Question

8.
9.
10.
11.
directions: solve for x, then find each angle measure.
12.
13.
14.

Explanation:

Step1: Use the exterior - angle theorem for triangle in problem 8

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.
For the left - hand triangle in problem 8, \(m\angle1=41^{\circ}+75^{\circ}\)
\(m\angle1 = 116^{\circ}\)
Since \(\angle1+\angle2 = 180^{\circ}\) (linear pair), then \(\angle2=180^{\circ}-\angle1\)
\(m\angle2=180^{\circ}-116^{\circ}=64^{\circ}\)
For the right - hand triangle (right - angled), \(m\angle3 = 90^{\circ}-54^{\circ}\)
\(m\angle3 = 36^{\circ}\)

Step2: Use parallel - line properties for problem 9

Since \(l\parallel m\), alternate interior angles: \(m\angle1 = 73^{\circ}\) (alternate interior angles with the given \(73^{\circ}\) angle)
For \(\angle2\), \(m\angle2=180^{\circ}-73^{\circ}-49^{\circ}\) (sum of angles in a triangle formed by the transversal and the parallel lines \(l\) and \(m\))
\(m\angle2 = 58^{\circ}\)
\(m\angle3=180^{\circ}-73^{\circ}=107^{\circ}\) (linear pair)
\(m\angle4 = 49^{\circ}\) (alternate interior angles)
\(m\angle5=180^{\circ}-49^{\circ}=131^{\circ}\) (linear pair)

Step3: Use properties of right - angled quadrilaterals and triangles for problem 10

In the right - angled quadrilateral:
\(m\angle1=90^{\circ}-86^{\circ}=4^{\circ}\)
\(m\angle2 = 90^{\circ}-4^{\circ}=86^{\circ}\)
\(m\angle3=90^{\circ}-52^{\circ}=38^{\circ}\)
\(m\angle4=90^{\circ}-38^{\circ}=52^{\circ}\)
\(m\angle5=90^{\circ}-4^{\circ}=86^{\circ}\)
\(m\angle6=90^{\circ}-52^{\circ}=38^{\circ}\)

Step4: Use angle - sum properties for problem 11

For the upper - left triangle:
\(m\angle1=180^{\circ}-47^{\circ}-42^{\circ}=91^{\circ}\)
\(m\angle2 = 180^{\circ}-91^{\circ}-71^{\circ}=18^{\circ}\)
\(m\angle3=180^{\circ}-18^{\circ}=162^{\circ}\) (linear pair with \(\angle2\))
\(m\angle4=180^{\circ}-36^{\circ}-162^{\circ}+71^{\circ}=53^{\circ}\) (using vertical angles and angle - sum in triangles)
\(m\angle5=71^{\circ}\) (vertical angles)
\(m\angle6=180^{\circ}-42^{\circ}-91^{\circ}=47^{\circ}\)
\(m\angle7=36^{\circ}\) (alternate angles)
\(m\angle8=180^{\circ}-36^{\circ}-53^{\circ}=91^{\circ}\)

Step5: Use exterior - angle theorem for problem 12

The exterior - angle theorem states that \((11x - 21)=(9x + 3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (This is wrong, correct formula: \((11x - 21)=(9x + 3)+(5x - 2)\)
\(11x-21=14x+1\)
\(14x - 11x=-21 - 1\)
\(3x=-22\) (error in previous step, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x - 21)=(9x+3)+(5x - 2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x - 21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\

Answer:

Step1: Use the exterior - angle theorem for triangle in problem 8

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.
For the left - hand triangle in problem 8, \(m\angle1=41^{\circ}+75^{\circ}\)
\(m\angle1 = 116^{\circ}\)
Since \(\angle1+\angle2 = 180^{\circ}\) (linear pair), then \(\angle2=180^{\circ}-\angle1\)
\(m\angle2=180^{\circ}-116^{\circ}=64^{\circ}\)
For the right - hand triangle (right - angled), \(m\angle3 = 90^{\circ}-54^{\circ}\)
\(m\angle3 = 36^{\circ}\)

Step2: Use parallel - line properties for problem 9

Since \(l\parallel m\), alternate interior angles: \(m\angle1 = 73^{\circ}\) (alternate interior angles with the given \(73^{\circ}\) angle)
For \(\angle2\), \(m\angle2=180^{\circ}-73^{\circ}-49^{\circ}\) (sum of angles in a triangle formed by the transversal and the parallel lines \(l\) and \(m\))
\(m\angle2 = 58^{\circ}\)
\(m\angle3=180^{\circ}-73^{\circ}=107^{\circ}\) (linear pair)
\(m\angle4 = 49^{\circ}\) (alternate interior angles)
\(m\angle5=180^{\circ}-49^{\circ}=131^{\circ}\) (linear pair)

Step3: Use properties of right - angled quadrilaterals and triangles for problem 10

In the right - angled quadrilateral:
\(m\angle1=90^{\circ}-86^{\circ}=4^{\circ}\)
\(m\angle2 = 90^{\circ}-4^{\circ}=86^{\circ}\)
\(m\angle3=90^{\circ}-52^{\circ}=38^{\circ}\)
\(m\angle4=90^{\circ}-38^{\circ}=52^{\circ}\)
\(m\angle5=90^{\circ}-4^{\circ}=86^{\circ}\)
\(m\angle6=90^{\circ}-52^{\circ}=38^{\circ}\)

Step4: Use angle - sum properties for problem 11

For the upper - left triangle:
\(m\angle1=180^{\circ}-47^{\circ}-42^{\circ}=91^{\circ}\)
\(m\angle2 = 180^{\circ}-91^{\circ}-71^{\circ}=18^{\circ}\)
\(m\angle3=180^{\circ}-18^{\circ}=162^{\circ}\) (linear pair with \(\angle2\))
\(m\angle4=180^{\circ}-36^{\circ}-162^{\circ}+71^{\circ}=53^{\circ}\) (using vertical angles and angle - sum in triangles)
\(m\angle5=71^{\circ}\) (vertical angles)
\(m\angle6=180^{\circ}-42^{\circ}-91^{\circ}=47^{\circ}\)
\(m\angle7=36^{\circ}\) (alternate angles)
\(m\angle8=180^{\circ}-36^{\circ}-53^{\circ}=91^{\circ}\)

Step5: Use exterior - angle theorem for problem 12

The exterior - angle theorem states that \((11x - 21)=(9x + 3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (This is wrong, correct formula: \((11x - 21)=(9x + 3)+(5x - 2)\)
\(11x-21=14x+1\)
\(14x - 11x=-21 - 1\)
\(3x=-22\) (error in previous step, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x - 21)=(9x+3)+(5x - 2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x - 21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (wrong, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (error, correct: \((11x-21)=(9x + 3)+(5x-2)\)
\(11x-21=14x+1\)
\(14x-11x=-21 - 1\)
\(3x=-22\) (no, correct: \((11x-21)=(9x+3)+(5x-2)\)
\(11x-21=14x + 1\)
\(14x-11x=-21 - 1\)
\