Question

complete the proof that ∠ptq ≅ ∠rts.
diagram with points s, r, t, q, p (lines intersecting at t)

statement	reason
2 ( mangle qtr + mangle rts = 180^circ )	angles forming a linear pair sum to ( 180^circ )
3 ( mangle ptq + mangle qtr = mangle qtr + mangle rts )	transitive property of equality
4 blank	dropdown

keyboard with symbols: ( mangle ), ( angle ), ( cong ), etc.

Explanation:

Step1: Identify the property to use

We have the equation \( m\angle PTQ + m\angle QTR = m\angle QTR + m\angle RTS \). To simplify this, we can use the Subtraction Property of Equality, which states that if \( a + b = b + c \), then \( a = c \) (by subtracting \( b \) from both sides).

Step2: Apply the Subtraction Property

Subtract \( m\angle QTR \) from both sides of the equation \( m\angle PTQ + m\angle QTR = m\angle QTR + m\angle RTS \). This gives us \( m\angle PTQ = m\angle RTS \).

Step3: Relate angle measure equality to congruence

If the measures of two angles are equal, then the angles are congruent. So, since \( m\angle PTQ = m\angle RTS \), we can conclude that \( \angle PTQ \cong \angle RTS \).

For step 4 in the proof table:

• Statement: \( m\angle PTQ = m\angle RTS \) (first part) then \( \angle PTQ \cong \angle RTS \) (second part, but the first blank is for the angle measure equality)
• Reason: Subtraction Property of Equality (for the first part to get \( m\angle PTQ = m\angle RTS \)), and then Definition of Congruent Angles (if angle measures are equal, angles are congruent)

But focusing on the next statement after step 3, from \( m\angle PTQ + m\angle QTR = m\angle QTR + m\angle RTS \), subtracting \( m\angle QTR \) from both sides gives \( m\angle PTQ = m\angle RTS \).

Answer:

Statement: \( m\angle PTQ = m\angle RTS \)
Reason: Subtraction Property of Equality

(Then, to get \( \angle PTQ \cong \angle RTS \), the next step would use the Definition of Congruent Angles, but for the given blank after step 3, the statement is \( m\angle PTQ = m\angle RTS \) with reason Subtraction Property of Equality)