Question

question 1 5.1-5.2 proofs
proof:
∠r is a rt. ∠. → m∠r = 90° → 90° + m∠s + m∠t = 180°

m∠r + m∠s + m∠t = 180° → m∠s + m∠t = 90°

∠s and ∠t are compl.

options: subtraction prop., def of rt. ∠, substitution, def of comp. angles, triangle angle - sum thm, given

Explanation:

Step1: Identify the first missing reason

The first step is "∠R is a rt. ∠." The reason for this is "Given" because it's a statement provided to start the proof.

Step2: Identify the reason for \( m\angle R = 90^\circ \)

The definition of a right angle (Def of rt. ∠) states that a right angle measures \( 90^\circ \), so this reason is "Def of rt. ∠".

Step3: Identify the reason for \( 90^\circ + m\angle S + m\angle T = 180^\circ \)

We substitute \( m\angle R = 90^\circ \) into the equation from the triangle angle - sum theorem. The triangle angle - sum theorem (Triangle Angle - Sum Thm) says that the sum of the interior angles of a triangle is \( 180^\circ \), so we use "Substitution" here (substituting \( m\angle R = 90^\circ \) into \( m\angle R+m\angle S + m\angle T=180^\circ \)).

Step4: Identify the reason for \( m\angle R + m\angle S + m\angle T = 180^\circ \)

This is the Triangle Angle - Sum Theorem (Triangle Angle - Sum Thm), which states that the sum of the angles in a triangle is \( 180^\circ \).

Step5: Identify the reason for \( m\angle S + m\angle T = 90^\circ \)

We start with \( 90^\circ+m\angle S + m\angle T = 180^\circ \) and subtract \( 90^\circ \) from both sides. This is the Subtraction Property of Equality (Subtraction Prop.), so the reason is "Subtraction Prop.".

Step6: Identify the reason for \( \angle S \) and \( \angle T \) are compl.

The definition of complementary angles (Def of comp. angles) states that two angles are complementary if their sum is \( 90^\circ \). Since \( m\angle S + m\angle T = 90^\circ \), the reason is "Def of comp. angles".

Answer:

• For "∠R is a rt. ∠.": Given
• For \( m\angle R = 90^\circ \): Def of rt. ∠
• For \( 90^\circ + m\angle S + m\angle T = 180^\circ \): Substitution
• For \( m\angle R + m\angle S + m\angle T = 180^\circ \): Triangle Angle - Sum Thm
• For \( m\angle S + m\angle T = 90^\circ \): Subtraction Prop.
• For \( \angle S \) and \( \angle T \) are compl.: Def of comp. angles