Question

use the figure shown for items 3 and 4. 3. what is m∠2 + m∠3? 4. if m∠4 = 35°, find m∠2 and m∠3. 5. in the figure shown, t || x and k || w. if m∠3 = 20°, list all the angles that are 20° and all the angles that are 160°.

Explanation:

3.

Step1: Observe angle - relationship

Since $\angle1 = 90^{\circ}$ and $\angle1+\angle2+\angle3 = 180^{\circ}$ (a straight - line angle is $180^{\circ}$).
$m\angle1 + m\angle2 + m\angle3=180^{\circ}$

Step2: Solve for $m\angle2 + m\angle3$

Substitute $m\angle1 = 90^{\circ}$ into the equation: $m\angle2 + m\angle3=180^{\circ}-m\angle1$.
$m\angle2 + m\angle3 = 180 - 90=90^{\circ}$

Step1: Use the property of vertical angles

$\angle2$ and $\angle4$ are vertical angles. Vertical angles are equal. So if $m\angle4 = 35^{\circ}$, then $m\angle2=m\angle4 = 35^{\circ}$.
$m\angle2 = 35^{\circ}$

Step2: Use the relationship from Item 3

We know from Item 3 that $m\angle2 + m\angle3=90^{\circ}$. Substitute $m\angle2 = 35^{\circ}$ into the equation: $m\angle3=90^{\circ}-m\angle2$.
$m\angle3=90 - 35 = 55^{\circ}$

Step1: Identify corresponding and vertical angles for $20^{\circ}$ angles

Given $m\angle3 = 20^{\circ}$. Vertical angles are equal, so $\angle1=\angle3 = 20^{\circ}$, $\angle5=\angle3 = 20^{\circ}$, $\angle9=\angle3 = 20^{\circ}$, $\angle13=\angle3 = 20^{\circ}$. Also, corresponding angles for parallel lines $t\parallel x$ and $k\parallel w$ are equal. So the angles that are $20^{\circ}$ are $\angle1,\angle3,\angle5,\angle9,\angle13$.

Step2: Identify supplementary and vertical angles for $160^{\circ}$ angles

Since $\angle3 = 20^{\circ}$, its supplementary angle $\angle2 = 180 - 20=160^{\circ}$. Vertical - angle and corresponding - angle relationships for parallel lines give that the angles that are $160^{\circ}$ are $\angle2,\angle4,\angle6,\angle10,\angle14,\angle16$.

Answer:

$90^{\circ}$

4.