Question

1. in $\triangle pqr$, if $m\angle p = (x - 12)\degree$, $m\angle q = (5x - 27)\degree$, and $m\angle r = (2x + 3)\degree$, find $x$ and the measure of each angle.

$x = \underline{\quad\quad}$
$m\angle p = \underline{\quad\quad}$
$m\angle q = \underline{\quad\quad}$
$m\angle r = \underline{\quad\quad}$

1. in $\triangle abc$, if $m\angle a$ is six less than seven times $x$, $m\angle b$ is five less than three times $x$, and $m\angle c$ is five less than four times $x$, find $x$ and the measure of each angle.

$x = \underline{\quad\quad}$
$m\angle a = \underline{\quad\quad}$
$m\angle b = \underline{\quad\quad}$
$m\angle c = \underline{\quad\quad}$

Explanation:

Problem 8:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \( 180^\circ \). So, \( m\angle P + m\angle Q + m\angle R = 180^\circ \).

Step2: Substitute angle expressions

Substitute \( m\angle P=(x - 12)^\circ \), \( m\angle Q=(5x - 27)^\circ \), \( m\angle R=(2x + 3)^\circ \) into the equation:
\( (x - 12)+(5x - 27)+(2x + 3)=180 \).

Step3: Simplify and solve for \( x \)

Combine like terms: \( x + 5x + 2x - 12 - 27 + 3 = 180 \)
\( 8x - 36 = 180 \)
Add 36 to both sides: \( 8x = 216 \)
Divide by 8: \( x = 27 \).

Step4: Find each angle

• \( m\angle P = 27 - 12 = 15^\circ \)
• \( m\angle Q = 5(27) - 27 = 135 - 27 = 108^\circ \)
• \( m\angle R = 2(27) + 3 = 54 + 3 = 57^\circ \)

Problem 9:

Step1: Define angle expressions

• \( m\angle A = 7x - 6 \) (six less than seven times \( x \))
• \( m\angle B = 3x - 5 \) (five less than three times \( x \))
• \( m\angle C = 5x - 4 \) (five less than four times \( x \))

Step2: Use triangle angle sum

\( (7x - 6)+(3x - 5)+(5x - 4)=180 \).

Step3: Simplify and solve for \( x \)

Combine like terms: \( 7x + 3x + 5x - 6 - 5 - 4 = 180 \)
\( 15x - 15 = 180 \)
Add 15 to both sides: \( 15x = 195 \)
Divide by 15: \( x = 13 \).

Step4: Find each angle

• \( m\angle A = 7(13) - 6 = 91 - 6 = 85^\circ \)
• \( m\angle B = 3(13) - 5 = 39 - 5 = 34^\circ \)
• \( m\angle C = 5(13) - 4 = 65 - 4 = 61^\circ \)

Final Answers:

Problem 8:

\( x = \boldsymbol{27} \)
\( m\angle P = \boldsymbol{15^\circ} \)
\( m\angle Q = \boldsymbol{108^\circ} \)
\( m\angle R = \boldsymbol{57^\circ} \)

Problem 9:

\( x = \boldsymbol{13} \)
\( m\angle A = \boldsymbol{85^\circ} \)
\( m\angle B = \boldsymbol{34^\circ} \)
\( m\angle C = \boldsymbol{61^\circ} \)

Answer:

Step1: Define angle expressions

• \( m\angle A = 7x - 6 \) (six less than seven times \( x \))
• \( m\angle B = 3x - 5 \) (five less than three times \( x \))
• \( m\angle C = 5x - 4 \) (five less than four times \( x \))

Step2: Use triangle angle sum

\( (7x - 6)+(3x - 5)+(5x - 4)=180 \).

Step3: Simplify and solve for \( x \)

Combine like terms: \( 7x + 3x + 5x - 6 - 5 - 4 = 180 \)
\( 15x - 15 = 180 \)
Add 15 to both sides: \( 15x = 195 \)
Divide by 15: \( x = 13 \).

Step4: Find each angle

• \( m\angle A = 7(13) - 6 = 91 - 6 = 85^\circ \)
• \( m\angle B = 3(13) - 5 = 39 - 5 = 34^\circ \)
• \( m\angle C = 5(13) - 4 = 65 - 4 = 61^\circ \)

Final Answers:

Problem 8:

\( x = \boldsymbol{27} \)
\( m\angle P = \boldsymbol{15^\circ} \)
\( m\angle Q = \boldsymbol{108^\circ} \)
\( m\angle R = \boldsymbol{57^\circ} \)

Problem 9:

\( x = \boldsymbol{13} \)
\( m\angle A = \boldsymbol{85^\circ} \)
\( m\angle B = \boldsymbol{34^\circ} \)
\( m\angle C = \boldsymbol{61^\circ} \)