Question

find the measure of each missing angle.

m∠1=

m∠2=

m∠3=

m∠4=

m∠5=

Explanation:

Step1: Find \(m\angle1\) and \(m\angle5\)

Use the linear - pair property. If two angles form a linear pair, their sum is \(180^{\circ}\).
For \(\angle1\) and the \(144^{\circ}\) angle: \(m\angle1 + 144^{\circ}=180^{\circ}\), so \(m\angle1=180^{\circ}- 144^{\circ}=36^{\circ}\).
For \(\angle5\) and the \(144^{\circ}\) angle: \(m\angle5 + 144^{\circ}=180^{\circ}\), so \(m\angle5=180^{\circ}-144^{\circ}=36^{\circ}\).

Step2: Find \(m\angle3\)

Use the linear - pair property. For \(\angle3\) and the \(56^{\circ}\) angle: \(m\angle3+56^{\circ}=180^{\circ}\), so \(m\angle3 = 180^{\circ}-56^{\circ}=124^{\circ}\).

Step3: Find \(m\angle2\)

Use the triangle - angle sum property (the sum of angles in a triangle is \(180^{\circ}\)).
In the smaller triangle with angles \(m\angle1 = 36^{\circ}\), \(m\angle3=124^{\circ}\), and \(\angle2\).
\(m\angle1+m\angle2+m\angle3 = 180^{\circ}\).
Substitute \(m\angle1 = 36^{\circ}\) and \(m\angle3=124^{\circ}\) into the equation: \(36^{\circ}+m\angle2 + 124^{\circ}=180^{\circ}\).
\(m\angle2=180^{\circ}-(36^{\circ}+124^{\circ})=180^{\circ}-160^{\circ}=20^{\circ}\).

Step4: Find \(m\angle4\)

Use the vertical - angle property (vertical angles are equal). \(\angle4\) and the \(56^{\circ}\) angle are vertical angles, so \(m\angle4 = 56^{\circ}\).

Answer:

\(m\angle1 = 36^{\circ}\), \(m\angle2=20^{\circ}\), \(m\angle3 = 124^{\circ}\), \(m\angle4=56^{\circ}\), \(m\angle5 = 36^{\circ}\)