Question

if $\angle 1 \cong \angle 2 \cong \angle 3$, $\angle 4 \cong \angle 5$, and $m\angle 4 = m\angle 3 + 10$, what is $m\angle 5$?
$m\angle 5 = \square ^\circ$

Explanation:

Step1: Recall the sum of exterior angles of a polygon

The sum of the exterior angles of any polygon is \( 360^\circ \). Here, we have a polygon with exterior angles \( \angle1, \angle2, \angle3, \angle4, \angle5, \) and \( \angle6 \). We know \( \angle6 = 90^\circ \) (right angle). Let \( m\angle1 = m\angle2 = m\angle3 = x \) (since they are congruent) and \( m\angle4 = m\angle5 = y \) (since they are congruent). Also, \( y = x + 10 \) (given \( m\angle4 = m\angle3 + 10 \)).

Step2: Set up the equation for the sum of exterior angles

The sum of exterior angles is \( m\angle1 + m\angle2 + m\angle3 + m\angle4 + m\angle5 + m\angle6 = 360^\circ \). Substituting the known values: \( x + x + x + y + y + 90^\circ = 360^\circ \). Simplify this to \( 3x + 2y + 90^\circ = 360^\circ \), so \( 3x + 2y = 270^\circ \).

Step3: Substitute \( y = x + 10 \) into the equation

Substitute \( y \) in the equation \( 3x + 2y = 270^\circ \): \( 3x + 2(x + 10) = 270^\circ \). Expand: \( 3x + 2x + 20 = 270 \). Combine like terms: \( 5x + 20 = 270 \). Subtract 20 from both sides: \( 5x = 250 \). Divide by 5: \( x = 50 \).

Step4: Find \( m\angle5 \)

Since \( y = x + 10 \) and \( x = 50 \), then \( y = 50 + 10 = 60 \). So \( m\angle5 = 60^\circ \).

Answer:

\( 60 \)