Question

1. find the measure of each missing angle. 3 points

63°
2
38°
3
1
m∠1 =
m∠2 =
m∠3 =

1. find x and m∠a. 5 points

b
(7x - 26)°
(5x - 27)°
a
(10x - 23)°
c
x =
m∠a =
4 points

Explanation:

Step1: Find $\angle1$ in the right - triangle

In the right - triangle with one angle $38^{\circ}$, using the fact that the sum of angles in a triangle is $180^{\circ}$ and one angle is $90^{\circ}$. So $m\angle1=180^{\circ}-90^{\circ} - 38^{\circ}=52^{\circ}$.

Step2: Find $\angle3$ in the large triangle

In the large triangle, one angle is $63^{\circ}$ and the non - adjacent interior angle to the exterior angle related to $\angle3$ is $38^{\circ}$. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So $m\angle3=38^{\circ}+63^{\circ}=101^{\circ}$.

Step3: Find $\angle2$

Since $\angle2$ and $\angle3$ are supplementary (linear pair), $m\angle2 = 180^{\circ}-m\angle3$. So $m\angle2=180^{\circ}-101^{\circ}=79^{\circ}$.

Step4: Solve for $x$ in the second triangle

Using the exterior - angle property of a triangle, $(10x - 23)^{\circ}=(5x - 27)^{\circ}+(7x - 26)^{\circ}$.
Expand the right - hand side: $10x-23=5x - 27+7x - 26$.
Combine like terms: $10x-23=12x-53$.
Move the $x$ terms to one side: $12x - 10x=53 - 23$.
$2x=30$, so $x = 15$.

Step5: Find $m\angle A$

Substitute $x = 15$ into the expression for $\angle A$. $m\angle A=(5x - 27)^{\circ}$.
$m\angle A=(5\times15 - 27)^{\circ}=(75 - 27)^{\circ}=48^{\circ}$.

Answer:

$m\angle1 = 52^{\circ}$
$m\angle2 = 79^{\circ}$
$m\angle3 = 101^{\circ}$
$x = 15$
$m\angle A = 48^{\circ}$