 5 \cdot \angle M + 4 \cdot \angle M = 10 \cdot \angle M .
So our equation becomes 10M=180 10 \cdot \angle M = 180^\circ .
We're getting closer to finding the measure of angle M!

STEP 7

To **isolate** M \angle M , we **divide** both sides of the equation by **10**: 10M10=18010 \frac{10 \cdot \angle M}{10} = \frac{180^\circ}{10} .
This gives us M=18 \angle M = 18^\circ .
Awesome, we found one of the angles!

STEP 8

We know that N=5M \angle N = 5 \cdot \angle M .
Since we found that M=18 \angle M = 18^\circ , we can **substitute** that value in: N=518=90 \angle N = 5 \cdot 18^\circ = 90^\circ .
Look at that, a right angle!

STEP 9

Similarly, we know that O=4M \angle O = 4 \cdot \angle M .
Substituting M=18 \angle M = 18^\circ , we get O=418=72 \angle O = 4 \cdot 18^\circ = 72^\circ .
We've found all the angles!

STEP 10

M=18 \angle M = 18^\circ N=90 \angle N = 90^\circ O=72 \angle O = 72^\circ

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