Question

complete the proof that $m\angle s + m\angle t = 90^\circ$.
triangle with right angle at u, vertices s, t, u

	statement	reason
1	$\overline{tu} \perp \overline{su}$
2	$m\angle u = 90^\circ$
3	$m\angle s + m\angle t + m\angle u = 180^\circ$
4	$m\angle s + m\angle t + 90^\circ = 180^\circ$
5	$m\angle s + m\angle t = 90^\circ$

Explanation:

Step1: Identify Right Angle

The symbol at $\angle U$ indicates a right angle, so $\overline{TU} \perp \overline{SU}$ (Given by the diagram's right angle mark).

Step2: Define Right Angle Measure

By definition of perpendicular lines, $m\angle U = 90^\circ$ (Perpendicular lines form right angles, which measure $90^\circ$).

Step3: Triangle Angle Sum

The sum of the interior angles of a triangle is $180^\circ$, so $m\angle S + m\angle T + m\angle U = 180^\circ$ (Triangle Angle Sum Theorem).

Step4: Substitute $\boldsymbol{m\angle U}$

Substitute $m\angle U = 90^\circ$ into the angle - sum equation: $m\angle S + m\angle T+90^\circ = 180^\circ$ (Substitution Property of Equality).

Step5: Solve for $\boldsymbol{m\angle S + m\angle T}$

Subtract $90^\circ$ from both sides of the equation $m\angle S + m\angle T + 90^\circ=180^\circ$: $m\angle S + m\angle T=180^\circ - 90^\circ = 90^\circ$ (Subtraction Property of Equality).

Answer:

The proof is completed by using the right - angle definition, Triangle Angle Sum Theorem, and properties of equality. The final result $m\angle S + m\angle T = 90^\circ$ is proven.