Question

1. (l36) from the given statement below, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

given: \\( \angle a \cong \angle x ; \overline{b c} \cong \overline{y z} ; \angle c \cong \angle z \\)
prove: \\( \triangle a b c \cong \triangle x y z \\)
a. definition of congruent triangles
b. angle-angle-side theorem
c. side-angle-side postulate
d. side-side-side postulate
e. angle-side-angle postulate

1. (l31) in \\( \triangle l m n \\), the exterior angle adjacent to \\( \angle l \\) has a measure of \\( 5 x+12 \\).

if \\( m \angle m=3 x-2 \\) and \\( m \angle n=50 \\), find the measure of \\( \angle l \\).
a. 56
b. 78
c. 94
d. 123
e. none of the above

Explanation:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In \(\triangle LMN\), the exterior angle adjacent to \(\angle L\) is \(m\angle L=(5x + 12)\), and the two non - adjacent interior angles are \(m\angle M=(3x-2)\) and \(m\angle N = 50\). So, we have the equation \(5x+12=(3x - 2)+50\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side of the equation: \((3x - 2)+50=3x+48\).
The equation becomes \(5x+12 = 3x+48\).
Subtract \(3x\) from both sides: \(5x-3x+12=3x - 3x+48\), which gives \(2x+12 = 48\).
Subtract 12 from both sides: \(2x+12-12=48 - 12\), so \(2x=36\).
Divide both sides by 2: \(x = 18\).

Step3: Find the measure of \(\angle L\)

Substitute \(x = 18\) into the expression for \(m\angle L\).
\(m\angle L=5x+12\).
\(m\angle L=5\times18 + 12\).
First, calculate \(5\times18=90\).
Then \(90+12=102\). Wait, no, let's check the equation again.
Wait, the exterior - angle formula: \(m\angle L\) (exterior angle) \(=m\angle M+m\angle N\).
\(5x + 12=(3x-2)+50\).
\(5x+12=3x + 48\).
\(5x-3x=48 - 12\).
\(2x=36\), \(x = 18\).
\(m\angle L=5x+12\).
\(m\angle L=5\times18+12=90 + 12=102\). But this is wrong. Wait, no, the problem might have a typo. Wait, if we assume the formula \(m\angle L\) (exterior angle) \(=m\angle M+m\angle N\)
\(5x+12=(3x - 2)+50\)
\(5x+12=3x+48\)
\(5x-3x=48 - 12\)
\(2x=36\)
\(x = 18\)
\(m\angle L=5x+12=5\times18+12=102\). But looking at the options, maybe the formula was misapplied. Wait, another way: if \(\angle L\) is an interior angle, and the exterior angle adjacent to \(\angle L\) is \(E\) (exterior angle), \(E=\angle M+\angle N\). If \(E = 5x + 12\), \(\angle M=3x-2\), \(\angle N = 50\)
\(5x+12=(3x-2)+50\)
\(5x+12=3x+48\)
\(2x=36\)
\(x = 18\)
If \(\angle L\) is an interior angle, and the sum of angles in a triangle is \(180^{\circ}\), \(\angle L+\angle M+\angle N=180\)
\(\angle L=180-(3x - 2)-50=180-3x + 2-50=132-3x\)
Also, if the exterior angle of \(\angle L\) is \(5x + 12\), then \(\angle L=180-(5x + 12)=168-5x\)
Set \(132-3x=168-5x\)
\(5x-3x=168 - 132\)
\(2x=36\)
\(x = 18\)
\(\angle L=132-3x=132-3\times18=132 - 54=78\)

Answer:

B. \(78\)