Question

1. find the perimeter of $\triangle mnp$.

1. find the measure of each missing angle.

$m\angle 1 = \underline{\quad\quad}$
$m\angle 2 = \underline{\quad\quad}$
$m\angle 3 = \underline{\quad\quad}$
$m\angle 4 = \underline{\quad\quad}$
$m\angle 5 = \underline{\quad\quad}$

1. find the measure of each missing angle.

$m\angle 1 = \underline{\quad\quad}$
$m\angle 2 = \underline{\quad\quad}$
$m\angle 3 = \underline{\quad\quad}$
$m\angle 4 = \underline{\quad\quad}$
$m\angle 5 = \underline{\quad\quad}$

1. find the value of $x$.

1. find the value of $x$.

1. find $m\angle wyz$.

1. find $m\angle acb$.

Explanation:

Problem 9

Step1: Identify midsegment properties

Segments QR, QS, RS are midsegments of $\triangle MNP$. Midsegment length = $\frac{1}{2}$ of parallel side.

Step2: Solve for $x$ using QR

$25 = \frac{1}{2}(5x-34)$
Multiply both sides by 2: $50 = 5x - 34$
$5x = 50 + 34 = 84$
$x = \frac{84}{5} = 16.8$

Step3: Find side lengths

$MN = 5x - 34 = 5(16.8)-34 = 84-34=50$
$NP = 2(x+4) = 2(16.8+4)=2(20.8)=41.6$
$MP = 2\times22 = 44$

Step4: Calculate perimeter

Perimeter = $MN + NP + MP$

Step1: Find $\angle 1$ and $\angle 4$

The triangle is isosceles, so $m\angle 1 = m\angle 4$. Sum of angles in triangle: $m\angle 1 + m\angle 4 + 144^\circ = 180^\circ$
$2m\angle 1 = 180-144=36^\circ$
$m\angle 1 = m\angle 4 = 18^\circ$

Step2: Find $\angle 5$

$\angle 5$ is supplementary to $144^\circ$: $m\angle 5 = 180-144=36^\circ$

Step3: Find $\angle 2$ and $\angle 3$

Small triangle is isosceles, $m\angle 2 = m\angle 3$. Sum: $m\angle 2 + m\angle 3 + 36^\circ = 180^\circ$
$2m\angle 2 = 180-36-56=88^\circ$
$m\angle 2 = m\angle 3 = 44^\circ$

Step1: Find $\angle 3$

Large right triangle: $m\angle 3 = 180-90-35=55^\circ$

Step2: Find $\angle 1$

Small right triangle, $\angle 1$ is complementary to $\angle 3$: $m\angle 1 = 90-55=35^\circ$

Step3: Find $\angle 2$

$\angle 2$ is complementary to $35^\circ$: $m\angle 2 = 90-35=55^\circ$

Step4: Find $\angle 4$

$\angle 4$ is supplementary to $\angle 3$: $m\angle 4 = 180-55=125^\circ$

Step5: Find $\angle 5$

$\angle 5$ is supplementary to $\angle 4$: $m\angle 5 = 180-125=55^\circ$

Answer:

$50 + 41.6 + 44 = 135.6$

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Problem 10