Question

given: $overline{rt}perpoverline{st}$
prove: $mangle r + mangle s=90$

1. $\triangle rst$ is a right triangle.
2. $angle r$ and $angle s$ are complementary.
3. click here to insert
4. click here to insert
5. the acute angles of any triangle are complementary.
6. click here to insert

Explanation:

Step1: Identify given information

$\overline{RT}\perp\overline{ST}$ is given.

Step2: Determine the measure of $\angle T$

Since $\overline{RT}\perp\overline{ST}$, by the definition of perpendicular lines, $\angle T = 90^{\circ}$.

Step3: Classify the triangle

As $\triangle RST$ has a $90^{\circ}$ angle ($\angle T$), it is a right - triangle.

Step4: Recall angle relationship in right - triangles

In a right - triangle, the two non - right (acute) angles are complementary. So $\angle R$ and $\angle S$ are complementary.

Step5: Use the definition of complementary angles

Since $\angle R$ and $\angle S$ are complementary, by the definition of complementary angles, $m\angle R + m\angle S = 90^{\circ}$.

Answer:

1. Statement: $\overline{RT}\perp\overline{ST}$; Reason: Given
2. Statement: $\angle T = 90^{\circ}$; Reason: Definition of perpendicular lines (If two lines are perpendicular, the angle formed between them is a right - angle, which measures $90^{\circ}$)
3. Statement: $\triangle RST$ is a right - triangle; Reason: A triangle with one right - angle is a right - triangle
4. Statement: $\angle R$ and $\angle S$ are complementary; Reason: The acute angles of any right - triangle are complementary
5. Statement: $m\angle R + m\angle S=90^{\circ}$; Reason: Definition of complementary angles (Two angles are complementary if the sum of their measures is $90^{\circ}$)