Question

missing angle.
4.
$m\angle1 = \underline{35^\circ}$
$m\angle2 = \underline{65^\circ}$
$m\angle3 = \underline{29^\circ}$
$m\angle4 = \underline{115^\circ}$
$m\angle5 = \underline{65^\circ}$
$m\angle6 = \underline{36}$
$m\angle7 = \underline{144}$

Explanation:

Step1: Find $\boldsymbol{m\angle1}$

Use triangle angle sum ($180^\circ$):
$$m\angle1 = 180^\circ - 80^\circ - 65^\circ = 35^\circ$$

Step2: Find $\boldsymbol{m\angle4}$

Linear pair with $65^\circ$:
$$m\angle4 = 180^\circ - 65^\circ = 115^\circ$$

Step3: Find $\boldsymbol{m\angle2}$

Vertical angle to $65^\circ$:
$$m\angle2 = 65^\circ$$

Step4: Find $\boldsymbol{m\angle3}$

Use triangle angle sum with $\angle1, \angle4$:
$$m\angle3 = 180^\circ - 35^\circ - 115^\circ = 29^\circ$$

Step5: Find $\boldsymbol{m\angle5}$

Vertical angle to $65^\circ$:
$$m\angle5 = 65^\circ$$

Step6: Find $\boldsymbol{m\angle6}$

Use triangle angle sum with $\angle3, \angle5$:
$$m\angle6 = 180^\circ - 29^\circ - 65^\circ = 86^\circ$$

Step7: Find $\boldsymbol{m\angle7}$

Use quadrilateral angle sum ($360^\circ$):
$$m\angle7 = 360^\circ - 90^\circ - 61^\circ - 86^\circ = 123^\circ$$

Answer:

$m\angle1 = 35^\circ$
$m\angle2 = 65^\circ$
$m\angle3 = 29^\circ$
$m\angle4 = 115^\circ$
$m\angle5 = 65^\circ$
$m\angle6 = 86^\circ$
$m\angle7 = 123^\circ$