The figure shows adjacent angles BAC and CAD. Adjacent angles BAC and CAD sharing common ray AC Given: m∠BAD = 129° m∠BAC = (2x −1)° m∠CAD = (3x + 5)° Part A: Using the angle addition postulate, write and solve an equation for x. Show all your work. (6 points) Part B: Find the m∠BAC. Show all your work. (4 points)

The Angle Addition Postulate states that the measure of the sum of two adjacent angles is equal to measure of the larger angle they form together.

If the adjacent angles BAC and CAD share a common ray AC, then according to the angle addition postulate, the sum of their measurers is equal to the measure of angle BAD:

[tex]m\angle BAC + m\angle CAD = m\angle BAD[/tex]

Given:

m∠BAD = 129°
m∠BAC = (2x - 1)°
m∠CAD = (3x + 5)°

Substitute the given values into the equation:

[tex](2x - 1)^{\circ}+ (3x + 5)^{\circ} = 129^{\circ}[/tex]

Now, solve for x:

[tex]\begin{aligned}(2x - 1)+ (3x + 5) &= 129\\2x-1+3x+5&=129\\5x+4&=129\\5x&=125\\x&=25\end{aligned}[/tex]

Therefore, the value of x is:

[tex]\Large\boxed{\boxed{x=25}}[/tex]

[tex]\hrulefill[/tex]

Part B

To find the measure of angle BAC, we can substitute x = 25 into the angle expression:

[tex]m\angle BAC = (2x - 1)^{\circ}\\\\m\angle BAC = (2(25) - 1)^{\circ}\\\\m\angle BAC = (50 - 1)^{\circ}\\\\m\angle BAC=49^{\circ}[/tex]

Therefore, the measure of angle BAC is:

[tex]\Large\boxed{\boxed{m\angle BAC=49^{\circ}}}[/tex]

msm555 msm555 · Answer 2 · 2024-01-19T03:51:00+01:00

Answer:

[tex] x = 25 [/tex]

[tex] m\angle BAC = 49^\circ [/tex]

Step-by-step explanation:

Let's use the angle addition postulate, which states that the measure of an angle formed by two adjacent angles is the sum of the measures of the two angles.

Given:

[tex] m\angle BAC = (2x - 1)^\circ [/tex]
[tex] m\angle CAD = (3x + 5)^\circ [/tex]
[tex] m\angle BAD = 129^\circ [/tex]

According to the angle addition postulate, we can write an equation:

[tex] m\angle BAC + m\angle CAD = m\angle BAD [/tex]

Substitute the given expressions into the equation:

[tex] (2x - 1) + (3x + 5) = 129 [/tex]

Combine like terms:

[tex] 5x + 4 = 129 [/tex]

Now, solve for [tex] x [/tex]:

[tex] 5x + 4-4 = 129-4 [/tex]

[tex] 5x = 125 [/tex]

[tex]x =\dfrac{125}{5}[/tex]

[tex] x = 25 [/tex]

Now that we have found the value of [tex] x [/tex], we can find [tex] m\angle BAC [/tex] using the expression [tex] (2x - 1) [/tex]:

[tex] m\angle BAC = (2 \times 25 - 1) [/tex]

[tex] m\angle BAC = 49 [/tex]

So, the solution is [tex] x = 25 [/tex] and [tex] m\angle BAC = 49^\circ [/tex].