Question

7.
m∠1=

m∠2=

m∠3=

m∠4=

m∠5=

m∠6=

m∠7=

Explanation:

Step1: Find \(m\angle1\)

Use the fact that the sum of angles in a right - triangle is \(180^{\circ}\). In the right - triangle with \(\angle1\), we know one angle is \(90^{\circ}\) and the other non - right angle related to \(144^{\circ}\) (linear pair: \(180 - 144=36^{\circ}\)). So \(m\angle1=180-(90 + 36)=54^{\circ}\) (sum of angles in a triangle \(=180^{\circ}\)). Wait, no, another approach: The triangle with \(\angle1\) is a triangle. The angle adjacent to \(144^{\circ}\) is \(180 - 144 = 36^{\circ}\). Using the sum of angles in a triangle (\(180^{\circ}\)), \(m\angle1=180-(90 + 36)=54^{\circ}\). Wait, wrong. Wait, the triangle with \(\angle1\): sum of angles in a triangle \(180^{\circ}\). The right - angle is \(90^{\circ}\), and the angle adjacent to \(144^{\circ}\) (linear pair) is \(180 - 144=36^{\circ}\). So \(m\angle1 = 180-(90 + 36)=54^{\circ}\). No, wait, another way. The triangle with \(\angle1\): \(\angle1+90+(180 - 144)=180\). \(\angle1+90 + 36=180\), \(\angle1=180-(90 + 36)=54^{\circ}\).

Step2: Find \(m\angle2\)

\(\angle2\) and \(95^{\circ}\) are supplementary (linear pair). So \(m\angle2=180 - 95=85^{\circ}\)

Step3: Find \(m\angle3\)

In the triangle with \(\angle2\) and \(\angle3\), using the sum of angles in a triangle (\(180^{\circ}\)). \(m\angle3=180-(m\angle2 + 36)\). Since \(m\angle2 = 85^{\circ}\) and the angle adjacent to \(144^{\circ}\) is \(36^{\circ}\). \(m\angle3=180-(85 + 36)=59^{\circ}\)

Step4: Find \(m\angle4\)

\(\angle3\) and \(\angle4\) are vertical angles. So \(m\angle4=m\angle3 = 59^{\circ}\)

Step5: Find \(m\angle5\)

In the right - triangle with \(\angle5\), \(m\angle5=180-(90 + 59)=31^{\circ}\)

Step6: Find \(m\angle6\)

In the triangle with \(\angle4\) and \(38^{\circ}\), using the sum of angles in a triangle (\(180^{\circ}\)). \(m\angle6=180-(m\angle4+38)\). Since \(m\angle4 = 59^{\circ}\), \(m\angle6=180-(59 + 38)=83^{\circ}\)

Step7: Find \(m\angle7\)

In the triangle with \(\angle6\) (we found \(m\angle6 = 83^{\circ}\)) and the angle adjacent to \(144^{\circ}\) (\(36^{\circ}\)), using the sum of angles in a triangle (\(180^{\circ}\)). \(m\angle7=180-(83 + 36)=61^{\circ}\)

Answer:

\(m\angle1 = 54^{\circ}\), \(m\angle2 = 85^{\circ}\), \(m\angle3 = 59^{\circ}\), \(m\angle4 = 59^{\circ}\), \(m\angle5 = 31^{\circ}\), \(m\angle6 = 83^{\circ}\), \(m\angle7 = 61^{\circ}\)