Question

theorem of interior angles of a triangle theorem
the sum of the interior angles of a triangle is 180°.

1. determine x in each of the following cases.

a)
b)
c)
d)
e)
f)

Explanation:

a) Step1: Find triangle's missing interior angle

Let the missing interior angle be $y$.
$180^\circ = 64^\circ + 82^\circ + y$
$y = 180^\circ - 64^\circ - 82^\circ = 34^\circ$

a) Step2: Solve for supplementary $x$

$x + y = 180^\circ$
$x = 180^\circ - 34^\circ = 146^\circ$

b) Step1: Identify triangle's equal angles

The triangle is isosceles, so the two base angles are equal: $\frac{180^\circ - 50^\circ}{2} = 65^\circ$

b) Step2: Solve for supplementary $x$

$x + 65^\circ = 180^\circ$
$x = 180^\circ - 65^\circ = 115^\circ$

c) Step1: Set up isosceles angle sum

The triangle is equilateral (all sides equal), so all angles are equal:
$2x + 10 = 60^\circ$

c) Step2: Solve for $x$

$2x = 60^\circ - 10^\circ = 50^\circ$
$x = \frac{50^\circ}{2} = 25^\circ$

d) Step1: Find triangle's missing interior angle

Let the missing interior angle be $y$.
$y + 126^\circ = 180^\circ \implies y = 54^\circ$

d) Step2: Solve for $x$ via angle sum

$180^\circ = x + 74^\circ + 54^\circ$
$x = 180^\circ - 74^\circ - 54^\circ = 52^\circ$

e) Step1: Find triangle's interior angle

Let the interior angle adjacent to $4x+15$ be $y$.
$y + (4x+15) = 180^\circ \implies y = 165^\circ - 4x$

e) Step2: Set up angle sum equation

$180^\circ = (2x+10) + (3x-5) + (165-4x)$
$180^\circ = x + 170^\circ$

e) Step3: Solve for $x$

$x = 180^\circ - 170^\circ = 10^\circ$

f) Step1: Find triangle's two interior angles

Let the angles be $y$ and $z$:
$y + 130^\circ = 180^\circ \implies y = 50^\circ$
$z + 140^\circ = 180^\circ \implies z = 40^\circ$

f) Step2: Solve for $x$ via angle sum

$180^\circ = 50^\circ + 40^\circ + (2x+4)$
$2x+4 = 90^\circ$
$2x = 86^\circ$
$x = 43^\circ$

Answer:

a) $\boldsymbol{146^\circ}$
b) $\boldsymbol{115^\circ}$
c) $\boldsymbol{25^\circ}$
d) $\boldsymbol{52^\circ}$
e) $\boldsymbol{10^\circ}$
f) $\boldsymbol{43^\circ}$