In the figure, line l_1 is parallel to line l_2. Point A lies on line l_1, while points B … — GRE General Test

In the figure, line $l_1$ is parallel to line $l_2$ . Point $A$ lies on line $l_1$ , while points $B$ and $C$ lie on line $l_2$ such that $B$ is to the left of $C$ . Line segments $AB$ and $AC$ extend from line $l_1$ to line $l_2$ to form $\triangle ABC$ . The measure of interior angle $\angle BAC$ is $(2x + 10)^\circ$ , the measure of interior angle $\angle ABC$ is $(3x - 15)^\circ$ , and the measure of the exterior angle at vertex $C$ along line $l_2$ is $(6x - 35)^\circ$ . What is the degree measure of angle $\angle BAC$ ?

A
$30^\circ$
B
$35^\circ$
C
$50^\circ$
$70^\circ$ Cevap
E
$75^\circ$

Cevap

The degree measure of angle

\angle BAC

70^\circ

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent (remote) interior angles. Setting up the equation

(2x + 10) + (3x - 15) = 6x - 35

gives

5x - 5 = 6x - 35

, which simplifies to

x = 30

. Substituting

x = 30

into the expression for

\angle BAC

(2x + 10)^\circ

, yields

2(30) + 10 = 70^\circ

Adım Adım Çözüm

Apply the Exterior Angle Theorem to set up an algebraic equation.

(2x + 10) + (3x - 15) = 6x - 35

The measure of an exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Simplify and solve the linear equation for

x

5x - 5 = 6x - 35 \implies x = 30

Combining like terms yields

5x - 5 = 6x - 35

. Subtracting

5x

from both sides gives

-5 = x - 35

, so

x = 30

Substitute

x = 30

into the expression for angle

\angle BAC

\text{m}\angle BAC = 2(30) + 10 = 70^\circ

The question asks for the degree measure of angle

\angle BAC

, which is given by

(2x + 10)^\circ

Anahtar Kavram

Exterior Angle Theorem and Angle Relationships in Parallel Lines

Soru

Cevap

Adım Adım Çözüm

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