Question

6). find the measure of each angle.
7). find the measure of angle #1
8). find the value of x. must show work to get credit.
9). find ( mangle1 )

Explanation:

6)

Step1: Find \(m\angle1\)

Since \(129^{\circ}\) and \(\angle1\) are supplementary (they form a linear - pair), \(m\angle1 = 180^{\circ}-129^{\circ}=51^{\circ}\)

Step2: Find \(m\angle2\)

\(\angle2\) and \(90^{\circ}\) are supplementary (they form a linear - pair), \(m\angle2=180^{\circ}-90^{\circ}=90^{\circ}\)

Step3: Find \(m\angle3\)

Using the triangle angle - sum property (the sum of angles in a triangle is \(180^{\circ}\)), for the triangle with \(\angle1 = 51^{\circ}\) and \(\angle2 = 90^{\circ}\), \(m\angle3=180^{\circ}-(51^{\circ}+90^{\circ}) = 39^{\circ}\)

Step4: Find \(m\angle7\)

\(\angle7\) and \(121^{\circ}\) are supplementary (they form a linear - pair), \(m\angle7=180^{\circ}-121^{\circ}=59^{\circ}\)

Step5: Find \(m\angle5\)

\(\angle5\) and \(47^{\circ}\) are vertical angles. Vertical angles are equal, so \(m\angle5 = 47^{\circ}\)

Step6: Find \(m\angle4\)

Using the triangle angle - sum property for the triangle with \(\angle5 = 47^{\circ}\) and \(\angle7 = 59^{\circ}\), \(m\angle4=180^{\circ}-(47^{\circ}+59^{\circ})=74^{\circ}\)

Step7: Find \(m\angle6\)

\(\angle6\) and \(\angle4\) are vertical angles. Vertical angles are equal, so \(m\angle6 = 74^{\circ}\)

7)

Step1: Use the triangle angle - sum property

The sum of angles in a triangle is \(180^{\circ}\). Given two angles \(76^{\circ}\) and \(59^{\circ}\), \(m\angle1=180^{\circ}-(76^{\circ}+59^{\circ})=45^{\circ}\)

8)

Step1: Use the triangle angle - sum property

The sum of angles in a triangle is \(180^{\circ}\). So \((10x - 11)+(3x - 2)+(3x + 1)=180\)

Step2: Simplify the left - hand side

\(10x-11 + 3x-2+3x + 1=(10x+3x+3x)+(-11 - 2 + 1)=16x-12\)

Step3: Solve the equation \(16x-12 = 180\)

Add \(12\) to both sides: \(16x=180 + 12=192\)
Divide both sides by \(16\): \(x=\frac{192}{16}=12\)

9)

Step1: Find the adjacent angle to \(152^{\circ}\)

The adjacent angle to \(152^{\circ}\) (they form a linear - pair) is \(180^{\circ}-152^{\circ}=28^{\circ}\)

Step2: Use the triangle angle - sum property

The sum of angles in a triangle is \(180^{\circ}\). Given an angle \(115^{\circ}\) and the angle \(28^{\circ}\) (from Step 1), \(m\angle1=180^{\circ}-(115^{\circ}+28^{\circ})=37^{\circ}\)

Answer:

\(m\angle1 = 51^{\circ}\), \(m\angle2 = 90^{\circ}\), \(m\angle3 = 39^{\circ}\), \(m\angle4 = 74^{\circ}\), \(m\angle5 = 47^{\circ}\), \(m\angle6 = 74^{\circ}\), \(m\angle7 = 59^{\circ}\)

\(m\angle1 = 45^{\circ}\)

\(x = 12\)

\(m\angle1 = 37^{\circ}\)