Question

which is an expression for ( mangle 3 )? select all that apply.
a ( 180^circ - (9x + 9)^circ + (9x + 1)^circ )
b ( 180^circ - (9x - 4)^circ )
c ( 180^circ - (9x + 1)^circ )
d ( 180^circ + (9x + 1)^circ )
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for a triangle, ( mangle 1 = (9x + 1)^circ ), ( mangle 2 = (9x + 9)^circ ), and ( mangle 4 = (9x - 4)^circ ). write an expression for ( mangle 3 ). then find ( mangle 3 ).
(the figure is not to scale.)

Explanation:

Step1: Use triangle exterior angle property

The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles. Here, $\angle 3$ is an exterior angle, so:
$$m\angle 3 = m\angle 1 + m\angle 4$$

Step2: Substitute given angle expressions

Replace $m\angle 1$ with $(2x+1)^\circ$ and $m\angle 4$ with $(6x-4)^\circ$:
$$m\angle 3 = (2x+1) + (6x-4)$$

Step3: Simplify the expression

Combine like terms:
$$m\angle 3 = 2x + 6x + 1 - 4 = 8x - 3$$

Step4: Use straight angle to find $x$

$\angle 2$ and $\angle 3$ form a straight angle, so $m\angle 2 + m\angle 3 = 180^\circ$. Substitute the expressions:
$$(3x+8) + (8x-3) = 180$$
$$11x + 5 = 180$$
$$11x = 175$$
$$x = \frac{175}{11}$$

Step5: Calculate $m\angle 3$

Substitute $x = \frac{175}{11}$ into $m\angle 3 = 8x - 3$:
$$m\angle 3 = 8\times\frac{175}{11} - 3 = \frac{1400}{11} - \frac{33}{11} = \frac{1367}{11} \approx 124.27$$

Step6: Verify valid expressions for $m\angle 3$

From straight angle: $m\angle 3 = 180^\circ - m\angle 2 = 180^\circ - (3x+8)^\circ$. From exterior angle: $m\angle 3 = m\angle1 + m\angle4 = (2x+1)^\circ + (6x-4)^\circ = 8x-3$, which is also $180^\circ - (3x+8)^\circ$.

Answer:

Valid expressions for $m\angle 3$:

• $180^\circ - (3x + 8)^\circ$
• $(2x + 1)^\circ + (6x - 4)^\circ$

Measure of $\angle 3$: $\boldsymbol{\frac{1367}{11}^\circ}$ (or approximately $124.3^\circ$)